Subgroups — #31008944

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568631	C₂	1	1	yes	maximal, characteristic
2	#22562017	C₂	1	2	yes	maximal, complemented
3	#22562014	C₂	1	2	yes	maximal, complemented
4	#22562010	C₂	1	2	yes	maximal, complemented
5	#22562006	C₂	1	2	yes	maximal, complemented
6	#22561964	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561824	C₂	1	1	yes	maximal, characteristic
9	#22561822	C₂	1	1	yes	maximal, characteristic
10	#22561806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₄	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	#16863864	C₄	1	1	yes	characteristic
15	C₂×#14523983	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16863353	C₄	1	2	yes	—
18	#16863352	C₄	1	1	yes	characteristic
19	#16863350	C₂²	1	2	yes	—
20	#16863348	C₄	1	2	yes	—
21	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₆₇	C₂²	1	2	yes	—
24	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
26	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
27	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
31	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
34	#16863172	C₂²	1	2	yes	—
35	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
36	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	#405×C₆₇	C₂²	1	1	yes	characteristic
39	#346×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
42	#14524051	—	2	4	no	—
43	#14524049	—	2	2	no	—
44	#14524049	—	2	2	no	—
45	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761611	C₂×C₄	1	2	yes	—
47	C₂×#11761611	C₂×C₄	1	2	yes	—
48	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₆₇	C₂³	1	2	yes	—
51	#14524008	C₈	1	1	yes	characteristic
52	#14524008	C₈	1	1	yes	characteristic
53	#14524005	C₈	1	1	yes	characteristic
54	#14524005	C₈	1	1	yes	characteristic
55	(C₂²⋊D₆₇)⋊C₄	C₈	1	2	yes	—
56	(C₂²⋊D₆₇)⋊C₄	C₈	1	2	yes	—
57	#14524000	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₆₇	C₂×C₄	1	2	yes	—
60	#14523983	—	2	2	no	—
61	C₂×C₆₇⋊C₁₆	—	2	2	no	—
62	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
65	#14523954	C₂×C₄	1	2	yes	—
66	#14523953	C₂×C₄	1	2	yes	—
67	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
68	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
69	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
72	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
76	M₃₂×C₆₇	—	2	2	no	—
77	C₂×C₁₆×C₆₇	—	2	2	no	—
78	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
81	#97×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	C₂³	1	2	yes	—
83	#11761628	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
86	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₆₇	C₂×C₈	1	2	yes	—
88	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
89	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
91	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
93	#11761612	—	2	2	no	—
94	#11761611	—	2	4	no	—
95	#11761611	—	2	4	no	—
96	C₂×C₆₇⋊C₈	—	2	2	no	—
97	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₈×D₆₇	—	2	4	no	—
101	C₈×D₆₇	—	2	2	no	—
102	C₈×D₆₇	—	2	2	no	—
103	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
104	Dic₆₇⋊C₄	C₂×C₈	1	2	yes	—
105	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
106	(C₂×D₆₇)⋊C₄	C₂×C₈	1	2	yes	—
107	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
108	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
109	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₆₇⋊C₁₆	—	2	1	no	—
114	C₆₇⋊C₁₆	—	2	1	no	—
