Subgroups — #30826491

Order 17152.

640 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569285	C₂	1	2	yes	maximal, complemented
2	#22559123	C₂	1	2	yes	maximal, complemented
3	#22559094	C₂	1	2	yes	maximal, complemented
4	#22559033	C₂	1	2	yes	maximal, complemented
5	D₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558678	C₂	1	2	yes	maximal, complemented
8	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
10	#16863474	C₂²	1	4	yes	—
11	#16863468	—	2	4	no	—
12	D₄×C₆₇⋊C₈	C₂²	1	4	yes	—
13	#16863453	C₂²	1	2	yes	—
14	#16863449	C₂²	1	1	yes	characteristic
15	#16863449	C₂²	1	1	yes	characteristic
16	#16863363	—	2	4	no	—
17	#16863149	C₂²	1	2	yes	—
18	#16863147	C₂²	1	2	yes	—
19	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863117	C₂²	1	1	yes	characteristic
22	#16863117	C₂²	1	1	yes	characteristic
23	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	#532×C₆₇	C₂²	1	2	yes	—
26	#500×C₆₇	C₄	1	1	yes	characteristic
27	#493×C₆₇	C₄	1	1	yes	characteristic
28	#486×C₆₇	C₄	1	2	yes	—
29	#466×C₆₇	C₄	1	2	yes	—
30	#466×C₆₇	C₄	1	2	yes	—
31	C₄×D₈×C₆₇	C₂²	1	2	yes	—
32	C₄×D₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	4	no	—
35	C₂²×C₆₇⋊C₈	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
37	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
44	#14523954	—	2	2	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
55	#127×C₆₇	C₂×C₄	1	2	yes	—
56	#124×C₆₇	—	2	2	no	—
57	#121×C₆₇	—	2	2	no	—
58	#121×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	#107×C₆₇	C₂×C₄	1	2	yes	—
63	#107×C₆₇	C₂×C₄	1	2	yes	—
64	#102×C₆₇	C₂×C₄	1	4	yes	—
65	#102×C₆₇	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	#11761612	—	8	4	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	8	4	no	—
75	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
76	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
77	#41×C₆₇	—	8	4	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	4	no	—
80	C₂×D₄×C₆₇	—	2	2	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	D₈×C₆₇	—	2	4	no	—
88	D₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₂²⋊C₄×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	#41	1	1	yes	characteristic
102	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂³×C₆₇	—	2	4	no	—
104	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₆₇	—	8	2	no	—
106	D₄×C₆₇	—	2	4	no	—
107	D₄×C₆₇	—	2	4	no	—
108	D₄×C₆₇	—	8	2	no	—
109	D₄×C₆₇	—	8	2	no	—
110	D₄×C₆₇	—	8	4	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	#131	1	2	yes	—
127	C₈×C₆₇	#131	1	2	yes	—
128	C₆₇⋊C₈	—	2	2	no	—
129	C₆₇⋊C₈	—	2	2	no	—
130	C₆₇⋊C₈	—	4	1	no	—
131	C₆₇⋊C₈	—	4	1	no	—
132	C₆₇⋊C₈	—	4	4	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	4	4	no	—
137	C₂²×C₆₇	—	4	4	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	4	4	no	—
145	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
147	C₄×C₆₇	#576	1	2	yes	—
148	#17970	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	4	1	no	—
150	C₂×C₆₇	—	8	4	no	—
151	C₂×C₆₇	#2931	1	1	yes	characteristic
152	C₂×C₆₇	#2948	1	1	yes	characteristic
153	C₂×C₆₇	#2957	1	1	yes	characteristic
154	#3169	D₆₇	1	1	yes	characteristic
155	#2981	—	67	2	no	—
156	#1594	—	67	2	no	—
157	#1589	—	67	2	no	—
158	#1580	—	67	2	no	—
159	C₈×D₈	—	67	2	no	—
160	#1568	—	67	1	no	—
161	#1496	—	67	2	no	—
162	#1450	—	67	1	no	—
163	C₆₇	#17970	1	1	yes	characteristic, complemented, Sylow
