Subgroups — C₆₇×#21900 #30701151

Order 17152.

1106 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1986	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1980	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1897	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1806	C₂	1	4	yes	maximal, complemented
8	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
9	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
10	#533×C₆₇	C₂²	1	1	yes	characteristic
11	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
12	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	#529×C₆₇	—	2	1	no	—
15	#529×C₆₇	—	2	1	no	—
16	#529×C₆₇	—	2	1	no	—
17	#529×C₆₇	—	2	1	no	—
18	#529×C₆₇	C₄	1	2	yes	complemented
19	#529×C₆₇	C₄	1	2	yes	complemented
20	#525×C₆₇	—	2	1	no	—
21	#525×C₆₇	—	2	1	no	—
22	#525×C₆₇	C₂²	1	2	yes	—
23	C₄×#41×C₆₇	—	2	1	no	—
24	C₄×#41×C₆₇	—	2	1	no	—
25	C₄×#41×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#432×C₆₇	—	2	1	no	—
29	#432×C₆₇	—	2	1	no	—
30	#431×C₆₇	—	2	1	no	—
31	#431×C₆₇	—	2	1	no	—
32	#423×C₆₇	C₂²	1	2	yes	—
33	C₂×#104×C₆₇	C₂²	1	8	yes	—
34	#407×C₆₇	C₂²	1	2	yes	—
35	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
36	#384×C₆₇	—	2	1	no	—
37	#382×C₆₇	—	2	1	no	—
38	#380×C₆₇	—	2	1	no	—
39	#380×C₆₇	—	2	1	no	—
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
42	#339×C₆₇	C₂²	1	8	yes	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
54	#126×C₆₇	—	2	2	no	—
55	#126×C₆₇	—	2	2	no	—
56	#125×C₆₇	—	2	1	no	—
57	#125×C₆₇	—	2	1	no	—
58	#125×C₆₇	C₂×C₄	1	2	yes	—
59	#124×C₆₇	—	2	1	no	—
60	#124×C₆₇	—	2	1	no	—
61	#124×C₆₇	C₂×C₄	1	2	yes	—
62	#123×C₆₇	—	2	2	no	—
63	#123×C₆₇	—	2	2	no	—
64	#123×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	1	no	—
66	C₂²⋊Q₈×C₆₇	—	2	1	no	—
67	C₂²⋊Q₈×C₆₇	—	2	1	no	—
68	C₂²⋊Q₈×C₆₇	—	2	1	no	—
69	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
71	#121×C₆₇	—	2	1	no	—
72	#121×C₆₇	—	2	1	no	—
73	#121×C₆₇	—	2	1	no	—
74	#121×C₆₇	—	2	1	no	—
75	#121×C₆₇	C₂×C₄	1	2	yes	—
76	#121×C₆₇	C₂×C₄	1	2	yes	—
77	#120×C₆₇	—	2	2	no	—
78	C₄×Q₈×C₆₇	—	2	1	no	—
79	C₄×Q₈×C₆₇	—	2	1	no	—
80	C₄×Q₈×C₆₇	—	2	1	no	—
81	C₄×Q₈×C₆₇	—	2	1	no	—
82	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	1	no	—
91	C₄×D₄×C₆₇	—	2	1	no	—
92	C₄×D₄×C₆₇	—	2	1	no	—
93	C₄×D₄×C₆₇	—	2	1	no	—
94	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
96	#117×C₆₇	—	2	1	no	—
97	#117×C₆₇	—	2	1	no	—
98	#117×C₆₇	—	2	2	no	—
99	#117×C₆₇	—	2	2	no	—
100	#117×C₆₇	—	2	2	no	—
101	#117×C₆₇	—	4	4	no	—
102	#117×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	—	2	2	no	—
108	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₄⋊C₈×C₆₇	—	2	2	no	—
122	C₄⋊C₈×C₆₇	—	2	2	no	—
123	#104×C₆₇	—	4	8	no	—
124	C₂²⋊C₈×C₆₇	—	2	2	no	—
125	C₂²⋊C₈×C₆₇	—	2	2	no	—
126	#97×C₆₇	—	2	2	no	—
127	C₄×C₈×C₆₇	—	2	2	no	—
128	#95×C₆₇	—	2	2	no	—
129	#95×C₆₇	—	2	2	no	—
130	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
131	#41×C₆₇	—	2	4	no	—
132	#41×C₆₇	—	2	4	no	—
133	#41×C₆₇	—	4	4	no	—
134	#41×C₆₇	—	4	4	no	—
135	C₂×Q₈×C₆₇	—	2	1	no	—
136	C₂×Q₈×C₆₇	—	2	1	no	—
137	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₆₇	—	2	2	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	1	no	—
142	C₂×D₄×C₆₇	—	2	1	no	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	—	4	2	no	—
152	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₆₇	—	2	4	no	—
171	M₁₆×C₆₇	—	2	4	no	—
172	C₂×C₈×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	1	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	4	4	no	—
184	C₄⋊C₄×C₆₇	—	4	4	no	—
185	C₄⋊C₄×C₆₇	—	4	4	no	—
186	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	2	no	—
190	C₂²⋊C₄×C₆₇	—	2	2	no	—
191	C₂²⋊C₄×C₆₇	—	2	2	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	2	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	2	2	no	—
197	C₂²⋊C₄×C₆₇	—	4	4	no	—
198	C₄²×C₆₇	—	2	2	no	—
199	C₄²×C₆₇	—	2	2	no	—
200	C₄²×C₆₇	—	2	2	no	—
201	C₄²×C₆₇	—	2	8	no	—
202	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	4	1	no	—
222	C₂³×C₆₇	—	4	1	no	—