115	C₆₇⋊C₁₆	—	2	2	no	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₆₇	M₁₆	1	4	yes	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₆₇	#41	1	2	yes	—
121	C₂×C₈×C₆₇	#41	1	2	yes	—
122	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
126	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
127	C₁₆×C₆₇	—	2	1	no	—
128	C₁₆×C₆₇	—	2	1	no	—
129	C₁₆×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₆₇	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₆₇	#117	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₆₇	—	2	4	no	—
139	C₂×Dic₆₇	—	2	2	no	—
140	C₂×Dic₆₇	—	4	1	no	—
141	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
143	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
146	D₂₆₈	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	2	2	no	—
149	C₄×D₆₇	—	2	4	no	—
150	C₄×D₆₇	—	4	2	no	—
151	Dic₁₃₄	—	2	2	no	—
152	C₈×C₆₇	#117	1	2	yes	—
153	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
154	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	2	1	no	—
158	C₆₇⋊C₈	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	#431	1	2	yes	—
162	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
163	C₂×D₆₇	—	2	2	no	—
164	C₂×D₆₇	—	2	2	no	—
165	C₂×D₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	#431	1	2	yes	—
169	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
170	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
171	Dic₆₇	—	2	1	no	—
172	Dic₆₇	—	2	1	no	—
173	Dic₆₇	—	2	1	no	—
174	Dic₆₇	—	2	2	no	—
175	Dic₆₇	#504	1	2	yes	—
176	#33423	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	2	no	—
178	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
180	C₂×C₆₇	#3407	1	1	yes	characteristic
181	D₆₇	—	4	2	no	—
182	C₄×#131	—	67	1	no	—
183	#2164	—	67	2	no	—
184	#2164	—	67	2	no	—
185	#2163	—	67	2	no	—
186	#2162	—	67	2	no	—
187	#2150	—	67	1	no	—
188	C₂×C₄⋊C₁₆	—	67	1	no	—
189	#2113	—	67	1	no	—
190	#2113	—	67	1	no	—
191	#2108	—	67	1	no	—
192	#2108	D₆₇	1	1	yes	characteristic, complemented
193	C₆₇	#33423	1	1	yes	characteristic, complemented, Sylow
194	C₂×#131	—	67	1	no	—
195	C₂×#131	—	67	1	no	—
196	C₄×#41	—	67	1	no	—
197	C₂×M₃₂	—	67	1	no	—
198	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
199	C₂²×C₁₆	—	67	1	no	—
200	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
201	#405	—	67	1	no	—
202	#405	—	67	2	no	—
203	#405	—	67	2	no	—
204	#405	C₂×D₆₇	1	1	yes	characteristic
205	C₄×M₁₆	—	67	1	no	—
206	C₄×M₁₆	—	67	2	no	—
207	C₂×C₄×C₈	—	67	1	no	—
208	C₂×C₄×C₈	—	67	2	no	—
209	C₄⋊C₁₆	—	67	1	no	—
210	C₄⋊C₁₆	—	67	1	no	—
211	C₄⋊C₁₆	—	67	1	no	—
212	C₄⋊C₁₆	—	67	1	no	—
213	C₄⋊C₁₆	—	67	2	no	—
214	C₄⋊C₁₆	—	67	2	no	—
215	C₂²⋊C₁₆	—	67	2	no	—
216	C₂²⋊C₁₆	—	67	2	no	—
217	C₂²⋊C₁₆	—	67	2	no	—
218	C₂²⋊C₁₆	—	67	2	no	—
219	#346	—	67	2	no	—
220	#346	C₂×D₆₇	1	2	yes	—
221	C₄×C₁₆	—	67	2	no	—
222	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
223	C₂×#41	—	67	1	no	—
224	#131	—	134	4	no	—
225	#131	—	134	2	no	—
226	#131	—	134	2	no	—
227	C₂×M₁₆	—	67	1	no	—
228	C₂×M₁₆	—	67	2	no	—
229	C₂×M₁₆	—	67	2	no	—
230	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
231	C₂²×C₈	—	67	1	no	—
232	C₂²×C₈	—	67	2	no	—
233	C₂²×C₈	—	67	2	no	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×Q₈	—	67	1	no	—
236	C₄×Q₈	—	67	1	no	—
237	C₄×D₄	—	67	1	no	—
238	C₄×D₄	—	67	1	no	—
239	C₄×D₄	—	67	2	no	—
240	C₄×D₄	—	67	2	no	—
241	#117	—	67	2	no	—
242	#117	C₄×D₆₇	1	1	yes	characteristic
243	C₂×C₄²	—	67	1	no	—
244	C₂×C₄²	—	67	2	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	134	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	67	1	no	—
249	C₂×C₁₆	—	67	2	no	—
250	C₂×C₁₆	—	67	1	no	—