164	#532	C₂×D₆₇	1	2	yes	—
165	#500	Dic₆₇	1	1	yes	characteristic
166	#493	Dic₆₇	1	1	yes	characteristic
167	#486	Dic₆₇	1	2	yes	—
168	#466	Dic₆₇	1	2	yes	—
169	#466	Dic₆₇	1	2	yes	—
170	C₄×D₈	C₂×D₆₇	1	2	yes	—
171	C₄×D₈	C₂×D₆₇	1	2	yes	—
172	#436	—	67	4	no	—
173	#435	—	134	4	no	—
174	C₈×D₄	—	67	4	no	—
175	#432	—	67	2	no	—
176	#431	—	67	1	no	—
177	#431	—	67	1	no	—
178	#408	—	134	4	no	—
179	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
180	#335	—	67	2	no	—
181	#334	—	67	2	no	—
182	D₄⋊C₈	—	67	4	no	—
183	D₄⋊C₈	—	67	4	no	—
184	#322	—	67	1	no	—
185	#322	—	67	1	no	—
186	C₈²	—	67	1	no	—
187	C₈²	—	67	1	no	—
188	C₂×#41	—	2	2	no	—
189	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
190	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
191	C₂×M₁₆	—	134	4	no	—
192	C₂²×C₈	—	134	4	no	—
193	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
196	#127	C₂×Dic₆₇	1	2	yes	—
197	#124	—	2	2	no	—
198	#121	—	2	2	no	—
199	#121	C₂×Dic₆₇	1	4	yes	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	C₂²×D₆₇	1	4	yes	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#107	C₂×Dic₆₇	1	2	yes	—
204	#107	C₂×Dic₆₇	1	2	yes	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	67	4	no	—
207	C₄⋊C₈	—	134	1	no	—
208	#102	C₂×Dic₆₇	1	4	yes	—
209	#102	C₂×Dic₆₇	1	4	yes	—
210	C₂²⋊C₈	—	67	2	no	—
211	C₂²⋊C₈	—	134	4	no	—
212	C₂²⋊C₈	—	134	4	no	—
213	C₂²⋊C₈	—	134	2	no	—
214	C₂²⋊C₈	—	134	1	no	—
215	#97	—	134	2	no	—
216	C₄×C₈	—	67	1	no	—
217	C₄×C₈	—	67	1	no	—
218	C₄×C₈	—	67	1	no	—
219	C₄×C₈	—	67	1	no	—
220	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
221	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
224	#41	—	8	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	#11761627	1	1	yes	characteristic
234	D₈	—	2	4	no	—
235	D₈	—	2	4	no	—
236	M₁₆	—	536	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	536	4	no	—
240	C₂×C₈	—	67	2	no	—
241	C₂×C₈	—	67	2	no	—
242	C₂×C₈	—	134	4	no	—
243	C₂×C₈	—	134	1	no	—
244	C₂×C₈	—	134	1	no	—
245	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
254	C₄²	D₄×D₆₇	1	1	yes	characteristic
255	C₄²	#11761627	1	1	yes	characteristic
256	C₄²	C₂³×D₆₇	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
274	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	268	1	no	—
281	C₈	—	268	1	no	—
282	C₈	—	268	4	no	—
283	C₈	—	134	2	no	—
284	C₈	—	134	2	no	—
285	C₈	#14524050	1	2	yes	—
286	C₈	#14524050	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	4	no	—
299	C₄	C₂×#14524050	1	1	yes	characteristic
300	C₄	C₂×#14524050	1	1	yes	characteristic
301	C₄	#16864197	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	#22568921	1	1	yes	characteristic
305	C₂	#22569061	1	1	yes	characteristic
306	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3169	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3169	D₆₇
hypocenter	C₆₇	#17970
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22569285	C₂	semidirect
2. #22559123	C₂	semidirect
3. #22559094	C₂	semidirect
4. #22559033	C₂	semidirect
5. D₈×C₆₇⋊C₈	C₂	semidirect
6. #22558972	C₂	semidirect
7. #22558678	C₂	semidirect
8. #22558492	C₂	semidirect
163. C₆₇	#17970	semidirect

Maximal subgroups

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

subgroup	index
#22569285	2
#22559123	2
#22559094	2
#22559033	2
D₈×C₆₇⋊C₈	2
#22558972	2
#22558678	2
#22558492	2
C₆₇×#3169	2
#17970	67