Subgroups — #30869710

Order 17152.

846 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568921	C₂	1	2	yes	maximal, complemented
2	#22561038	C₂	1	2	yes	maximal, complemented
3	#22560538	C₂	1	8	yes	maximal, complemented
4	#22560347	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524050	C₂²	1	4	yes	—
8	#16863474	—	2	2	no	—
9	#16863474	—	2	2	no	—
10	D₄×C₆₇⋊C₈	—	2	2	no	—
11	D₄×C₆₇⋊C₈	—	2	2	no	—
12	#16863420	C₂²	1	2	yes	—
13	C₂×#14523970	C₂²	1	8	yes	—
14	#16863363	C₂²	1	4	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863161	C₂²	1	4	yes	—
18	#16863150	C₂²	1	4	yes	—
19	C₂×#135×C₆₇	C₂²	1	4	yes	—
20	#532×C₆₇	C₂²	1	2	yes	—
21	#495×C₆₇	C₄	1	4	yes	complemented
22	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
23	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
24	#492×C₆₇	C₄	1	2	yes	complemented
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#14524050	—	4	8	no	—
27	C₂×#11761612	—	2	4	no	—
28	C₂×#11761612	C₂³	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	4	no	—
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	—	2	1	no	—
33	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	#14523970	—	4	8	no	—
37	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
39	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
40	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
42	C₂×#41×C₆₇	C₂³	1	4	yes	—
43	#135×C₆₇	—	4	8	no	—
44	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
46	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
49	#127×C₆₇	C₂×C₄	1	2	yes	—
50	#124×C₆₇	C₂×C₄	1	4	yes	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	C₄×D₄×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×Dic₆₇	—	2	2	no	—
60	#11761612	—	2	8	no	—
61	#11761612	—	4	8	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	8	no	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₄×Dic₆₇	—	2	4	no	—
68	C₄×Dic₆₇	—	2	4	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	#41×C₆₇	—	2	8	no	—
71	#41×C₆₇	—	2	8	no	—
72	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	2	no	—
75	C₂×D₄×C₆₇	—	2	2	no	—
76	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₇	—	4	4	no	—
82	SD₁₆×C₆₇	—	4	8	no	—
83	D₈×C₆₇	—	4	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂²⋊C₄×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
97	Q₈×C₆₇	—	2	8	no	—
98	D₄×C₆₇	—	2	8	no	—
99	D₄×C₆₇	—	4	8	no	—
100	D₄×C₆₇	—	4	4	no	—
101	D₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
114	C₂×Dic₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₈×C₆₇	—	2	2	no	—
118	C₈×C₆₇	—	2	2	no	—
119	C₆₇⋊C₈	—	2	2	no	—
120	C₆₇⋊C₈	—	2	2	no	—
121	C₆₇⋊C₈	—	4	4	no	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	4	4	no	—
125	C₂²×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₆₇	#540	1	2	yes	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	4	4	no	—
132	C₄×C₆₇	#443	1	2	yes	—
133	C₄×C₆₇	#444	1	2	yes	—
134	C₄×C₆₇	#518	1	2	yes	—
135	C₄×C₆₇	#535	1	2	yes	—
136	Dic₆₇	—	4	4	no	—
137	#22280	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	8	4	no	—
140	C₂×C₆₇	#2328	1	1	yes	characteristic
141	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
142	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
143	#3147	D₆₇	1	1	yes	characteristic
144	#2931	—	67	2	no	—
145	#1962	—	67	2	no	—
146	#1876	—	67	8	no	—
147	#1837	—	67	1	no	—
148	#1744	—	67	1	no	—
149	C₆₇	#22280	1	1	yes	characteristic, complemented, Sylow
150	C₂×#135	C₂×D₆₇	1	4	yes	—
151	C₂×#131	—	67	4	no	—
152	#532	C₂×D₆₇	1	2	yes	—
153	#495	Dic₆₇	1	4	yes	complemented
154	#494	Dic₆₇	1	1	yes	characteristic, complemented
155	#493	Dic₆₇	1	1	yes	characteristic, complemented
