Subgroups — #30770883

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₂×#16863181	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863179	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558298	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524051	C₄	1	1	yes	characteristic
9	C₂×#14524049	C₂²	1	1	yes	characteristic
10	#16863864	C₄	1	1	yes	characteristic
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16863353	—	2	1	no	—
14	#16863352	C₄	1	1	yes	characteristic
15	#16863350	—	2	1	no	—
16	#16863348	—	2	1	no	—
17	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₆₇	—	2	1	no	—
20	#16863181	C₄	1	4	yes	complemented
21	#16863179	C₄	1	4	yes	complemented
22	#16863171	—	2	1	no	—
23	#16863171	—	2	1	no	—
24	#16863171	—	2	1	no	—
25	C₁₆×Dic₆₇	—	2	1	no	—
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	#405×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
30	#14524051	—	2	2	no	—
31	#14524051	C₈	1	4	yes	—
32	#14524049	—	2	1	no	—
33	#14524049	—	2	1	no	—
34	#14524049	C₂×C₄	1	2	yes	—
35	#14524049	C₂×C₄	1	2	yes	—
36	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
37	C₂×#11761611	—	2	1	no	—
38	C₂×#11761611	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₆₇	—	2	1	no	—
41	C₂×C₈×D₆₇	—	2	1	no	—
42	#14524008	C₈	1	2	yes	—
43	#14524005	C₈	1	2	yes	—
44	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
45	#14524000	—	2	1	no	—
46	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₆₇	—	2	1	no	—
48	#14523983	—	2	1	no	—
49	#14523983	C₂×C₄	1	2	yes	complemented
50	C₂×C₆₇⋊C₁₆	—	2	1	no	—
51	C₂×C₆₇⋊C₁₆	—	2	1	no	—
52	C₂×C₆₇⋊C₁₆	—	2	1	no	—
53	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	#14523954	—	2	1	no	—
55	#14523953	—	2	1	no	—
56	C₄×C₆₇⋊C₈	—	2	1	no	—
57	C₈×Dic₆₇	—	2	1	no	—
58	C₈×Dic₆₇	D₄	1	1	yes	characteristic
59	C₈×Dic₆₇	D₄	1	1	yes	characteristic
60	C₈×Dic₆₇	D₄	1	1	yes	characteristic
61	C₈×Dic₆₇	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₆₇	—	2	1	no	—
66	M₃₂×C₆₇	C₂×C₄	1	2	yes	complemented
67	C₂×C₁₆×C₆₇	—	2	1	no	—
68	C₂×C₁₆×C₆₇	—	2	1	no	—
69	C₂×C₁₆×C₆₇	—	2	1	no	—
70	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
71	#97×C₆₇	—	2	1	no	—
72	C₄×C₈×C₆₇	—	2	1	no	—
73	#11761628	—	2	1	no	—
74	#11761628	—	2	1	no	—
75	#11761628	C₄²	1	2	yes	—
76	#11761628	C₂×C₈	1	2	yes	—
77	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
78	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₆₇	—	2	1	no	—
80	C₂²×Dic₆₇	M₁₆	1	2	yes	—
81	C₂×C₄×D₆₇	—	2	1	no	—
82	C₂×C₄×D₆₇	—	2	2	no	—
83	#11761612	—	2	1	no	—
84	#11761612	C₄²	1	2	yes	—
85	#11761611	—	2	2	no	—
86	#11761611	—	2	2	no	—
87	#11761611	—	2	4	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
93	C₈×D₆₇	—	2	2	no	—
94	C₈×D₆₇	—	2	2	no	—
95	C₈×D₆₇	—	2	4	no	—
96	Dic₆₇⋊C₄	—	2	2	no	—
97	(C₂×D₆₇)⋊C₄	—	2	2	no	—
98	C₄×Dic₆₇	—	2	2	no	—
99	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
102	C₆₇⋊C₁₆	—	2	2	no	—
103	C₆₇⋊C₁₆	—	2	2	no	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	M₁₆×C₆₇	—	2	1	no	—
106	M₁₆×C₆₇	C₂×C₈	1	2	yes	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
115	C₂²⋊C₄×C₆₇	—	2	1	no	—
116	C₄²×C₆₇	—	2	1	no	—
117	C₁₆×C₆₇	—	2	2	no	—
