Subgroups — C₆₇×#60900 #30740151

Order 17152.

1942 conjugacy classes of subgroups.

The subgroup lattice has 610 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	2	yes	maximal, complemented
2	C₆₇×#3250	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3248	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3247	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3246	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3245	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3076	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3050	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3035	C₂	1	4	yes	maximal, complemented
11	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	4	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	#533×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	C₂²	1	2	yes	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	#532×C₆₇	C₂²	1	2	yes	—
19	#532×C₆₇	C₂²	1	2	yes	—
20	#529×C₆₇	—	2	2	no	—
21	#529×C₆₇	—	2	2	no	—
22	#529×C₆₇	—	2	2	no	—
23	#529×C₆₇	—	2	2	no	—
24	#529×C₆₇	C₂²	1	4	yes	—
25	#529×C₆₇	C₂²	1	4	yes	—
26	#525×C₆₇	—	2	2	no	—
27	#525×C₆₇	—	2	2	no	—
28	#525×C₆₇	C₂²	1	4	yes	—
29	C₄×#41×C₆₇	—	2	2	no	—
30	C₄×#41×C₆₇	—	2	2	no	—
31	C₄×#41×C₆₇	C₂²	1	4	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#479×C₆₇	C₂²	1	8	yes	complemented
34	#478×C₆₇	C₂²	1	8	yes	complemented
35	#477×C₆₇	C₂²	1	4	yes	complemented
36	#476×C₆₇	C₂²	1	4	yes	complemented
37	#475×C₆₇	C₂²	1	4	yes	complemented
38	#474×C₆₇	C₂²	1	4	yes	complemented
39	#467×C₆₇	C₂²	1	4	yes	—
40	#467×C₆₇	C₂²	1	4	yes	—
41	#466×C₆₇	C₂²	1	4	yes	—
42	#466×C₆₇	C₂²	1	4	yes	—
43	#465×C₆₇	C₂²	1	8	yes	—
44	#465×C₆₇	C₂²	1	8	yes	—
45	#464×C₆₇	C₂²	1	8	yes	complemented
46	#463×C₆₇	C₂²	1	8	yes	complemented
47	#462×C₆₇	C₂²	1	4	yes	complemented
48	#461×C₆₇	C₂²	1	4	yes	complemented
49	#460×C₆₇	C₂²	1	4	yes	complemented
50	#459×C₆₇	C₂²	1	4	yes	complemented
51	#427×C₆₇	C₂²	1	4	yes	—
52	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
53	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
56	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
57	#416×C₆₇	C₂²	1	8	yes	—
58	C₂×#41×C₆₇	—	2	2	no	—
59	C₂×#41×C₆₇	—	2	2	no	—
60	C₂×#41×C₆₇	C₂³	1	4	yes	—
61	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	#135×C₆₇	—	4	8	no	—
65	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
67	C₂×D₈×C₆₇	C₂³	1	4	yes	—
68	C₂²×C₈×C₆₇	—	4	4	no	—
69	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	#127×C₆₇	C₂³	1	2	yes	—
77	#126×C₆₇	—	2	4	no	—
78	#126×C₆₇	—	2	4	no	—
79	#125×C₆₇	—	2	2	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	—	2	2	no	—
84	#124×C₆₇	C₂³	1	4	yes	—
85	#123×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	4	no	—
87	#123×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	2	no	—
89	C₂²⋊Q₈×C₆₇	—	2	2	no	—
90	C₂²⋊Q₈×C₆₇	—	2	2	no	—
91	C₂²⋊Q₈×C₆₇	—	2	2	no	—
92	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
93	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
94	#121×C₆₇	—	2	2	no	—
95	#121×C₆₇	—	2	2	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	C₂³	1	4	yes	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	#120×C₆₇	—	2	4	no	—
101	C₄×Q₈×C₆₇	—	2	2	no	—
102	C₄×Q₈×C₆₇	—	2	2	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	—	2	2	no	—
105	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
106	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
107	C₄×D₄×C₆₇	—	2	4	no	—
108	C₄×D₄×C₆₇	—	2	4	no	—
109	C₄×D₄×C₆₇	—	2	4	no	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	4	no	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	—	2	2	no	—
117	C₄×D₄×C₆₇	C₂³	1	4	yes	—
118	C₄×D₄×C₆₇	C₂³	1	4	yes	—
119	#117×C₆₇	—	2	2	no	—
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	—	2	4	no	—
122	#117×C₆₇	—	2	4	no	—
