Subgroups — C₆₇×#18085 #30697336

Order 17152.

612 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2979	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1590	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1485	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1484	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	C₄	1	1	yes	characteristic
15	#492×C₆₇	C₄	1	1	yes	characteristic
16	#487×C₆₇	C₄	1	1	yes	characteristic
17	#486×C₆₇	C₄	1	1	yes	characteristic
18	#465×C₆₇	C₄	1	2	yes	—
19	#465×C₆₇	C₄	1	2	yes	—
20	#446×C₆₇	—	2	1	no	—
21	#446×C₆₇	C₂²	1	2	yes	—
22	C₈×Q₈×C₆₇	—	2	1	no	—
23	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
26	#436×C₆₇	—	2	1	no	—
27	#436×C₆₇	C₂²	1	2	yes	—
28	C₈×D₄×C₆₇	—	2	1	no	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	—
30	#433×C₆₇	—	2	2	no	—
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#431×C₆₇	C₂²	1	1	yes	characteristic
34	#408×C₆₇	—	2	2	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#334×C₆₇	C₂²	1	1	yes	characteristic
40	#334×C₆₇	C₂²	1	1	yes	characteristic
41	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
42	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	#322×C₆₇	C₂²	1	2	yes	—
46	C₈²×C₆₇	C₂²	1	2	yes	—
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×M₁₆×C₆₇	—	2	1	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
59	#125×C₆₇	—	2	1	no	—
60	#124×C₆₇	—	2	1	no	—
61	C₂²⋊Q₈×C₆₇	—	2	1	no	—
62	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
63	#121×C₆₇	—	2	1	no	—
64	#121×C₆₇	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₆₇	—	2	1	no	—
66	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
67	C₄×D₄×C₆₇	—	2	1	no	—
68	C₄×D₄×C₆₇	C₂³	1	2	yes	—
69	#117×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	C₂×C₄	1	2	yes	—
72	#106×C₆₇	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	C₂×C₄	1	2	yes	—
82	#103×C₆₇	C₂×C₄	1	2	yes	—
83	#102×C₆₇	C₂×C₄	1	2	yes	—
84	#102×C₆₇	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈×C₆₇	—	2	1	no	—
86	C₂²⋊C₈×C₆₇	—	2	1	no	—
87	C₂²⋊C₈×C₆₇	—	2	2	no	—
88	C₂²⋊C₈×C₆₇	—	2	2	no	—
89	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
90	#97×C₆₇	—	2	2	no	—
91	#97×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂³	1	2	yes	—
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	2	yes	—
101	#41×C₆₇	—	8	1	no	—
102	#41×C₆₇	—	8	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
114	SD₁₆×C₆₇	—	2	4	no	—
115	SD₁₆×C₆₇	—	4	2	no	—
116	M₁₆×C₆₇	—	2	4	no	—
117	M₁₆×C₆₇	—	8	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	4	1	no	—
125	C₂×C₈×C₆₇	—	8	2	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂²⋊C₄×C₆₇	—	2	2	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₆₇	#41	1	1	yes	characteristic
147	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₆₇	—	2	2	no	—
151	Q₈×C₆₇	—	2	2	no	—
152	Q₈×C₆₇	—	4	2	no	—
153	Q₈×C₆₇	—	8	1	no	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	8	1	no	—
158	D₄×C₆₇	—	8	2	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
175	C₈×C₆₇	—	2	4	no	—
176	C₈×C₆₇	—	2	1	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	—	2	1	no	—
179	C₈×C₆₇	—	2	2	no	—
180	C₈×C₆₇	—	2	2	no	—
181	C₈×C₆₇	—	4	1	no	—
182	C₈×C₆₇	—	4	1	no	—
183	C₈×C₆₇	—	4	1	no	—
184	C₈×C₆₇	—	4	1	no	—
185	C₈×C₆₇	#131	1	2	yes	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	2	1	no	—
188	C₂²×C₆₇	—	2	1	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	2	1	no	—
195	C₄×C₆₇	—	2	1	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	#443	1	2	yes	—
200	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
201	C₄×C₆₇	#568	1	1	yes	characteristic
202	C₄×C₆₇	#577	1	1	yes	characteristic
203	C₄×C₆₇	#577	1	1	yes	characteristic
204	#18085	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂×C₆₇	—	4	1	no	—
206	C₂×C₆₇	—	8	2	no	—
207	C₂×C₆₇	#2935	1	1	yes	characteristic
208	C₂×C₆₇	#2949	1	1	yes	characteristic
209	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
210	#3167	C₂×C₆₇	1	1	yes	characteristic
211	#2986	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#2979	C₂×C₆₇	1	1	yes	characteristic
213	#1590	C₂×C₆₇	1	2	yes	complemented
214	#1588	C₂×C₆₇	1	2	yes	complemented
215	#1585	C₂×C₆₇	1	2	yes	complemented
216	C₈×SD₁₆	C₂×C₆₇	1	2	yes	complemented
217	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1485	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1484	C₂×C₆₇	1	1	yes	characteristic
220	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₆₇	#18085	1	1	yes	characteristic, direct product factor, Sylow
222	#533	C₂²×C₆₇	1	1	yes	characteristic
223	#532	C₂²×C₆₇	1	1	yes	characteristic
224	#498	C₄×C₆₇	1	1	yes	characteristic
225	#492	C₄×C₆₇	1	1	yes	characteristic
