Subgroups — C₆₇×#17928 #30697179

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1589	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1581	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1480	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₆₇	—	2	1	no	—
11	C₂²×M₁₆×C₆₇	—	2	1	no	—
12	#532×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#480×C₆₇	C₄	1	2	yes	—
15	#474×C₆₇	C₄	1	2	yes	—
16	#459×C₆₇	C₄	1	2	yes	—
17	#447×C₆₇	C₄	1	2	yes	—
18	C₄×D₈×C₆₇	C₂²	1	4	yes	—
19	#436×C₆₇	—	2	1	no	—
20	#436×C₆₇	—	2	1	no	—
21	#436×C₆₇	—	2	1	no	—
22	#436×C₆₇	C₂²	1	2	yes	—
23	#436×C₆₇	C₂²	1	2	yes	—
24	#436×C₆₇	C₂²	1	2	yes	complemented
25	C₈×D₄×C₆₇	—	2	1	no	—
26	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	#423×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#408×C₆₇	—	2	1	no	—
32	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#335×C₆₇	C₂²	1	4	yes	complemented
35	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
36	D₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
37	#322×C₆₇	C₂²	1	2	yes	complemented
38	C₈²×C₆₇	C₂²	1	2	yes	complemented
39	C₂×#41×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
43	#131×C₆₇	—	4	4	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	4	4	no	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#124×C₆₇	—	2	1	no	—
56	#121×C₆₇	—	2	1	no	—
57	#121×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#107×C₆₇	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	#102×C₆₇	C₂×C₄	1	4	yes	—
75	#102×C₆₇	C₂×C₄	1	4	yes	—
76	C₂²⋊C₈×C₆₇	—	2	1	no	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	—	2	2	no	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	—	2	2	no	—
81	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂⁴×C₆₇	—	2	1	no	—
89	#41×C₆₇	—	4	2	no	—
90	#41×C₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	D₈×C₆₇	—	4	4	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	M₁₆×C₆₇	—	4	4	no	—
111	M₁₆×C₆₇	—	4	4	no	—
112	M₁₆×C₆₇	—	4	4	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	—	4	4	no	—
119	C₂×C₈×C₆₇	D₈	1	4	yes	complemented
120	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	#41	1	1	yes	characteristic
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	2	no	—
142	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
143	Q₈×C₆₇	—	4	2	no	—
144	D₄×C₆₇	—	2	2	no	—
145	D₄×C₆₇	—	2	4	no	—
146	D₄×C₆₇	—	2	2	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	4	2	no	—
149	D₄×C₆₇	—	4	4	no	—
150	D₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
165	C₂×C₄×C₆₇	#135	1	2	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₈×C₆₇	—	2	4	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	C₂×D₈	1	2	yes	—
176	C₈×C₆₇	C₂×D₈	1	2	yes	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	4	2	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	8	4	no	—
186	C₂²×C₆₇	C₄×D₈	1	2	yes	—
187	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	4	2	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	C₄×D₈	1	2	yes	—
194	C₄×C₆₇	#568	1	2	yes	—
195	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
197	#17928	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	8	2	no	—
200	C₂×C₆₇	—	8	2	no	—
201	C₂×C₆₇	#2930	1	1	yes	characteristic
202	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
203	C₂×C₆₇	#2955	1	1	yes	characteristic
204	#3100	C₂×C₆₇	1	1	yes	characteristic
205	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1589	C₂×C₆₇	1	4	yes	complemented
208	#1581	C₂×C₆₇	1	4	yes	complemented
209	#1571	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1486	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#1480	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₆₇	#17928	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#131	—	2	1	no	—
215	C₂²×M₁₆	—	2	1	no	—
216	#532	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
218	#480	C₄×C₆₇	1	2	yes	—
219	#474	C₄×C₆₇	1	2	yes	—
220	#459	C₄×C₆₇	1	2	yes	—
221	#447	C₄×C₆₇	1	2	yes	—
222	C₄×D₈	C₂²×C₆₇	1	4	yes	—
223	#436	—	2	1	no	—
224	#436	—	2	1	no	—
225	#436	—	2	1	no	—
226	#436	C₂²×C₆₇	1	2	yes	—
227	#436	C₂²×C₆₇	1	2	yes	—
228	#436	C₂²×C₆₇	1	2	yes	complemented
229	C₈×D₄	—	2	1	no	—
230	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
231	#431	C₂²×C₆₇	1	1	yes	characteristic
232	#431	C₂²×C₆₇	1	1	yes	characteristic
233	#423	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
235	#408	—	2	1	no	—
236	C₂×C₂²⋊C₈	—	2	1	no	—
237	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