223	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₆₇	—	2	4	no	—
227	Q₈×C₆₇	—	4	2	no	—
228	Q₈×C₆₇	—	4	2	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	4	2	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	4	2	no	—
234	D₄×C₆₇	—	4	2	no	—
235	D₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	4	4	no	—
263	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
276	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
277	C₈×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	2	1	no	—
279	C₂²×C₆₇	—	2	1	no	—
280	C₂²×C₆₇	—	2	1	no	—
281	C₂²×C₆₇	—	2	1	no	—
282	C₂²×C₆₇	—	2	1	no	—
283	C₂²×C₆₇	—	2	1	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	1	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₆₇	#535	1	1	yes	characteristic
293	C₂²×C₆₇	#536	1	1	yes	characteristic
294	C₄×C₆₇	—	2	1	no	—
295	C₄×C₆₇	—	2	1	no	—
296	C₄×C₆₇	—	2	1	no	—
297	C₄×C₆₇	—	2	1	no	—
298	C₄×C₆₇	—	2	1	no	—
299	C₄×C₆₇	—	2	1	no	—
300	C₄×C₆₇	—	2	1	no	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	4	2	no	—
303	C₄×C₆₇	—	4	2	no	—
304	C₄×C₆₇	—	4	2	no	—
305	C₄×C₆₇	—	8	4	no	—
306	C₄×C₆₇	#421	1	2	yes	—
307	C₄×C₆₇	#421	1	2	yes	—
308	C₄×C₆₇	#520	1	2	yes	—
309	C₄×C₆₇	#537	1	2	yes	—
310	#21900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂×C₆₇	—	2	1	no	—
312	C₂×C₆₇	—	2	1	no	—
313	C₂×C₆₇	—	4	1	no	—
314	C₂×C₆₇	—	4	1	no	—
315	C₂×C₆₇	—	8	2	no	—
316	C₂×C₆₇	#2311	1	1	yes	characteristic
317	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
318	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
319	#3444	C₂×C₆₇	1	1	yes	characteristic
320	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
321	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#1986	C₂×C₆₇	1	2	yes	complemented
323	#1980	C₂×C₆₇	1	2	yes	complemented
324	#1897	C₂×C₆₇	1	4	yes	complemented
325	#1806	C₂×C₆₇	1	4	yes	complemented
326	C₆₇	#21900	1	1	yes	characteristic, direct product factor, Sylow
327	#533	C₄×C₆₇	1	1	yes	characteristic, complemented
328	#533	C₄×C₆₇	1	1	yes	characteristic, complemented
329	#533	C₂²×C₆₇	1	1	yes	characteristic
330	#532	C₄×C₆₇	1	1	yes	characteristic, complemented
331	#532	C₄×C₆₇	1	1	yes	characteristic, complemented
332	#532	C₂²×C₆₇	1	1	yes	characteristic
333	#529	—	2	1	no	—
334	#529	—	2	1	no	—
335	#529	—	2	1	no	—
336	#529	—	2	1	no	—
337	#529	C₄×C₆₇	1	2	yes	complemented
338	#529	C₄×C₆₇	1	2	yes	complemented
339	#525	—	2	1	no	—
340	#525	—	2	1	no	—
341	#525	C₂²×C₆₇	1	2	yes	—
342	C₄×#41	—	2	1	no	—
343	C₄×#41	—	2	1	no	—
344	C₄×#41	C₂²×C₆₇	1	2	yes	—
345	C₂×#117	C₂²×C₆₇	1	2	yes	—
346	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
347	#432	—	2	1	no	—
348	#432	—	2	1	no	—
349	#431	—	2	1	no	—
350	#431	—	2	1	no	—
351	#423	C₂²×C₆₇	1	2	yes	—
352	C₂×#104	C₂²×C₆₇	1	8	yes	—
353	#407	C₂²×C₆₇	1	2	yes	—
354	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
355	#384	—	2	1	no	—
356	#382	—	2	1	no	—
357	#380	—	2	1	no	—
358	#380	—	2	1	no	—
359	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
360	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
361	#339	C₂²×C₆₇	1	8	yes	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	—	2	1	no	—
364	C₂×#41	C₂³×C₆₇	1	2	yes	—
365	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
366	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
369	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
370	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
372	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
373	#126	—	2	2	no	—
374	#126	—	2	2	no	—
375	#125	—	2	1	no	—
376	#125	—	2	1	no	—
377	#125	C₂×C₄×C₆₇	1	2	yes	—
378	#124	—	2	1	no	—
379	#124	—	2	1	no	—
380	#124	C₂×C₄×C₆₇	1	2	yes	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	#123	—	2	2	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
389	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
390	#121	—	2	1	no	—
391	#121	—	2	1	no	—
392	#121	—	2	1	no	—
393	#121	—	2	1	no	—
394	#121	C₂×C₄×C₆₇	1	2	yes	—
395	#121	C₂×C₄×C₆₇	1	2	yes	—
396	#120	—	2	2	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