156	#492	Dic₆₇	1	2	yes	complemented
157	#436	—	134	2	no	—
158	#436	—	134	2	no	—
159	C₈×D₄	—	134	2	no	—
160	C₈×D₄	—	134	2	no	—
161	#423	—	67	2	no	—
162	C₂×#104	—	67	8	no	—
163	#408	—	67	4	no	—
164	C₂×C₄×C₈	—	67	1	no	—
165	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	C₄³	—	67	1	no	—
167	#341	—	67	4	no	—
168	#336	—	67	4	no	—
169	C₂×#41	C₂²×D₆₇	1	4	yes	—
170	#135	—	4	8	no	—
171	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
172	C₂×SD₁₆	C₂×Dic₆₇	1	8	yes	—
173	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
174	#131	—	268	8	no	—
175	C₂×M₁₆	—	67	4	no	—
176	C₂×M₁₆	—	134	4	no	—
177	C₂²×C₈	—	67	2	no	—
178	C₂²×C₈	—	134	4	no	—
179	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
180	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
181	#127	C₂×Dic₆₇	1	2	yes	—
182	#124	C₂×Dic₆₇	1	4	yes	—
183	#121	—	2	2	no	—
184	#121	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	2	2	no	—
187	C₂×C₄²	—	67	2	no	—
188	C₂×C₄²	—	134	1	no	—
189	C₂×C₄²	—	134	1	no	—
190	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊C₈	—	134	2	no	—
192	C₄⋊C₈	—	134	2	no	—
193	#104	—	268	8	no	—
194	C₂²⋊C₈	—	134	4	no	—
195	C₂²⋊C₈	—	134	4	no	—
196	C₄×C₈	—	67	1	no	—
197	C₄×C₈	—	67	2	no	—
198	C₄×C₈	—	67	1	no	—
199	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
200	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
201	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
202	#41	—	2	8	no	—
203	#41	—	2	8	no	—
204	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	134	2	no	—
211	C₂²×C₄	#11761627	1	1	yes	characteristic
212	C₂²×C₄	#11761627	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
214	Q₁₆	—	4	4	no	—
215	SD₁₆	—	4	8	no	—
216	D₈	—	4	4	no	—
217	M₁₆	—	268	8	no	—
218	M₁₆	—	134	8	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	268	8	no	—
221	C₂×C₈	—	67	2	no	—
222	C₂×C₈	—	67	2	no	—
223	C₂×C₈	—	134	4	no	—
224	C₂×C₈	—	134	4	no	—
225	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
226	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	268	2	no	—
232	C₄²	—	134	4	no	—
233	C₄²	—	134	4	no	—
234	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
236	C₄²	D₄×D₆₇	1	1	yes	characteristic
237	C₄²	D₄×D₆₇	1	1	yes	characteristic
238	C₂³	—	2	4	no	—
239	C₂³	C₂×#11761627	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	D₄	—	2	8	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	268	2	no	—
251	C₂×C₄	—	268	2	no	—
252	C₂×C₄	—	134	4	no	—
253	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₄	#14524088	1	4	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	268	4	no	—
263	C₈	—	134	2	no	—
264	C₈	—	134	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
271	C₂²	#16864000	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	268	4	no	—
276	C₄	#16863513	1	2	yes	—
277	C₄	#16863518	1	2	yes	—
278	C₄	#16863871	1	2	yes	—
279	C₄	#16863959	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	8	4	no	—
282	C₂	#22562935	1	1	yes	characteristic
283	C₂	C₂×#16863513	1	1	yes	characteristic
284	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#22280
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #22568921	C₂	semidirect
2. #22561038	C₂	semidirect
3. #22560538	C₂	semidirect
4. #22560347	C₂	semidirect
5. #22559954	C₂	semidirect
21. #495×C₆₇	C₄	semidirect
22. #494×C₆₇	C₄	semidirect
23. #493×C₆₇	C₄	semidirect
24. #492×C₆₇	C₄	semidirect
149. C₆₇	#22280	semidirect
153. #495	Dic₆₇	semidirect
154. #494	Dic₆₇	semidirect
155. #493	Dic₆₇	semidirect
156. #492	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568921	2
#22561038	2
#22560538	2
#22560347	2
#22559954	2
C₆₇×#3147	2
#22280	67