118	C₁₆×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂²⋊D₆₇	—	2	2	no	—
129	C₂²⋊D₆₇	—	2	2	no	—
130	C₂×Dic₆₇	—	2	1	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
133	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
134	D₂₆₈	—	2	1	no	—
135	D₂₆₈	C₄×C₈	1	2	yes	—
136	C₄×D₆₇	—	2	2	no	—
137	C₄×D₆₇	—	2	2	no	—
138	C₄×D₆₇	—	4	4	no	—
139	Dic₁₃₄	—	2	1	no	—
140	Dic₁₃₄	C₄×C₈	1	2	yes	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
144	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
145	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
146	C₆₇⋊C₈	—	2	2	no	—
147	C₆₇⋊C₈	C₂²⋊C₈	1	2	yes	—
148	C₆₇⋊C₈	C₂²⋊C₈	1	2	yes	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	C₄×M₁₆	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₆₇	#431	1	1	yes	characteristic
153	C₂×D₆₇	—	2	2	no	—
154	C₂×D₆₇	—	4	2	no	—
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
158	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₆₇	#431	1	1	yes	characteristic
160	Dic₆₇	—	2	2	no	—
161	Dic₆₇	—	2	2	no	—
162	Dic₆₇	—	2	2	no	—
163	#12604	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₆₇	#2117	1	2	yes	—
168	D₆₇	—	4	2	no	—
169	C₄×#131	—	67	1	no	—
170	C₂×#350	—	67	1	no	—
171	C₂×#350	—	67	1	no	—
172	C₄×M₃₂	—	67	1	no	—
173	C₂×#346	—	67	1	no	—
174	#1379	—	67	1	no	—
175	#1379	D₆₇	1	1	yes	characteristic, complemented
176	C₆₇	#12604	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	67	1	no	—
178	C₂×#131	—	67	1	no	—
179	C₄×#41	—	67	1	no	—
180	C₂×M₃₂	—	67	1	no	—
181	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
182	C₂²×C₁₆	—	67	1	no	—
183	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
184	#405	—	67	1	no	—
185	#405	—	134	1	no	—
186	#405	—	134	1	no	—
187	#405	C₂×D₆₇	1	1	yes	characteristic
188	C₄×M₁₆	—	67	1	no	—
189	C₄×M₁₆	—	134	1	no	—
190	C₂×C₄×C₈	—	67	1	no	—
191	C₂×C₄×C₈	—	134	1	no	—
192	#350	—	67	4	no	—
193	#350	—	67	4	no	—
194	#346	—	134	1	no	—
195	#346	—	134	1	no	—
196	#346	—	134	1	no	—
197	C₄×C₁₆	—	134	1	no	—
198	C₂×#41	—	67	1	no	—
199	#131	—	67	4	no	—
200	#131	—	67	2	no	—
201	#131	—	67	2	no	—
202	#131	—	134	2	no	—
203	#131	—	134	1	no	—
204	#131	—	134	1	no	—
205	C₂×M₁₆	—	67	1	no	—
206	C₂×M₁₆	—	134	1	no	—
207	C₂×M₁₆	—	134	1	no	—
208	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₈	—	67	1	no	—
210	C₂²×C₈	—	134	1	no	—
211	C₂²×C₈	—	134	1	no	—
212	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
213	C₄×Q₈	—	67	2	no	—
214	C₄×D₄	—	67	2	no	—
215	C₄×D₄	—	134	2	no	—
216	#117	—	134	1	no	—
217	#117	C₄×D₆₇	1	1	yes	characteristic
218	C₂×C₄²	—	67	1	no	—
219	C₂×C₄²	—	134	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	67	2	no	—
222	M₃₂	—	134	1	no	—
223	M₃₂	C₄×D₆₇	1	2	yes	complemented
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	67	2	no	—
228	C₂×C₁₆	—	134	1	no	—
229	C₂×C₁₆	—	134	1	no	—
230	C₂×C₁₆	—	134	1	no	—
231	C₂×C₁₆	C₄×D₆₇	1	2	yes	complemented
232	#97	—	2	1	no	—
233	#97	—	134	1	no	—
234	#97	—	134	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	67	1	no	—
237	C₄×C₈	—	67	1	no	—
238	C₄×C₈	—	67	1	no	—
239	C₄×C₈	—	67	1	no	—
240	C₄×C₈	—	134	1	no	—
241	C₄×C₈	—	134	1	no	—
242	#41	—	67	2	no	—
243	#41	—	67	2	no	—
244	#41	—	134	1	no	—
245	#41	—	134	1	no	—
