Subgroups — C₆₇×#17714 #30696965

Order 17152.

1252 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1551	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1548	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1545	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1543	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₄	1	2	yes	—
9	#532×C₆₇	—	2	1	no	—
10	#532×C₆₇	—	2	1	no	—
11	#532×C₆₇	—	2	1	no	—
12	#532×C₆₇	—	2	1	no	—
13	#532×C₆₇	C₄	1	2	yes	—
14	#532×C₆₇	C₄	1	2	yes	—
15	#532×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₄	1	2	yes	—
20	C₂×#121×C₆₇	—	2	1	no	—
21	C₂×#121×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	—	2	1	no	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#432×C₆₇	—	2	1	no	—
26	#432×C₆₇	C₂²	1	2	yes	complemented
27	#431×C₆₇	—	2	1	no	—
28	#431×C₆₇	—	2	1	no	—
29	#431×C₆₇	—	2	1	no	—
30	#431×C₆₇	C₂²	1	2	yes	—
31	#431×C₆₇	C₂²	1	2	yes	complemented
32	#431×C₆₇	C₂²	1	2	yes	complemented
33	#423×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
35	#407×C₆₇	—	2	1	no	—
36	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
37	C₄×M₁₆×C₆₇	—	2	1	no	—
38	C₂×C₄×C₈×C₆₇	—	2	1	no	—
39	#333×C₆₇	C₂²	1	8	yes	complemented
40	#331×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	4	8	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	1	no	—
58	C₄⋊Q₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
61	#127×C₆₇	—	2	1	no	—
62	#127×C₆₇	—	2	1	no	—
63	#127×C₆₇	C₂×C₄	1	2	yes	—
64	#127×C₆₇	C₂³	1	2	yes	—
65	#124×C₆₇	—	2	2	no	—
66	#124×C₆₇	—	2	2	no	—
67	#124×C₆₇	C₂×C₄	1	4	yes	—
68	#124×C₆₇	C₂×C₄	1	4	yes	—
69	#121×C₆₇	—	2	2	no	—
70	#121×C₆₇	—	2	2	no	—
71	#121×C₆₇	—	2	2	no	—
72	#121×C₆₇	—	2	2	no	—
73	#121×C₆₇	—	2	2	no	—
74	#121×C₆₇	—	2	2	no	—
75	#121×C₆₇	—	2	2	no	—
76	#121×C₆₇	—	2	2	no	—
77	C₄×D₄×C₆₇	—	2	2	no	—
78	C₄×D₄×C₆₇	—	2	2	no	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₄×D₄×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
104	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
105	C₂²⋊C₈×C₆₇	—	2	2	no	—
106	C₂²⋊C₈×C₆₇	—	2	2	no	—
107	C₂²⋊C₈×C₆₇	—	2	2	no	—
108	C₂²⋊C₈×C₆₇	—	2	2	no	—
109	#97×C₆₇	—	2	2	no	—
110	C₄×C₈×C₆₇	—	2	2	no	—
111	C₄×C₈×C₆₇	—	2	2	no	—
112	C₄×C₈×C₆₇	—	2	2	no	—
113	C₂⁴×C₆₇	—	2	2	no	—
114	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
115	#41×C₆₇	—	4	8	no	—
116	#41×C₆₇	—	4	8	no	—
117	#41×C₆₇	—	4	8	no	—
118	#41×C₆₇	—	4	8	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	4	4	no	—
121	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	4	4	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	4	no	—
133	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	4	2	no	—
140	C₂²×C₄×C₆₇	—	4	8	no	—
141	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	M₁₆×C₆₇	—	4	4	no	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	—	2	4	no	—
162	C₂×C₈×C₆₇	—	4	4	no	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	4	1	no	—
191	C₂³×C₆₇	—	4	4	no	—
192	C₂³×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	—	4	4	no	—
194	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	Q₈×C₆₇	—	4	4	no	—
198	Q₈×C₆₇	—	4	4	no	—
199	D₄×C₆₇	—	4	4	no	—
200	D₄×C₆₇	—	4	4	no	—
201	D₄×C₆₇	—	4	8	no	—
202	D₄×C₆₇	—	4	8	no	—
203	D₄×C₆₇	—	4	4	no	—
204	D₄×C₆₇	—	4	4	no	—
205	D₄×C₆₇	—	4	4	no	—
206	D₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	4	1	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	8	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	#100	1	4	yes	—
