Subgroups — #30749030

Order 17152.

386 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558648	C₂	1	2	yes	maximal
4	#22558100	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2965	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	—	2	1	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	#16863161	—	2	1	no	—
10	#16863149	—	2	1	no	—
11	#16863149	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	2	no	—
13	D₄⋊(C₆₇⋊C₈)	—	2	2	no	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	C₄	1	1	yes	characteristic
16	#446×C₆₇	C₄	1	1	yes	characteristic
17	#436×C₆₇	C₄	1	1	yes	characteristic
18	#436×C₆₇	C₄	1	1	yes	characteristic
19	#435×C₆₇	—	2	1	no	—
20	#435×C₆₇	—	2	1	no	—
21	#431×C₆₇	—	2	1	no	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	#14523979	—	2	4	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523954	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	1	no	—
34	C₂×M₁₆×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×D₄×C₆₇	—	2	1	no	—
40	C₄×D₄×C₆₇	—	2	1	no	—
41	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
43	#117×C₆₇	—	2	1	no	—
44	#117×C₆₇	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	#97×C₆₇	—	2	1	no	—
55	#97×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	#11761612	—	2	4	no	—
62	#11761612	—	4	2	no	—
63	#11761612	—	8	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	#41×C₆₇	—	4	2	no	—
67	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₆₇	—	2	1	no	—
69	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
76	M₁₆×C₆₇	—	4	2	no	—
77	M₁₆×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	4	1	no	—
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₇	#41	1	1	yes	characteristic
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
102	Q₈×C₆₇	—	2	2	no	—
103	D₄×C₆₇	—	2	2	no	—
104	D₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	—	4	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	4	1	no	—
128	C₈×C₆₇	—	4	1	no	—
129	C₈×C₆₇	#104	1	2	yes	—
130	C₆₇⋊C₈	—	4	4	no	—
131	C₆₇⋊C₈	—	8	2	no	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#380	1	1	yes	characteristic
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	4	1	no	—
143	C₄×C₆₇	—	4	1	no	—
144	C₄×C₆₇	—	4	1	no	—
145	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
146	C₄×C₆₇	#421	1	1	yes	characteristic
147	C₄×C₆₇	#430	1	2	yes	—
148	C₄×C₆₇	#472	1	1	yes	characteristic
149	C₄×C₆₇	#473	1	1	yes	characteristic
150	#8673	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	4	1	no	—
152	C₂×C₆₇	—	8	1	no	—
153	C₂×C₆₇	#1806	1	1	yes	characteristic
154	C₂×C₆₇	#1811	1	1	yes	characteristic
155	C₂×C₆₇	#1816	1	1	yes	characteristic
156	#2965	D₆₇	1	1	yes	characteristic
157	C₄×#108	—	67	1	no	—
158	#1567	—	67	1	no	—
159	#1489	—	67	2	no	—
160	#1299	—	67	2	no	—
161	C₆₇	#8673	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
163	#446	Dic₆₇	1	1	yes	characteristic
164	#446	Dic₆₇	1	1	yes	characteristic
165	#436	Dic₆₇	1	1	yes	characteristic
166	#436	Dic₆₇	1	1	yes	characteristic
167	#435	—	2	1	no	—
168	#435	—	2	1	no	—
169	#431	—	2	1	no	—
170	#431	C₂×D₆₇	1	1	yes	characteristic
171	C₂×#108	—	134	1	no	—
172	#423	—	67	2	no	—
173	C₄×M₁₆	—	2	1	no	—
174	C₄×M₁₆	—	67	2	no	—
175	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
176	#341	—	134	1	no	—
177	#335	—	134	1	no	—
178	#335	—	134	1	no	—
179	Q₈⋊C₈	—	134	2	no	—
180	D₄⋊C₈	—	134	2	no	—
181	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
189	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	D₂₆₈	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	Dic₁₃₄	1	1	yes	characteristic
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
198	#108	—	134	4	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	134	2	no	—
202	C₄⋊C₈	—	134	2	no	—
203	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
204	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	#97	—	134	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	134	2	no	—
213	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
217	#41	—	4	2	no	—
218	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
226	C₂²×C₄	#11761628	1	1	yes	characteristic
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	268	2	no	—
230	M₁₆	—	536	2	no	—
231	M₁₆	—	134	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	268	2	no	—
237	C₂×C₈	—	268	2	no	—
238	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
240	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
245	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
254	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
255	C₄²	#11761628	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	#14524022	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	#14524002	1	1	yes	characteristic
277	C₂×C₄	#14524017	1	1	yes	characteristic
278	C₂×C₄	#14524017	1	1	yes	characteristic
279	C₂×C₄	#14524022	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	268	4	no	—
286	C₈	—	536	2	no	—
287	C₈	#14523970	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	#16863256	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#14523970	1	1	yes	characteristic
302	C₄	#16863413	1	1	yes	characteristic
303	C₄	#16863445	1	2	yes	—
304	C₄	#16863666	1	1	yes	characteristic
305	C₄	#16863671	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	#22560229	1	1	yes	characteristic
309	C₂	#22560251	1	1	yes	characteristic
310	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2965	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#41×C₆₇	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2965	D₆₇
hypocenter	C₆₇	#8673
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
1. C₄×#14523979	C₂	semidirect
2. #22558968	C₂	semidirect
4. #22558100	C₂	semidirect
161. C₆₇	#8673	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523979	2
#22558968	2
#22558648	2
#22558100	2
C₆₇×#2965	2
#8673	67