123	#117×C₆₇	—	2	4	no	—
124	#117×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄²×C₆₇	—	2	4	no	—
126	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	D₄	1	2	yes	—
128	C₂×C₄²×C₆₇	D₄	1	2	yes	—
129	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	#107×C₆₇	C₂³	1	4	yes	—
135	#107×C₆₇	C₂³	1	4	yes	—
136	#106×C₆₇	C₂³	1	8	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	8	no	—
147	#41×C₆₇	—	4	8	no	—
148	#41×C₆₇	—	4	8	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₆₇	—	4	4	no	—
180	SD₁₆×C₆₇	—	4	8	no	—
181	D₈×C₆₇	—	4	4	no	—
182	C₂×C₈×C₆₇	—	2	8	no	—
183	C₂×C₈×C₆₇	—	4	2	no	—
184	C₂×C₈×C₆₇	—	4	4	no	—
185	C₂×C₈×C₆₇	—	4	2	no	—
186	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
187	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	4	no	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	—	2	4	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	C₂×D₄	1	2	yes	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₄²×C₆₇	C₂×D₄	1	2	yes	—
217	C₄²×C₆₇	C₂×D₄	1	2	yes	—
218	C₄²×C₆₇	C₂⁴	1	2	yes	—
219	C₄²×C₆₇	C₂⁴	1	2	yes	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₆₇	—	2	4	no	—
228	Q₈×C₆₇	—	2	4	no	—
229	Q₈×C₆₇	—	4	4	no	—
230	Q₈×C₆₇	—	4	4	no	—
231	D₄×C₆₇	—	2	4	no	—
232	D₄×C₆₇	—	2	4	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	4	8	no	—
236	D₄×C₆₇	—	4	8	no	—
237	D₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	1	no	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	8	no	—
256	C₂×C₄×C₆₇	—	4	8	no	—
257	C₂×C₄×C₆₇	—	4	8	no	—
258	C₂×C₄×C₆₇	#135	1	4	yes	complemented
259	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
267	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
268	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
269	C₈×C₆₇	—	2	4	no	—
270	C₈×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	2	8	no	—
272	C₂²×C₆₇	—	2	1	no	—
273	C₂²×C₆₇	—	2	1	no	—
274	C₂²×C₆₇	—	2	1	no	—
275	C₂²×C₆₇	—	2	1	no	—
276	C₂²×C₆₇	—	2	1	no	—
277	C₂²×C₆₇	—	2	1	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	4	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	#537	1	2	yes	—
282	C₂²×C₆₇	C₂×#135	1	2	yes	—
283	C₂²×C₆₇	#575	1	2	yes	—
284	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
285	C₄×C₆₇	—	2	1	no	—
286	C₄×C₆₇	—	2	1	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	4	4	no	—
291	C₄×C₆₇	—	4	4	no	—
292	C₄×C₆₇	—	4	4	no	—
293	C₄×C₆₇	#537	1	2	yes	—
294	C₄×C₆₇	C₂×#135	1	4	yes	—
295	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
296	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
297	#60900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
298	C₂×C₆₇	—	2	2	no	—
299	C₂×C₆₇	—	2	2	no	—
300	C₂×C₆₇	—	2	2	no	—
301	C₂×C₆₇	—	8	4	no	—
302	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
303	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
304	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
305	#3444	C₂×C₆₇	1	2	yes	complemented
306	#3250	C₂×C₆₇	1	4	yes	complemented
307	#3249	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#3248	C₂×C₆₇	1	4	yes	complemented
309	#3247	C₂×C₆₇	1	4	yes	complemented
310	#3246	C₂×C₆₇	1	2	yes	complemented
311	#3245	C₂×C₆₇	1	1	yes	characteristic, complemented
312	#3076	C₂×C₆₇	1	4	yes	complemented
313	#3050	C₂×C₆₇	1	4	yes	complemented
314	#3035	C₂×C₆₇	1	4	yes	complemented
315	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₆₇	#60900	1	1	yes	characteristic, direct product factor, Sylow
317	C₂×#135	C₂²×C₆₇	1	4	yes	—
318	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
319	#533	C₂²×C₆₇	1	2	yes	—
320	#533	C₂²×C₆₇	1	2	yes	—
321	#533	C₂²×C₆₇	1	2	yes	—
322	#532	C₂²×C₆₇	1	2	yes	—
323	#532	C₂²×C₆₇	1	2	yes	—
324	#532	C₂²×C₆₇	1	2	yes	—
325	#529	—	2	2	no	—
326	#529	—	2	2	no	—
327	#529	—	2	2	no	—
328	#529	—	2	2	no	—