228	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	#135	1	4	yes	—
238	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₈×C₆₇	—	4	4	no	—
242	C₈×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	2	1	no	—
244	C₂²×C₆₇	—	2	1	no	—
245	C₂²×C₆₇	—	2	1	no	—
246	C₂²×C₆₇	—	2	1	no	—
247	C₂²×C₆₇	—	2	1	no	—
248	C₂²×C₆₇	—	2	1	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	8	8	no	—
255	C₂²×C₆₇	C₂×#100	1	2	yes	—
256	C₂²×C₆₇	#416	1	2	yes	—
257	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	2	1	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	2	1	no	—
264	C₄×C₆₇	—	2	1	no	—
265	C₄×C₆₇	—	4	4	no	—
266	C₄×C₆₇	C₂×#100	1	2	yes	—
267	C₄×C₆₇	C₂×#100	1	2	yes	—
268	C₄×C₆₇	#416	1	2	yes	—
269	C₄×C₆₇	C₂×#135	1	2	yes	—
270	C₄×C₆₇	#575	1	2	yes	—
271	#17714	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₆₇	—	2	2	no	—
273	C₂×C₆₇	—	2	2	no	—
274	C₂×C₆₇	—	4	1	no	—
275	C₂×C₆₇	—	8	4	no	—
276	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
277	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
278	C₂×C₆₇	#2896	1	1	yes	characteristic
279	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
280	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1551	C₂×C₆₇	1	4	yes	complemented
283	#1548	C₂×C₆₇	1	2	yes	complemented
284	#1545	C₂×C₆₇	1	2	yes	complemented
285	#1543	C₂×C₆₇	1	4	yes	complemented
286	C₆₇	#17714	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×#41	C₄×C₆₇	1	2	yes	—
288	#532	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	C₄×C₆₇	1	2	yes	—
293	#532	C₄×C₆₇	1	2	yes	—
294	#532	C₂²×C₆₇	1	2	yes	—
295	#532	C₂²×C₆₇	1	2	yes	—
296	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#124	C₄×C₆₇	1	2	yes	—
299	C₂×#121	—	2	1	no	—
300	C₂×#121	—	2	1	no	—
301	C₂×C₄×D₄	—	2	1	no	—
302	C₂×C₄×D₄	—	2	1	no	—
303	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
304	#432	—	2	1	no	—
305	#432	C₂²×C₆₇	1	2	yes	complemented
306	#431	—	2	1	no	—
307	#431	—	2	1	no	—
308	#431	—	2	1	no	—
309	#431	C₂²×C₆₇	1	2	yes	—
310	#431	C₂²×C₆₇	1	2	yes	complemented
311	#431	C₂²×C₆₇	1	2	yes	complemented
312	#423	C₂²×C₆₇	1	2	yes	—
313	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
314	#407	—	2	1	no	—
315	C₂×C₂²⋊C₈	—	2	1	no	—
316	C₄×M₁₆	—	2	1	no	—
317	C₂×C₄×C₈	—	2	1	no	—
318	#333	C₂²×C₆₇	1	8	yes	complemented
319	#331	C₂²×C₆₇	1	8	yes	complemented
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	4	8	no	—
323	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
324	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
325	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
330	C₂³×C₄	—	2	2	no	—
331	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
332	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
333	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
334	C₂×M₁₆	—	2	2	no	—
335	C₂²×C₈	—	2	2	no	—
336	C₄⋊Q₈	—	2	1	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
339	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
340	#127	—	2	1	no	—
341	#127	—	2	1	no	—
342	#127	C₂×C₄×C₆₇	1	2	yes	—
343	#127	C₂³×C₆₇	1	2	yes	—
344	#124	—	2	2	no	—
345	#124	—	2	2	no	—
346	#124	C₂×C₄×C₆₇	1	4	yes	—
347	#124	C₂×C₄×C₆₇	1	4	yes	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₂²⋊C₄	—	2	2	no	—
368	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
370	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
