Subgroups — #30864364

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863504	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₆₈	C₂	1	2	yes	maximal
4	#22562928	C₂	1	2	yes	maximal
5	#22561080	C₂	1	2	yes	maximal
6	#22561070	C₂	1	2	yes	maximal
7	#22561029	C₂	1	2	yes	maximal
8	#22560373	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
11	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
12	C₂×#14524043	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₃₄	C₂²	1	4	yes	—
14	#16863776	C₄	1	8	yes	complemented
15	#16863504	—	2	8	no	—
16	C₄×Dic₂₆₈	—	2	8	no	—
17	C₂×#14523977	C₂²	1	2	yes	—
18	C₂×#14523975	C₂²	1	2	yes	—
19	C₂×#14523968	C₂²	1	4	yes	—
20	C₂×#14523968	C₂²	1	4	yes	—
21	#16863302	C₂²	1	4	yes	—
22	#16863281	—	2	8	no	—
23	#16863262	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#376×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₃₄	C₂³	1	4	yes	—
32	C₂×Dic₂₆₈	—	2	4	no	—
33	C₂×Dic₂₆₈	—	2	4	no	—
34	C₂×Dic₂₆₈	—	2	4	no	—
35	C₂×Dic₂₆₈	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	—
37	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	—
38	#14524043	—	2	4	no	—
39	#14524043	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₃₄	—	2	8	no	—
41	C₄×Dic₁₃₄	—	2	8	no	—
42	C₂×C₄⋊Dic₆₇	—	2	4	no	—
43	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
44	C₂×Dic₆₇⋊C₄	—	2	4	no	—
45	C₂×C₄×Dic₆₇	—	2	4	no	—
46	#14523977	—	2	4	no	—
47	#14523975	—	2	4	no	—
48	#14523968	—	2	8	no	—
49	#14523968	—	2	8	no	—
50	C₆₇⋊#95	—	2	4	no	—
51	C₆₇⋊#95	—	2	8	no	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	#97×C₆₇	—	2	2	no	—
60	#97×C₆₇	—	2	2	no	—
61	#97×C₆₇	D₄	1	4	yes	—
62	C₄×C₈×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₆₇	D₄	1	4	yes	—
65	#95×C₆₇	D₄	1	4	yes	—
66	C₂×Dic₁₃₄	—	2	4	no	—
67	C₂×Dic₁₃₄	—	2	4	no	—
68	C₂×Dic₁₃₄	—	2	4	no	—
69	C₂×Dic₁₃₄	—	2	4	no	—
70	C₂×Dic₁₃₄	—	2	4	no	—
71	C₂×Dic₁₃₄	C₂²×C₄	1	8	yes	—
72	C₂²×Dic₆₇	—	2	4	no	—
73	C₂²×Dic₆₇	—	2	4	no	—
74	Dic₂₆₈	—	2	16	no	—
75	Dic₂₆₈	—	2	16	no	—
76	C₄⋊Dic₆₇	—	2	16	no	—
77	C₄⋊Dic₆₇	—	2	4	no	—
78	C₄⋊Dic₆₇	—	2	4	no	—
79	Dic₆₇⋊C₄	—	4	8	no	—
80	C₄×Dic₆₇	—	4	8	no	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	#41	1	2	yes	—
96	C₂×C₈×C₆₇	#41	1	2	yes	—
97	C₄²×C₆₇	—	2	1	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	#41	1	2	yes	—
103	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₆₇	#127	1	4	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₆₇	—	2	8	no	—
117	C₂×Dic₆₇	—	4	4	no	—
118	C₂×Dic₆₇	—	4	4	no	—
119	C₂×Dic₆₇	—	4	4	no	—
120	C₂×Dic₆₇	—	4	8	no	—
121	Dic₁₃₄	—	2	8	no	—
122	Dic₁₃₄	—	2	8	no	—
123	Dic₁₃₄	—	4	4	no	—
124	Dic₁₃₄	—	4	4	no	—
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	C₄×D₄	1	4	yes	—
128	C₈×C₆₇	C₄×D₄	1	4	yes	—
129	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
131	C₂²×C₆₇	#532	1	1	yes	characteristic
132	C₂²×C₆₇	#578	1	4	yes	—
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₆₇	#532	1	1	yes	characteristic
139	C₄×C₆₇	#532	1	1	yes	characteristic
140	Dic₆₇	—	4	8	no	—
141	Dic₆₇	—	8	4	no	—
142	#21698	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
144	C₂×C₆₇	#2950	1	2	yes	—
145	C₂×C₆₇	#3156	1	2	yes	—
146	C₂×C₆₇	#3177	1	2	yes	—
147	C₂×#494	—	67	1	no	—
148	C₂×#441	—	67	2	no	—
149	C₂×C₄×Q₁₆	—	67	2	no	—
150	#2327	—	67	2	no	—
151	#1971	—	67	2	no	—
152	#1969	—	67	2	no	—
153	#1960	—	67	2	no	—
154	#1846	—	67	1	no	—
155	#1726	D₆₇	1	1	yes	characteristic
156	C₆₇	#21698	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₁₆	—	67	2	no	—
158	C₂²×Q₁₆	—	67	2	no	—
159	C₂×C₄⋊Q₈	—	67	2	no	—
160	C₂×C₄×Q₈	—	67	4	no	—
161	#494	—	67	8	no	—
162	#441	—	134	8	no	—
163	C₄×Q₁₆	—	134	8	no	—
164	C₂×#107	—	67	2	no	—
165	C₂×#106	—	67	2	no	—