329	#529	C₂²×C₆₇	1	4	yes	—
330	#529	C₂²×C₆₇	1	4	yes	—
331	#525	—	2	2	no	—
332	#525	—	2	2	no	—
333	#525	C₂²×C₆₇	1	4	yes	—
334	C₄×#41	—	2	2	no	—
335	C₄×#41	—	2	2	no	—
336	C₄×#41	C₂²×C₆₇	1	4	yes	—
337	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
338	#479	C₂²×C₆₇	1	8	yes	complemented
339	#478	C₂²×C₆₇	1	8	yes	complemented
340	#477	C₂²×C₆₇	1	4	yes	complemented
341	#476	C₂²×C₆₇	1	4	yes	complemented
342	#475	C₂²×C₆₇	1	4	yes	complemented
343	#474	C₂²×C₆₇	1	4	yes	complemented
344	#467	C₂²×C₆₇	1	4	yes	—
345	#467	C₂²×C₆₇	1	4	yes	—
346	#466	C₂²×C₆₇	1	4	yes	—
347	#466	C₂²×C₆₇	1	4	yes	—
348	#465	C₂²×C₆₇	1	8	yes	—
349	#465	C₂²×C₆₇	1	8	yes	—
350	#464	C₂²×C₆₇	1	8	yes	complemented
351	#463	C₂²×C₆₇	1	8	yes	complemented
352	#462	C₂²×C₆₇	1	4	yes	complemented
353	#461	C₂²×C₆₇	1	4	yes	complemented
354	#460	C₂²×C₆₇	1	4	yes	complemented
355	#459	C₂²×C₆₇	1	4	yes	complemented
356	#427	C₂²×C₆₇	1	4	yes	—
357	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
358	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
361	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
362	#416	C₂²×C₆₇	1	8	yes	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	—	2	2	no	—
365	C₂×#41	C₂³×C₆₇	1	4	yes	—
366	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
367	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
369	#135	—	4	8	no	—
370	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
371	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
372	C₂×D₈	C₂³×C₆₇	1	4	yes	—
373	C₂²×C₈	—	4	4	no	—
374	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
375	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
376	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
378	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
379	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
380	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
381	#127	C₂³×C₆₇	1	2	yes	—
382	#126	—	2	4	no	—
383	#126	—	2	4	no	—
384	#125	—	2	2	no	—
385	#125	—	2	2	no	—
386	#125	C₂³×C₆₇	1	4	yes	—
387	#124	—	2	2	no	—
388	#124	—	2	2	no	—
389	#124	C₂³×C₆₇	1	4	yes	—
390	#123	—	2	4	no	—
391	#123	—	2	4	no	—
392	#123	—	2	4	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
398	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
399	#121	—	2	2	no	—
400	#121	—	2	2	no	—
401	#121	—	2	2	no	—
402	#121	—	2	2	no	—
403	#121	C₂³×C₆₇	1	4	yes	—
404	#121	C₂³×C₆₇	1	4	yes	—
405	#120	—	2	4	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	2	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
411	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	4	no	—
416	C₄×D₄	—	2	4	no	—
417	C₄×D₄	—	2	4	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	2	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	C₂³×C₆₇	1	4	yes	—
423	C₄×D₄	C₂³×C₆₇	1	4	yes	—
424	#117	—	2	2	no	—
425	#117	—	2	2	no	—
426	#117	—	2	4	no	—
427	#117	—	2	4	no	—
428	#117	—	2	4	no	—
429	#117	C₂³×C₆₇	1	4	yes	—
430	C₂×C₄²	—	2	4	no	—
431	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₆₇	1	2	yes	—
433	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₆₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
439	#107	C₂³×C₆₇	1	4	yes	—
440	#107	C₂³×C₆₇	1	4	yes	—
441	#106	C₂³×C₆₇	1	8	yes	—
442	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
443	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
444	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
445	#103	C₂³×C₆₇	1	8	yes	—
446	#103	C₂³×C₆₇	1	8	yes	—
447	#102	C₂³×C₆₇	1	8	yes	—
448	#102	C₂³×C₆₇	1	8	yes	—
449	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
450	#41	—	2	8	no	—
451	#41	—	2	8	no	—
452	#41	—	4	8	no	—
453	#41	—	4	8	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
484	Q₁₆	—	4	4	no	—
485	SD₁₆	—	4	8	no	—
486	D₈	—	4	4	no	—
487	C₂×C₈	—	2	8	no	—
488	C₂×C₈	—	4	2	no	—