226	#487	C₄×C₆₇	1	1	yes	characteristic
227	#486	C₄×C₆₇	1	1	yes	characteristic
228	#465	C₄×C₆₇	1	2	yes	—
229	#465	C₄×C₆₇	1	2	yes	—
230	#446	—	2	1	no	—
231	#446	C₂²×C₆₇	1	2	yes	—
232	C₈×Q₈	—	2	1	no	—
233	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
234	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
235	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
236	#436	—	2	1	no	—
237	#436	C₂²×C₆₇	1	2	yes	—
238	C₈×D₄	—	2	1	no	—
239	C₈×D₄	C₂²×C₆₇	1	2	yes	—
240	#433	—	2	2	no	—
241	#431	C₂²×C₆₇	1	1	yes	characteristic
242	#431	C₂²×C₆₇	1	1	yes	characteristic
243	#431	C₂²×C₆₇	1	1	yes	characteristic
244	#408	—	2	2	no	—
245	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
247	#335	C₂²×C₆₇	1	1	yes	characteristic
248	#335	C₂²×C₆₇	1	1	yes	characteristic
249	#334	C₂²×C₆₇	1	1	yes	characteristic
250	#334	C₂²×C₆₇	1	1	yes	characteristic
251	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
252	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
253	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
254	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
255	#322	C₂²×C₆₇	1	2	yes	—
256	C₈²	C₂²×C₆₇	1	2	yes	—
257	C₂×#41	—	2	1	no	—
258	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
265	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
266	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
268	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
269	#125	—	2	1	no	—
270	#124	—	2	1	no	—
271	C₂²⋊Q₈	—	2	1	no	—
272	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
273	#121	—	2	1	no	—
274	#121	C₂×C₄×C₆₇	1	2	yes	—
275	C₄×Q₈	—	2	1	no	—
276	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	C₂³×C₆₇	1	2	yes	—
279	#117	—	2	1	no	—
280	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
281	#106	C₂×C₄×C₆₇	1	2	yes	—
282	#106	C₂×C₄×C₆₇	1	2	yes	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
286	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
291	#103	C₂×C₄×C₆₇	1	2	yes	—
292	#103	C₂×C₄×C₆₇	1	2	yes	—
293	#102	C₂×C₄×C₆₇	1	2	yes	—
294	#102	C₂×C₄×C₆₇	1	2	yes	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
300	#97	—	2	2	no	—
301	#97	—	2	1	no	—
302	C₄×C₈	—	2	2	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
306	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₆₇	1	2	yes	—
308	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₆₇	1	2	yes	—
311	#41	—	8	1	no	—
312	#41	—	8	1	no	—
313	C₂×Q₈	—	2	1	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
324	SD₁₆	—	2	4	no	—
325	SD₁₆	—	4	2	no	—
326	M₁₆	—	2	4	no	—
327	M₁₆	—	8	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	8	2	no	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
337	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
356	C₄²	#41×C₆₇	1	1	yes	characteristic
357	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
358	C₂³	—	2	2	no	—
359	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	2	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	8	1	no	—
364	D₄	—	2	2	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	1	no	—
368	D₄	—	8	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
380	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	2	1	no	—
387	C₈	—	2	1	no	—
388	C₈	—	2	1	no	—
389	C₈	—	2	2	no	—
390	C₈	—	2	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₈	#131×C₆₇	1	2	yes	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	#443×C₆₇	1	2	yes	—
410	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
411	C₄	#568×C₆₇	1	1	yes	characteristic
412	C₄	#577×C₆₇	1	1	yes	characteristic
413	C₄	#577×C₆₇	1	1	yes	characteristic
414	C₂	—	4	1	no	—
415	C₂	—	8	2	no	—
416	C₂	C₆₇×#2935	1	1	yes	characteristic
417	C₂	C₆₇×#2949	1	1	yes	characteristic
418	C₂	C₂×#443×C₆₇	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	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2986	C₂	semidirect
4. C₆₇×#1590	C₂	semidirect
5. C₆₇×#1588	C₂	semidirect
6. C₆₇×#1585	C₂	semidirect
7. C₈×SD₁₆×C₆₇	C₂	semidirect
8. C₆₇×#1564	C₂	semidirect
9. C₆₇×#1485	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
204. #18085	C₆₇	direct (N × H)
211. #2986	C₂×C₆₇	semidirect
213. #1590	C₂×C₆₇	semidirect
214. #1588	C₂×C₆₇	semidirect
215. #1585	C₂×C₆₇	semidirect
216. C₈×SD₁₆	C₂×C₆₇	semidirect
217. #1564	C₂×C₆₇	semidirect
218. #1485	C₂×C₆₇	semidirect
220. C₈×M₁₆	C₂×C₆₇	semidirect
221. C₆₇	#18085	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3167	2
C₆₇×#2986	2
C₆₇×#2979	2
C₆₇×#1590	2
C₆₇×#1588	2
C₆₇×#1585	2
C₈×SD₁₆×C₆₇	2
C₆₇×#1564	2
C₆₇×#1485	2
C₆₇×#1484	2
C₈×M₁₆×C₆₇	2
#18085	67

Subgroups — C67×#18085 #30697336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18085 #30697336