251	C₂×C₁₆	—	134	2	no	—
252	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
254	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
255	#97	—	67	2	no	—
256	#97	—	67	2	no	—
257	#97	C₄×D₆₇	1	2	yes	—
258	C₄×C₈	—	67	1	no	—
259	C₄×C₈	—	67	2	no	—
260	C₄×C₈	—	67	1	no	—
261	C₄×C₈	—	67	1	no	—
262	C₄×C₈	—	67	1	no	—
263	C₄×C₈	—	67	2	no	—
264	C₄×C₈	C₂²×D₆₇	1	2	yes	—
265	#41	—	134	2	no	—
266	#41	—	134	2	no	—
267	C₂×Q₈	—	67	1	no	—
268	C₂×D₄	—	67	2	no	—
269	C₂×D₄	—	67	1	no	—
270	C₂²×C₄	—	67	1	no	—
271	C₂²×C₄	—	67	2	no	—
272	C₂²×C₄	—	67	1	no	—
273	C₂²×C₄	—	67	2	no	—
274	C₂²×C₄	—	67	2	no	—
275	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
276	M₁₆	—	134	2	no	—
277	M₁₆	—	134	4	no	—
278	M₁₆	—	134	4	no	—
279	M₁₆	#11761611	1	4	yes	—
280	C₂×C₈	—	67	1	no	—
281	C₂×C₈	—	67	2	no	—
282	C₂×C₈	—	67	1	no	—
283	C₂×C₈	—	134	2	no	—
284	C₂×C₈	—	134	4	no	—
285	C₂×C₈	—	134	2	no	—
286	C₂×C₈	—	134	2	no	—
287	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
289	C₂×C₈	#11761628	1	2	yes	—
290	C₂×C₈	#11761628	1	2	yes	—
291	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
292	C₄⋊C₄	—	67	2	no	—
293	C₄⋊C₄	—	67	2	no	—
294	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
295	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
296	C₂²⋊C₄	—	67	2	no	—
297	C₂²⋊C₄	—	67	2	no	—
298	C₂²⋊C₄	C₈×D₆₇	1	2	yes	—
299	C₄²	—	67	1	no	—
300	C₄²	—	67	2	no	—
301	C₄²	—	67	1	no	—
302	C₄²	—	67	2	no	—
303	C₄²	—	67	1	no	—
304	C₄²	—	67	1	no	—
305	C₄²	C₂×C₄×D₆₇	1	2	yes	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	134	1	no	—
310	C₁₆	—	134	1	no	—
311	C₁₆	—	134	2	no	—
312	C₂³	—	67	2	no	—
313	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
314	Q₈	—	134	2	no	—
315	D₄	—	134	4	no	—
316	D₄	—	134	2	no	—
317	C₂×C₄	—	268	2	no	—
318	C₂×C₄	—	268	1	no	—
319	C₂×C₄	—	67	1	no	—
320	C₂×C₄	—	67	1	no	—
321	C₂×C₄	—	67	1	no	—
322	C₂×C₄	—	67	1	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	134	4	no	—
325	C₂×C₄	—	134	2	no	—
326	C₂×C₄	—	134	2	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	#14524000	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×#11761611	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
334	C₈	—	134	2	no	—
335	C₈	—	134	1	no	—
336	C₈	—	134	1	no	—
337	C₈	#14524000	1	2	yes	—
338	C₈	C₂×#11761611	1	4	yes	—
339	C₈	C₂×#11761628	1	1	yes	characteristic
340	C₈	C₂×#11761628	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	134	2	no	—
344	C₂²	—	134	2	no	—
345	C₂²	—	134	2	no	—
346	C₂²	#16863447	1	2	yes	—
347	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	67	2	no	—
351	C₄	—	134	2	no	—
352	C₄	—	134	1	no	—
353	C₄	—	134	1	no	—
354	C₄	—	134	1	no	—
355	C₄	#16863447	1	2	yes	—
356	C₄	C₂×#14524000	1	1	yes	characteristic
357	C₄	C₂²×#11761611	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	268	2	no	—
360	C₂	C₂×#16863447	1	1	yes	characteristic
361	C₂	#504×D₆₇	1	1	yes	characteristic
362	C₂	#22573310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2108	C₂
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#33423
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #22562017	C₂	semidirect
3. #22562014	C₂	semidirect
4. #22562010	C₂	semidirect
5. #22562006	C₂	semidirect
6. #22561964	C₂	semidirect
7. C₂×Dic₆₇⋊C₁₆	C₂	semidirect
10. #22561806	C₂	semidirect
11. C₆₇×#2108	C₂	semidirect
192. #2108	D₆₇	semidirect
193. C₆₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#22568631	2
#22562017	2
#22562014	2
#22562010	2
#22562006	2
#22561964	2
C₂×Dic₆₇⋊C₁₆	2
#22561824	2
#22561822	2
#22561806	2
C₆₇×#2108	2
#33423	67