380	C₄⋊C₈	—	2	2	no	—
381	C₄⋊C₈	—	2	2	no	—
382	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
383	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
384	C₂²⋊C₈	—	2	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	C₂²⋊C₈	—	2	2	no	—
388	#97	—	2	2	no	—
389	C₄×C₈	—	2	2	no	—
390	C₄×C₈	—	2	2	no	—
391	C₄×C₈	—	2	2	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
394	#41	—	4	8	no	—
395	#41	—	4	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	4	4	no	—
400	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	8	no	—
420	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
438	M₁₆	—	4	4	no	—
439	C₂×C₈	—	2	4	no	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	4	4	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
451	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
452	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
455	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
456	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
457	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
474	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
475	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
476	Q₈	—	4	4	no	—
477	Q₈	—	4	4	no	—
478	D₄	—	4	4	no	—
479	D₄	—	4	4	no	—
480	D₄	—	4	8	no	—
481	D₄	—	4	8	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	#100×C₆₇	1	4	yes	—
507	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
508	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
509	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
513	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	#135×C₆₇	1	4	yes	—
517	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
520	C₈	—	4	4	no	—
521	C₈	—	4	4	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	8	8	no	—
534	C₂²	C₂×#100×C₆₇	1	2	yes	—
535	C₂²	#416×C₆₇	1	2	yes	—
536	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
537	C₄	—	2	1	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	4	4	no	—
545	C₄	C₂×#100×C₆₇	1	2	yes	—
546	C₄	C₂×#100×C₆₇	1	2	yes	—
547	C₄	#416×C₆₇	1	2	yes	—
548	C₄	C₂×#135×C₆₇	1	2	yes	—
549	C₄	#575×C₆₇	1	2	yes	—
550	C₂	—	2	2	no	—
551	C₂	—	2	2	no	—
552	C₂	—	4	1	no	—
553	C₂	—	8	4	no	—
554	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
555	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
556	C₂	C₆₇×#2896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. C₂×#431×C₆₇	C₂	semidirect
4. C₆₇×#1551	C₂	semidirect
5. C₆₇×#1548	C₂	semidirect
6. C₆₇×#1545	C₂	semidirect
7. C₆₇×#1543	C₂	semidirect
26. #432×C₆₇	C₂²	semidirect
31. #431×C₆₇	C₂²	semidirect
32. #431×C₆₇	C₂²	semidirect
34. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
39. #333×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
271. #17714	C₆₇	direct (N × H)
280. #2920	C₂×C₆₇	semidirect
281. C₂×#431	C₂×C₆₇	semidirect
282. #1551	C₂×C₆₇	semidirect
283. #1548	C₂×C₆₇	semidirect
284. #1545	C₂×C₆₇	semidirect
285. #1543	C₂×C₆₇	semidirect
286. C₆₇	#17714	direct (N × H)
305. #432	C₂²×C₆₇	semidirect
310. #431	C₂²×C₆₇	semidirect
311. #431	C₂²×C₆₇	semidirect
313. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
318. #333	C₂²×C₆₇	semidirect
319. #331	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₆₇×#2920	2
C₂×#431×C₆₇	2
C₆₇×#1551	2
C₆₇×#1548	2
C₆₇×#1545	2
C₆₇×#1543	2
#17714	67

Subgroups — C67×#17714 #30696965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17714 #30696965