166	C₂×#103	—	67	4	no	—
167	C₂×#103	—	67	4	no	—
168	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
169	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
170	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
174	#391	—	67	4	no	—
175	#387	—	134	8	no	—
176	#382	—	67	2	no	—
177	#376	C₂×D₆₇	1	1	yes	characteristic
178	C₂²×Q₈	—	67	4	no	—
179	C₂×Q₁₆	—	67	8	no	—
180	C₂×Q₁₆	—	67	8	no	—
181	C₂×Q₁₆	—	134	4	no	—
182	C₂×Q₁₆	—	134	4	no	—
183	C₂×Q₁₆	—	134	4	no	—
184	C₂×Q₁₆	—	134	4	no	—
185	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₄⋊Q₈	—	67	8	no	—
190	C₄⋊Q₈	—	134	4	no	—
191	C₄×Q₈	—	134	8	no	—
192	C₄×Q₈	—	134	8	no	—
193	C₂×C₄⋊C₄	—	67	4	no	—
194	C₂×C₄⋊C₄	—	134	4	no	—
195	C₂×C₄⋊C₄	—	134	4	no	—
196	C₂×C₄²	—	134	4	no	—
197	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
200	#107	—	134	4	no	—
201	#106	—	134	4	no	—
202	#103	—	134	8	no	—
203	#103	—	134	8	no	—
204	#97	—	2	2	no	—
205	#97	—	2	2	no	—
206	#97	D₂₆₈	1	4	yes	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	C₄×D₆₇	1	4	yes	—
209	C₄×C₈	D₂₆₈	1	4	yes	—
210	#95	—	134	4	no	—
211	#95	—	134	8	no	—
212	#95	D₂₆₈	1	4	yes	—
213	C₂×Q₈	—	67	8	no	—
214	C₂×Q₈	—	134	4	no	—
215	C₂×Q₈	—	134	4	no	—
216	C₂×Q₈	—	134	4	no	—
217	C₂×Q₈	—	134	4	no	—
218	C₂×Q₈	—	134	4	no	—
219	C₂²×C₄	—	134	4	no	—
220	C₂²×C₄	—	134	4	no	—
221	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
222	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
223	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
224	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
225	Q₁₆	—	134	16	no	—
226	Q₁₆	—	134	16	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
233	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
234	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
235	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
236	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
237	C₂×C₈	#11761628	1	2	yes	—
238	C₂×C₈	#11761628	1	2	yes	—
239	C₄⋊C₄	—	268	8	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₄⋊C₄	—	134	4	no	—
242	C₄⋊C₄	—	134	16	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	268	8	no	—
246	C₄²	C₂×C₄×D₆₇	1	2	yes	—
247	C₄²	C₂×D₂₆₈	1	2	yes	—
248	C₄²	C₂×D₂₆₈	1	2	yes	—
249	C₄²	#11761628	1	2	yes	—
250	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
251	Q₈	—	268	4	no	—
252	Q₈	—	268	4	no	—
253	Q₈	—	134	8	no	—
254	Q₈	—	134	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	268	4	no	—
261	C₂×C₄	—	268	4	no	—
262	C₂×C₄	—	268	8	no	—
263	C₂×C₄	—	134	8	no	—
264	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
265	C₂×C₄	#14524040	1	4	yes	—
266	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
271	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	C₄×D₂₆₈	1	4	yes	—
275	C₈	C₄×D₂₆₈	1	4	yes	—
276	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
277	C₂²	C₂×#14524040	1	1	yes	characteristic
278	C₂²	#16863946	1	1	yes	characteristic
279	C₂²	#16864213	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	268	8	no	—
284	C₄	—	536	4	no	—
285	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
286	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
287	C₄	#16863946	1	1	yes	characteristic
288	C₄	#16863946	1	1	yes	characteristic
289	C₂	C₄×#14524040	1	1	yes	characteristic
290	C₂	#22569078	1	2	yes	—
291	C₂	#22570703	1	2	yes	—
292	C₂	#22570868	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21698
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
14. #16863776	C₄	semidirect
156. C₆₇	#21698	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
C₂×#16863504	2
C₂×C₄×Dic₂₆₈	2
#22562928	2
#22561080	2
#22561070	2
#22561029	2
#22560373	2
C₆₇×#1726	2
#21698	67