Subgroups — C₆₇×#11045 #30690296

Order 17152.

930 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#101×C₆₇	C₂	1	2	yes	maximal
3	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#343×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#423×C₆₇	—	2	4	no	—
12	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
13	C₂×#101×C₆₇	—	2	4	no	—
14	C₂×#101×C₆₇	—	2	4	no	—
15	C₂×#101×C₆₇	—	2	4	no	—
16	#391×C₆₇	—	2	1	no	—
17	#391×C₆₇	—	2	1	no	—
18	#391×C₆₇	—	2	2	no	—
19	C₂×#95×C₆₇	C₄	1	4	yes	—
20	#343×C₆₇	—	2	4	no	—
21	#336×C₆₇	—	2	1	no	—
22	#336×C₆₇	—	2	1	no	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	—	2	1	no	—
25	C₂²×Q₈×C₆₇	—	2	2	no	—
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂²×Q₈×C₆₇	—	2	1	no	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂²×Q₈×C₆₇	—	2	2	no	—
30	C₂²×Q₈×C₆₇	—	2	2	no	—
31	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
32	C₂²×Q₈×C₆₇	Q₈	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
48	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	D₄	1	2	yes	—
60	C₄⋊C₈×C₆₇	—	2	8	no	—
61	#101×C₆₇	—	2	16	no	—
62	#95×C₆₇	—	2	4	no	—
63	#95×C₆₇	—	2	4	no	—
64	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×Q₈×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	—	2	4	no	—
71	C₂×Q₈×C₆₇	—	2	4	no	—
72	C₂×Q₈×C₆₇	—	2	4	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	4	2	no	—
76	C₂×Q₈×C₆₇	—	4	2	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×Q₈×C₆₇	—	4	2	no	—
79	C₂×Q₈×C₆₇	—	4	2	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	4	no	—
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	#41	1	2	yes	—
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	2	4	no	—
110	M₁₆×C₆₇	—	4	2	no	—
111	M₁₆×C₆₇	—	4	2	no	—
112	M₁₆×C₆₇	—	4	2	no	—
113	M₁₆×C₆₇	—	4	2	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	4	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂³×C₆₇	#117	1	1	yes	characteristic
132	C₂³×C₆₇	C₄×D₄	1	2	yes	—
133	C₂³×C₆₇	C₄×D₄	1	2	yes	—
134	Q₈×C₆₇	—	4	2	no	—
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	4	no	—
137	Q₈×C₆₇	—	4	4	no	—
138	Q₈×C₆₇	—	4	2	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	4	no	—
154	C₂×C₄×C₆₇	#101	1	4	yes	complemented
155	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#120	1	2	yes	—
157	C₂×C₄×C₆₇	#121	1	2	yes	—
158	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#121	1	2	yes	—
161	C₂×C₄×C₆₇	#121	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
166	C₂×C₄×C₆₇	#123	1	2	yes	—
167	C₂×C₄×C₆₇	#123	1	2	yes	—
168	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#124	1	2	yes	—
170	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#126	1	2	yes	—
172	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
173	C₈×C₆₇	—	4	4	no	—
174	C₈×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	4	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	#380	1	1	yes	characteristic
181	C₂²×C₆₇	#385	1	2	yes	—
182	C₂²×C₆₇	#386	1	1	yes	characteristic
183	C₂²×C₆₇	#388	1	2	yes	—
184	C₂²×C₆₇	#390	1	1	yes	characteristic
185	C₂²×C₆₇	C₂×#101	1	2	yes	—
186	C₂²×C₆₇	#413	1	2	yes	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	#392	1	1	yes	characteristic
192	C₄×C₆₇	#393	1	1	yes	characteristic
193	C₄×C₆₇	#394	1	2	yes	—
194	C₄×C₆₇	#396	1	2	yes	—
195	C₄×C₆₇	#397	1	1	yes	characteristic
196	C₄×C₆₇	#399	1	1	yes	characteristic
197	#11045	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	2	2	no	—
200	C₂×C₆₇	#1439	1	1	yes	characteristic
201	C₂×C₆₇	#1801	1	2	yes	—
202	C₂×C₆₇	#1901	1	2	yes	—
203	C₂×C₆₇	#1966	1	1	yes	characteristic
204	C₂×C₆₇	#1967	1	1	yes	characteristic
205	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
206	C₂²×#101	C₂×C₆₇	1	2	yes	—
207	C₂×#391	C₂×C₆₇	1	1	yes	characteristic
208	C₂×#343	C₂×C₆₇	1	2	yes	—
209	#1728	C₂×C₆₇	1	1	yes	characteristic
210	C₆₇	#11045	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
213	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
216	#423	—	2	4	no	—
217	C₂×C₄⋊C₈	—	2	2	no	—
218	C₂×#101	—	2	4	no	—
219	C₂×#101	—	2	4	no	—
220	C₂×#101	—	2	4	no	—
221	#391	—	2	1	no	—