402	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
414	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
415	#117	—	2	1	no	—
416	#117	—	2	1	no	—
417	#117	—	2	2	no	—
418	#117	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	4	4	no	—
421	#117	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	2	no	—
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₄²	—	2	2	no	—
427	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₄⋊C₈	—	2	2	no	—
441	C₄⋊C₈	—	2	2	no	—
442	#104	—	4	8	no	—
443	C₂²⋊C₈	—	2	2	no	—
444	C₂²⋊C₈	—	2	2	no	—
445	#97	—	2	2	no	—
446	C₄×C₈	—	2	2	no	—
447	#95	—	2	2	no	—
448	#95	—	2	2	no	—
449	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
450	#41	—	2	4	no	—
451	#41	—	2	4	no	—
452	#41	—	4	4	no	—
453	#41	—	4	4	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
489	M₁₆	—	2	4	no	—
490	M₁₆	—	2	4	no	—
491	C₂×C₈	—	2	4	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	4	4	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄⋊C₄	—	4	4	no	—
505	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
506	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	4	4	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	8	no	—
521	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
522	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
523	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
524	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
525	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
526	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
527	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
528	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
529	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
530	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
531	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
532	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
535	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
543	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
544	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
545	Q₈	—	2	4	no	—
546	Q₈	—	4	2	no	—
547	Q₈	—	4	2	no	—
548	D₄	—	2	4	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
585	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
592	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
594	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
595	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
596	C₈	—	4	4	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
611	C₂²	#535×C₆₇	1	1	yes	characteristic
612	C₂²	#536×C₆₇	1	1	yes	characteristic
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	4	no	—
625	C₄	#421×C₆₇	1	2	yes	—
626	C₄	#421×C₆₇	1	2	yes	—
627	C₄	#520×C₆₇	1	2	yes	—
628	C₄	#537×C₆₇	1	2	yes	—
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	4	1	no	—
632	C₂	—	4	1	no	—
633	C₂	—	8	2	no	—
634	C₂	C₆₇×#2311	1	1	yes	characteristic
635	C₂	C₂×#421×C₆₇	1	1	yes	characteristic
636	C₂	C₂×#421×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2920	C₂	semidirect
3. C₆₇×#2285	C₂	semidirect
4. C₆₇×#1986	C₂	semidirect
5. C₆₇×#1980	C₂	semidirect
6. C₆₇×#1897	C₂	semidirect
7. C₆₇×#1806	C₂	semidirect
8. #533×C₆₇	C₄	semidirect
9. #533×C₆₇	C₄	semidirect
11. #532×C₆₇	C₄	semidirect
12. #532×C₆₇	C₄	semidirect
18. #529×C₆₇	C₄	semidirect
19. #529×C₆₇	C₄	semidirect
310. #21900	C₆₇	direct (N × H)
320. #2920	C₂×C₆₇	semidirect
321. #2285	C₂×C₆₇	semidirect
322. #1986	C₂×C₆₇	semidirect
323. #1980	C₂×C₆₇	semidirect
324. #1897	C₂×C₆₇	semidirect
325. #1806	C₂×C₆₇	semidirect
326. C₆₇	#21900	direct (N × H)
327. #533	C₄×C₆₇	semidirect
328. #533	C₄×C₆₇	semidirect
330. #532	C₄×C₆₇	semidirect
331. #532	C₄×C₆₇	semidirect
337. #529	C₄×C₆₇	semidirect
338. #529	C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3444	2
C₆₇×#2920	2
C₆₇×#2285	2
C₆₇×#1986	2
C₆₇×#1980	2
C₆₇×#1897	2
C₆₇×#1806	2
#21900	67

Subgroups — C67×#21900 #30701151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21900 #30701151