246	C₂×Q₈	—	67	1	no	—
247	C₂×D₄	—	67	1	no	—
248	C₂×D₄	—	134	1	no	—
249	C₂²×C₄	—	67	2	no	—
250	C₂²×C₄	—	134	2	no	—
251	C₂²×C₄	—	134	1	no	—
252	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	67	2	no	—
255	M₁₆	—	134	4	no	—
256	M₁₆	—	134	1	no	—
257	M₁₆	—	134	2	no	—
258	M₁₆	—	134	2	no	—
259	M₁₆	C₈×D₆₇	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	67	1	no	—
262	C₂×C₈	—	67	1	no	—
263	C₂×C₈	—	67	2	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	134	4	no	—
266	C₂×C₈	—	134	1	no	—
267	C₂×C₈	—	134	2	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
273	C₂×C₈	#11761627	1	1	yes	characteristic
274	C₂×C₈	#11761629	1	1	yes	characteristic
275	C₄⋊C₄	—	134	2	no	—
276	C₄⋊C₄	C₈×D₆₇	1	2	yes	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	134	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	67	1	no	—
281	C₄²	—	67	1	no	—
282	C₄²	—	67	2	no	—
283	C₄²	—	134	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	134	2	no	—
287	C₁₆	—	134	2	no	—
288	C₂³	—	134	1	no	—
289	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
290	Q₈	—	67	2	no	—
291	Q₈	—	134	1	no	—
292	D₄	—	67	2	no	—
293	D₄	—	134	2	no	—
294	D₄	—	134	2	no	—
295	D₄	—	134	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	268	4	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	67	2	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	134	1	no	—
302	C₂×C₄	—	134	2	no	—
303	C₂×C₄	—	134	2	no	—
304	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
306	C₂×C₄	#14524001	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	67	2	no	—
312	C₈	—	67	2	no	—
313	C₈	—	134	2	no	—
314	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
315	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
316	C₈	#14524005	1	1	yes	characteristic
317	C₈	#14524005	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	268	2	no	—
320	C₂²	—	134	2	no	—
321	C₂²	#16863348	1	1	yes	characteristic
322	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
323	C₂²	#16863448	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	134	2	no	—
326	C₄	—	134	2	no	—
327	C₄	—	134	2	no	—
328	C₄	#16863243	1	1	yes	characteristic
329	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
330	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
331	C₄	#16863448	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	268	2	no	—
335	C₂	#22559966	1	1	yes	characteristic
336	C₂	#22561835	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1379	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1379	D₆₇
hypocenter	C₆₇	#12604
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₆₇	C₂	semidirect
5. C₂×#16863171	C₂	semidirect
6. #22558298	C₂	semidirect
7. C₆₇×#1379	C₂	semidirect
20. #16863181	C₄	semidirect
21. #16863179	C₄	semidirect
49. #14523983	C₂×C₄	semidirect
53. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
66. M₃₂×C₆₇	C₂×C₄	semidirect
70. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
175. #1379	D₆₇	semidirect
176. C₆₇	#12604	semidirect
223. M₃₂	C₄×D₆₇	semidirect
231. C₂×C₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568631	2
C₂×#16863181	2
C₂×#16863179	2
M₃₂×Dic₆₇	2
C₂×#16863171	2
#22558298	2
C₆₇×#1379	2
#12604	67