Subgroups — C₆₇×#33018 #30712269

Order 17152.

1432 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#518×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#485×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#450×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2625	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1825	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#518×C₆₇	—	2	4	no	—
16	#518×C₆₇	—	2	4	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
24	#485×C₆₇	—	2	4	no	—
25	#450×C₆₇	—	2	4	no	—
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#400×C₆₇	—	2	1	no	—
36	#398×C₆₇	—	2	1	no	—
37	#397×C₆₇	—	2	2	no	—
38	#391×C₆₇	C₂²	1	2	yes	complemented
39	#389×C₆₇	C₂²	1	2	yes	complemented
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	complemented
42	#340×C₆₇	C₂²	1	2	yes	complemented
43	#336×C₆₇	C₂²	1	2	yes	complemented
44	#336×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	—	2	1	no	—
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	—	2	4	no	—
48	C₂²×D₄×C₆₇	—	2	4	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	D₄	1	4	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	—	2	2	no	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×SD₁₆×C₆₇	—	2	4	no	—
57	C₂×SD₁₆×C₆₇	—	2	4	no	—
58	C₂×SD₁₆×C₆₇	—	2	4	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²⋊Q₈×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	2	8	no	—
65	C₄×D₄×C₆₇	—	2	8	no	—
66	C₄×D₄×C₆₇	—	2	4	no	—
67	C₄×D₄×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	—	2	4	no	—
70	#117×C₆₇	—	2	2	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	D₄	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
89	C₂×C₄²×C₆₇	—	2	1	no	—
90	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
91	#107×C₆₇	—	2	2	no	—
92	#106×C₆₇	—	2	2	no	—
93	#103×C₆₇	—	2	2	no	—
94	#103×C₆₇	—	2	2	no	—
95	#102×C₆₇	—	2	4	no	—
96	#102×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	2	4	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	2	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	8	no	—
105	C₂⁴×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	8	1	no	—
108	C₂⁴×C₆₇	—	8	2	no	—
109	C₂⁴×C₆₇	—	8	1	no	—
110	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	8	no	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	2	16	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂×D₄×C₆₇	#41	1	4	yes	—
125	C₂×D₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	8	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	8	2	no	—
136	C₂²×C₄×C₆₇	—	8	2	no	—
137	C₂²×C₄×C₆₇	—	8	2	no	—
138	C₂²×C₄×C₆₇	—	8	2	no	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	SD₁₆×C₆₇	—	4	8	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	—	2	1	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	1	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	4	2	no	—
162	C₄⋊C₄×C₆₇	—	8	2	no	—
163	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	8	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	4	2	no	—
170	C₂²⋊C₄×C₆₇	—	8	1	no	—
171	C₄²×C₆₇	—	2	4	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	16	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	1	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	—	4	2	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	8	4	no	—
187	C₂³×C₆₇	—	8	4	no	—
188	C₂³×C₆₇	—	8	8	no	—
189	C₂³×C₆₇	—	8	8	no	—
190	C₂³×C₆₇	#120	1	2	yes	—
191	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₆₇	—	4	2	no	—
193	Q₈×C₆₇	—	4	1	no	—
194	Q₈×C₆₇	—	4	1	no	—
195	D₄×C₆₇	—	2	16	no	—
196	D₄×C₆₇	—	2	4	no	—
197	D₄×C₆₇	—	2	4	no	—
198	D₄×C₆₇	—	2	4	no	—
199	D₄×C₆₇	#123	1	8	yes	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	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	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	1	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	8	8	no	—