222	#391	—	2	1	no	—
223	#391	—	2	2	no	—
224	C₂×#95	C₄×C₆₇	1	4	yes	—
225	#343	—	2	4	no	—
226	#336	—	2	1	no	—
227	#336	—	2	1	no	—
228	C₂²×Q₈	—	2	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	C₂²×Q₈	—	2	2	no	—
231	C₂²×Q₈	—	2	2	no	—
232	C₂²×Q₈	—	2	1	no	—
233	C₂²×Q₈	—	2	1	no	—
234	C₂²×Q₈	—	2	2	no	—
235	C₂²×Q₈	—	2	2	no	—
236	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
237	C₂²×Q₈	Q₈×C₆₇	1	2	yes	—
238	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
239	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
240	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
253	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₆₇	1	2	yes	—
265	C₄⋊C₈	—	2	8	no	—
266	#101	—	2	16	no	—
267	#95	—	2	4	no	—
268	#95	—	2	4	no	—
269	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
270	C₂×Q₈	—	2	4	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	2	4	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×Q₈	—	4	2	no	—
284	C₂×Q₈	—	4	2	no	—
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	4	no	—
298	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
300	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
301	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	#41×C₆₇	1	2	yes	—
309	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	#41×C₆₇	1	2	yes	—
311	M₁₆	—	2	4	no	—
312	M₁₆	—	2	4	no	—
313	M₁₆	—	2	4	no	—
314	M₁₆	—	2	4	no	—
315	M₁₆	—	4	2	no	—
316	M₁₆	—	4	2	no	—
317	M₁₆	—	4	2	no	—
318	M₁₆	—	4	2	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	2	4	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	4	2	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
335	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
336	C₂³	#117×C₆₇	1	1	yes	characteristic
337	C₂³	C₄×D₄×C₆₇	1	2	yes	—
338	C₂³	C₄×D₄×C₆₇	1	2	yes	—
339	Q₈	—	4	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	4	no	—
342	Q₈	—	4	4	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	8	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	8	4	no	—
359	C₂×C₄	#101×C₆₇	1	4	yes	complemented
360	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
361	C₂×C₄	#120×C₆₇	1	2	yes	—
362	C₂×C₄	#121×C₆₇	1	2	yes	—
363	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
364	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
365	C₂×C₄	#121×C₆₇	1	2	yes	—
366	C₂×C₄	#121×C₆₇	1	2	yes	—
367	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
368	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
370	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
371	C₂×C₄	#123×C₆₇	1	2	yes	—
372	C₂×C₄	#123×C₆₇	1	2	yes	—
373	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
374	C₂×C₄	#124×C₆₇	1	2	yes	—
375	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
376	C₂×C₄	#126×C₆₇	1	2	yes	—
377	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
378	C₈	—	4	4	no	—
379	C₈	—	4	4	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	4	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	#380×C₆₇	1	1	yes	characteristic
386	C₂²	#385×C₆₇	1	2	yes	—
387	C₂²	#386×C₆₇	1	1	yes	characteristic
388	C₂²	#388×C₆₇	1	2	yes	—
389	C₂²	#390×C₆₇	1	1	yes	characteristic
390	C₂²	C₂×#101×C₆₇	1	2	yes	—
391	C₂²	#413×C₆₇	1	2	yes	—
392	C₄	—	2	4	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	4	no	—
395	C₄	—	8	2	no	—
396	C₄	#392×C₆₇	1	1	yes	characteristic
397	C₄	#393×C₆₇	1	1	yes	characteristic
398	C₄	#394×C₆₇	1	2	yes	—
399	C₄	#396×C₆₇	1	2	yes	—
400	C₄	#397×C₆₇	1	1	yes	characteristic
401	C₄	#399×C₆₇	1	1	yes	characteristic
402	C₂	—	2	2	no	—
403	C₂	—	2	2	no	—
404	C₂	C₆₇×#1439	1	1	yes	characteristic
405	C₂	C₆₇×#1801	1	2	yes	—
406	C₂	C₆₇×#1901	1	2	yes	—
407	C₂	C₆₇×#1966	1	1	yes	characteristic
408	C₂	C₆₇×#1967	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
154. C₂×C₄×C₆₇	#101	semidirect
197. #11045	C₆₇	direct (N × H)
210. C₆₇	#11045	direct (N × H)
359. C₂×C₄	#101×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₂²×#101×C₆₇	2
C₂×#391×C₆₇	2
C₂×#343×C₆₇	2
C₆₇×#1728	2
#11045	67

Subgroups — C67×#11045 #30690296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11045 #30690296