489	C₂×C₈	—	4	4	no	—
490	C₂×C₈	—	4	2	no	—
491	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
492	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	4	no	—
516	C₄²	—	2	4	no	—
517	C₄²	C₂×D₄×C₆₇	1	2	yes	—
518	C₄²	C₂×D₄×C₆₇	1	2	yes	—
519	C₄²	C₂×D₄×C₆₇	1	2	yes	—
520	C₄²	C₂×D₄×C₆₇	1	2	yes	—
521	C₄²	C₂×D₄×C₆₇	1	2	yes	—
522	C₄²	C₂×D₄×C₆₇	1	2	yes	—
523	C₄²	C₂⁴×C₆₇	1	2	yes	—
524	C₄²	C₂⁴×C₆₇	1	2	yes	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	4	no	—
529	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
530	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
531	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
532	Q₈	—	2	4	no	—
533	Q₈	—	2	4	no	—
534	Q₈	—	4	4	no	—
535	Q₈	—	4	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	4	4	no	—
539	D₄	—	4	8	no	—
540	D₄	—	4	8	no	—
541	D₄	—	4	8	no	—
542	D₄	—	4	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	8	no	—
563	C₂×C₄	#135×C₆₇	1	4	yes	complemented
564	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
570	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
571	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
572	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
573	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
574	C₈	—	2	4	no	—
575	C₈	—	2	4	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	#537×C₆₇	1	2	yes	—
587	C₂²	C₂×#135×C₆₇	1	2	yes	—
588	C₂²	#575×C₆₇	1	2	yes	—
589	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	#537×C₆₇	1	2	yes	—
599	C₄	C₂×#135×C₆₇	1	4	yes	—
600	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
601	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
602	C₂	—	2	2	no	—
603	C₂	—	2	2	no	—
604	C₂	—	2	2	no	—
605	C₂	—	8	4	no	—
606	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
607	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
608	C₂	C₂×#575×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
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
1. C₆₇×#3444	C₂	semidirect
2. C₆₇×#3250	C₂	semidirect
3. C₆₇×#3249	C₂	semidirect
4. C₆₇×#3248	C₂	semidirect
5. C₆₇×#3247	C₂	semidirect
6. C₆₇×#3246	C₂	semidirect
7. C₆₇×#3245	C₂	semidirect
8. C₆₇×#3076	C₂	semidirect
9. C₆₇×#3050	C₂	semidirect
10. C₆₇×#3035	C₂	semidirect
11. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
33. #479×C₆₇	C₂²	semidirect
34. #478×C₆₇	C₂²	semidirect
35. #477×C₆₇	C₂²	semidirect
36. #476×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
38. #474×C₆₇	C₂²	semidirect
45. #464×C₆₇	C₂²	semidirect
46. #463×C₆₇	C₂²	semidirect
47. #462×C₆₇	C₂²	semidirect
48. #461×C₆₇	C₂²	semidirect
49. #460×C₆₇	C₂²	semidirect
50. #459×C₆₇	C₂²	semidirect
258. C₂×C₄×C₆₇	#135	semidirect
297. #60900	C₆₇	direct (N × H)
305. #3444	C₂×C₆₇	semidirect
306. #3250	C₂×C₆₇	semidirect
307. #3249	C₂×C₆₇	semidirect
308. #3248	C₂×C₆₇	semidirect
309. #3247	C₂×C₆₇	semidirect
310. #3246	C₂×C₆₇	semidirect
311. #3245	C₂×C₆₇	semidirect
312. #3076	C₂×C₆₇	semidirect
313. #3050	C₂×C₆₇	semidirect
314. #3035	C₂×C₆₇	semidirect
315. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
316. C₆₇	#60900	direct (N × H)
338. #479	C₂²×C₆₇	semidirect
339. #478	C₂²×C₆₇	semidirect
340. #477	C₂²×C₆₇	semidirect
341. #476	C₂²×C₆₇	semidirect
342. #475	C₂²×C₆₇	semidirect
343. #474	C₂²×C₆₇	semidirect
350. #464	C₂²×C₆₇	semidirect
351. #463	C₂²×C₆₇	semidirect
352. #462	C₂²×C₆₇	semidirect
353. #461	C₂²×C₆₇	semidirect
354. #460	C₂²×C₆₇	semidirect
355. #459	C₂²×C₆₇	semidirect
563. C₂×C₄	#135×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×#3250	2
C₆₇×#3249	2
C₆₇×#3248	2
C₆₇×#3247	2
C₆₇×#3246	2
C₆₇×#3245	2
C₆₇×#3076	2
C₆₇×#3050	2
C₆₇×#3035	2
C₂²×C₄⋊C₈×C₆₇	2
#60900	67

Subgroups — C67×#60900 #30740151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60900 #30740151