224	C₂×C₄×C₆₇	—	8	8	no	—
225	C₂×C₄×C₆₇	#120	1	2	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
229	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
230	C₈×C₆₇	—	4	2	no	—
231	C₈×C₆₇	—	4	1	no	—
232	C₈×C₆₇	—	4	1	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	8	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	2	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	4	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	8	16	no	—
246	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
247	C₂²×C₆₇	#525	1	2	yes	—
248	C₂²×C₆₇	#576	1	1	yes	characteristic
249	C₂²×C₆₇	#577	1	2	yes	—
250	C₂²×C₆₇	#577	1	1	yes	characteristic
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	4	2	no	—
253	C₄×C₆₇	—	4	2	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	4	2	no	—
256	C₄×C₆₇	—	8	1	no	—
257	C₄×C₆₇	—	8	2	no	—
258	C₄×C₆₇	—	8	1	no	—
259	C₄×C₆₇	C₂×#123	1	2	yes	—
260	C₄×C₆₇	#525	1	1	yes	characteristic
261	C₄×C₆₇	#525	1	1	yes	characteristic
262	#33018	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	4	2	no	—
264	C₂×C₆₇	—	4	8	no	—
265	C₂×C₆₇	#2849	1	1	yes	characteristic
266	C₂×C₆₇	#3009	1	1	yes	characteristic
267	C₂×C₆₇	#3012	1	1	yes	characteristic
268	C₂×C₆₇	#3118	1	2	yes	—
269	C₂×C₆₇	#3123	1	2	yes	—
270	C₂×#518	C₂×C₆₇	1	1	yes	characteristic
271	C₂×#485	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#450	C₂×C₆₇	1	1	yes	characteristic
273	#2625	C₂×C₆₇	1	1	yes	characteristic, complemented
274	#1877	C₂×C₆₇	1	2	yes	complemented
275	#1877	C₂×C₆₇	1	2	yes	complemented
276	#1864	C₂×C₆₇	1	2	yes	complemented
277	#1862	C₂×C₆₇	1	2	yes	complemented
278	#1825	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#1818	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1786	C₂×C₆₇	1	1	yes	characteristic
281	C₆₇	#33018	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
284	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
285	#518	—	2	4	no	—
286	#518	—	2	4	no	—
287	C₂×C₄×D₄	—	2	1	no	—
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
290	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
291	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
293	C₂²×C₂²⋊C₄	—	2	2	no	—
294	#485	—	2	4	no	—
295	#450	—	2	4	no	—
296	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
298	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
299	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#102	C₂²×C₆₇	1	2	yes	—
301	C₂×#102	C₂²×C₆₇	1	2	yes	—
302	C₂×#102	C₂²×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
305	#400	—	2	1	no	—
306	#398	—	2	1	no	—
307	#397	—	2	2	no	—
308	#391	C₂²×C₆₇	1	2	yes	complemented
309	#389	C₂²×C₆₇	1	2	yes	complemented
310	#380	C₂²×C₆₇	1	1	yes	characteristic
311	#340	C₂²×C₆₇	1	2	yes	complemented
312	#340	C₂²×C₆₇	1	2	yes	complemented
313	#336	C₂²×C₆₇	1	2	yes	complemented
314	#336	C₂²×C₆₇	1	2	yes	complemented
315	C₂⁵	—	2	1	no	—
316	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
317	C₂²×D₄	—	2	4	no	—
318	C₂²×D₄	—	2	4	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	D₄×C₆₇	1	4	yes	—
322	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	—	2	2	no	—
324	C₂³×C₄	—	2	2	no	—
325	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
326	C₂×SD₁₆	—	2	4	no	—
327	C₂×SD₁₆	—	2	4	no	—
328	C₂×SD₁₆	—	2	4	no	—
329	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₄×D₄	—	2	8	no	—
335	C₄×D₄	—	2	8	no	—
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	2	4	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	4	no	—
340	#117	—	2	2	no	—
341	#117	—	2	2	no	—
342	#117	—	2	2	no	—
343	#117	D₄×C₆₇	1	4	yes	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	—	2	16	no	—
353	C₂×C₂²⋊C₄	—	2	1	no	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
361	#107	—	2	2	no	—
362	#106	—	2	2	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	#102	—	2	4	no	—
366	#102	—	2	4	no	—