238	#335	C₂²×C₆₇	1	4	yes	complemented
239	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
240	D₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
241	#322	C₂²×C₆₇	1	2	yes	complemented
242	C₈²	C₂²×C₆₇	1	2	yes	complemented
243	C₂×#41	—	2	1	no	—
244	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	—	2	1	no	—
246	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
247	#131	—	4	4	no	—
248	C₂×M₁₆	—	2	1	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	4	4	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
257	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
258	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
259	#124	—	2	1	no	—
260	#121	—	2	1	no	—
261	#121	C₂×C₄×C₆₇	1	2	yes	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
266	C₄×D₄	C₂³×C₆₇	1	2	yes	—
267	C₄×D₄	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
269	C₂×C₂²⋊C₄	—	2	1	no	—
270	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
271	#107	C₂×C₄×C₆₇	1	4	yes	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
276	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
277	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
278	#102	C₂×C₄×C₆₇	1	4	yes	—
279	#102	C₂×C₄×C₆₇	1	4	yes	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	2	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
286	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
287	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₆₇	1	2	yes	—
289	C₄×C₈	C₂³×C₆₇	1	2	yes	—
290	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
292	C₂⁴	—	2	1	no	—
293	#41	—	4	2	no	—
294	#41	—	4	2	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	1	no	—
300	C₂×D₄	—	4	2	no	—
301	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
302	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
312	D₈	—	4	4	no	—
313	M₁₆	—	4	2	no	—
314	M₁₆	—	4	4	no	—
315	M₁₆	—	4	4	no	—
316	M₁₆	—	4	4	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	4	no	—
323	C₂×C₈	D₈×C₆₇	1	4	yes	complemented
324	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
325	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
332	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₄²	#41×C₆₇	1	1	yes	characteristic
340	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	D₄	—	2	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	4	no	—
354	D₄	—	4	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
369	C₂×C₄	#135×C₆₇	1	2	yes	—
370	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
373	C₈	—	2	4	no	—
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	2	no	—
379	C₈	C₂×D₈×C₆₇	1	2	yes	—
380	C₈	C₂×D₈×C₆₇	1	2	yes	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	8	4	no	—
390	C₂²	C₄×D₈×C₆₇	1	2	yes	—
391	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	4	2	no	—
396	C₄	—	4	2	no	—
397	C₄	C₄×D₈×C₆₇	1	2	yes	—
398	C₄	#568×C₆₇	1	2	yes	—
399	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
400	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	8	2	no	—
403	C₂	—	8	2	no	—
404	C₂	C₆₇×#2930	1	1	yes	characteristic
405	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
406	C₂	C₆₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3100	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2931	C₂	semidirect
3. C₂×#436×C₆₇	C₂	semidirect
4. C₆₇×#1589	C₂	semidirect
5. C₆₇×#1581	C₂	semidirect
6. C₆₇×#1571	C₂	semidirect
7. C₆₇×#1486	C₂	semidirect
8. C₆₇×#1480	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
24. #436×C₆₇	C₂²	semidirect
26. C₈×D₄×C₆₇	C₂²	semidirect
34. #335×C₆₇	C₂²	semidirect
36. D₄⋊C₈×C₆₇	C₂²	semidirect
37. #322×C₆₇	C₂²	semidirect
38. C₈²×C₆₇	C₂²	semidirect
119. C₂×C₈×C₆₇	D₈	semidirect
197. #17928	C₆₇	direct (N × H)
205. #2931	C₂×C₆₇	semidirect
206. C₂×#436	C₂×C₆₇	semidirect
207. #1589	C₂×C₆₇	semidirect
208. #1581	C₂×C₆₇	semidirect
209. #1571	C₂×C₆₇	semidirect
210. #1486	C₂×C₆₇	semidirect
211. #1480	C₂×C₆₇	semidirect
212. #1450	C₂×C₆₇	semidirect
213. C₆₇	#17928	direct (N × H)
228. #436	C₂²×C₆₇	semidirect
230. C₈×D₄	C₂²×C₆₇	semidirect
238. #335	C₂²×C₆₇	semidirect
240. D₄⋊C₈	C₂²×C₆₇	semidirect
241. #322	C₂²×C₆₇	semidirect
242. C₈²	C₂²×C₆₇	semidirect
323. C₂×C₈	D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3100	2
C₆₇×#2931	2
C₂×#436×C₆₇	2
C₆₇×#1589	2
C₆₇×#1581	2
C₆₇×#1571	2
C₆₇×#1486	2
C₆₇×#1480	2
C₆₇×#1450	2
#17928	67

Subgroups — C67×#17928 #30697179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17928 #30697179