367	#102	—	2	4	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	#95	—	2	1	no	—
371	#95	—	2	2	no	—
372	#95	—	2	2	no	—
373	#95	—	2	2	no	—
374	C₂⁴	—	2	8	no	—
375	C₂⁴	—	2	2	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	8	1	no	—
378	C₂⁴	—	8	2	no	—
379	C₂⁴	—	8	1	no	—
380	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	8	no	—
388	C₂×D₄	—	2	16	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
394	C₂×D₄	#41×C₆₇	1	4	yes	—
395	C₂×D₄	#41×C₆₇	1	4	yes	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	8	no	—
401	C₂²×C₄	—	2	8	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	8	2	no	—
406	C₂²×C₄	—	8	2	no	—
407	C₂²×C₄	—	8	2	no	—
408	C₂²×C₄	—	8	2	no	—
409	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
413	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
414	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
415	SD₁₆	—	4	8	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₄⋊C₄	—	8	2	no	—
433	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
434	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₂²⋊C₄	—	8	1	no	—
441	C₄²	—	2	4	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	8	4	no	—
457	C₂³	—	8	4	no	—
458	C₂³	—	8	8	no	—
459	C₂³	—	8	8	no	—
460	C₂³	#120×C₆₇	1	2	yes	—
461	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
462	Q₈	—	4	2	no	—
463	Q₈	—	4	1	no	—
464	Q₈	—	4	1	no	—
465	D₄	—	2	16	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	#123×C₆₇	1	8	yes	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	8	8	no	—
494	C₂×C₄	—	8	8	no	—
495	C₂×C₄	#120×C₆₇	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
498	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
499	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
500	C₈	—	4	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	8	16	no	—
516	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
517	C₂²	#525×C₆₇	1	2	yes	—
518	C₂²	#576×C₆₇	1	1	yes	characteristic
519	C₂²	#577×C₆₇	1	2	yes	—
520	C₂²	#577×C₆₇	1	1	yes	characteristic
521	C₄	—	2	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	8	1	no	—
527	C₄	—	8	1	no	—
528	C₄	—	8	2	no	—
529	C₄	C₂×#123×C₆₇	1	2	yes	—
530	C₄	#525×C₆₇	1	1	yes	characteristic
531	C₄	#525×C₆₇	1	1	yes	characteristic
532	C₂	—	4	2	no	—
533	C₂	—	4	8	no	—
534	C₂	C₆₇×#2849	1	1	yes	characteristic
535	C₂	C₆₇×#3009	1	1	yes	characteristic
536	C₂	C₆₇×#3012	1	1	yes	characteristic
537	C₂	C₆₇×#3118	1	2	yes	—
538	C₂	C₆₇×#3123	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#485×C₆₇	C₂	semidirect
4. C₆₇×#2625	C₂	semidirect
5. C₆₇×#1877	C₂	semidirect
6. C₆₇×#1877	C₂	semidirect
7. C₆₇×#1864	C₂	semidirect
8. C₆₇×#1862	C₂	semidirect
9. C₆₇×#1825	C₂	semidirect
10. C₆₇×#1818	C₂	semidirect
26. C₂×#107×C₆₇	C₂²	semidirect
27. C₂×#106×C₆₇	C₂²	semidirect
28. C₂×#103×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
38. #391×C₆₇	C₂²	semidirect
39. #389×C₆₇	C₂²	semidirect
41. #340×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #336×C₆₇	C₂²	semidirect
44. #336×C₆₇	C₂²	semidirect
262. #33018	C₆₇	direct (N × H)
271. C₂×#485	C₂×C₆₇	semidirect
273. #2625	C₂×C₆₇	semidirect
274. #1877	C₂×C₆₇	semidirect
275. #1877	C₂×C₆₇	semidirect
276. #1864	C₂×C₆₇	semidirect
277. #1862	C₂×C₆₇	semidirect
278. #1825	C₂×C₆₇	semidirect
279. #1818	C₂×C₆₇	semidirect
281. C₆₇	#33018	direct (N × H)
296. C₂×#107	C₂²×C₆₇	semidirect
297. C₂×#106	C₂²×C₆₇	semidirect
298. C₂×#103	C₂²×C₆₇	semidirect
299. C₂×#103	C₂²×C₆₇	semidirect
308. #391	C₂²×C₆₇	semidirect
309. #389	C₂²×C₆₇	semidirect
311. #340	C₂²×C₆₇	semidirect
312. #340	C₂²×C₆₇	semidirect
313. #336	C₂²×C₆₇	semidirect
314. #336	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×#518×C₆₇	2
C₂×#485×C₆₇	2
C₂×#450×C₆₇	2
C₆₇×#2625	2
C₆₇×#1877	2
C₆₇×#1864	2
C₆₇×#1862	2
C₆₇×#1825	2
C₆₇×#1818	2
C₆₇×#1786	2
#33018	67

Subgroups — C67×#33018 #30712269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33018 #30712269