Subgroups — #22563911

Order 8576.

608 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×#14524043	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14524020	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863328	C₂	1	2	yes	maximal, complemented
5	#16863317	C₂	1	2	yes	maximal, complemented
6	#16863294	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863294	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863271	C₂	1	2	yes	maximal
9	#16863266	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863256	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
13	C₂³×Dic₆₇	C₂²	1	2	yes	—
14	#14524043	—	2	4	no	—
15	#14524020	—	2	8	no	—
16	C₂×C₄⋊Dic₆₇	—	2	4	no	—
17	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
26	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
28	C₆₇⋊#95	C₂²	1	2	yes	—
29	C₆₇⋊#95	C₂²	1	2	yes	—
30	C₆₇⋊#95	C₂²	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	#95×C₆₇	C₂²	1	1	yes	characteristic
37	#95×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×Dic₁₃₄	—	2	4	no	—
39	C₂×Dic₁₃₄	—	2	4	no	—
40	C₂×Dic₁₃₄	—	4	2	no	—
41	C₂²×Dic₆₇	—	2	4	no	—
42	C₂²×Dic₆₇	—	2	4	no	—
43	C₂²×Dic₆₇	—	4	2	no	—
44	C₂²×Dic₆₇	C₂³	1	2	yes	—
45	C₂²×Dic₆₇	C₂³	1	2	yes	—
46	C₂²×Dic₆₇	C₂³	1	2	yes	—
47	C₂²×Dic₆₇	C₂³	1	2	yes	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	—	2	4	no	—
50	C₄⋊Dic₆₇	—	2	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	—	2	4	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	D₄	1	4	yes	—
56	Dic₆₇⋊C₄	—	2	4	no	—
57	Dic₆₇⋊C₄	—	2	4	no	—
58	C₂²⋊Dic₆₇	—	2	4	no	—
59	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	Q₈	1	4	yes	complemented
63	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₆₇	D₄	1	4	yes	—
70	C₂²⋊C₄×C₆₇	Q₈	1	4	yes	complemented
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	—	2	1	no	—
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	C₂×D₄	1	2	yes	—
78	C₂³×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂³×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	8	no	—
92	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₆₇	#41	1	2	yes	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	2	2	no	—
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	—	2	2	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	4	8	no	—
112	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
113	Dic₁₃₄	—	4	8	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	C₄⋊Q₈	1	4	yes	complemented
121	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	complemented
134	Dic₆₇	—	2	4	no	—
135	Dic₆₇	—	4	2	no	—
136	Dic₆₇	—	4	2	no	—
137	Dic₆₇	—	4	2	no	—
138	C₂×C₆₇	—	2	4	no	—
139	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₆₇	#544	1	1	yes	characteristic
142	C₂×C₆₇	D₄²	1	1	yes	characteristic
143	C₂×C₆₇	#547	1	1	yes	characteristic
144	C₂×C₆₇	D₄×Q₈	1	1	yes	characteristic
145	C₂×C₆₇	#554	1	1	yes	characteristic
146	#2440	—	67	1	no	maximal, Sylow
147	C₆₇	#2440	1	1	yes	characteristic, complemented, Sylow
148	C₂×C₄⋊Q₈	—	67	1	no	—
149	C₂×C₂²⋊Q₈	—	67	2	no	—
150	C₂²×C₄⋊C₄	—	67	1	no	—
151	#398	—	67	2	no	—
152	#395	—	67	2	no	—
153	#389	—	67	1	no	—
154	#389	—	67	1	no	—
155	#385	—	67	2	no	—
156	#384	—	67	1	no	—
157	#380	—	67	1	no	—
158	C₄×C₂²⋊C₄	D₆₇	1	1	yes	characteristic
159	C₂²×Q₈	—	67	2	no	—
160	C₂³×C₄	—	67	2	no	—
161	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	134	4	no	—
163	C₂²⋊Q₈	—	134	8	no	—
164	C₂×C₄⋊C₄	—	67	2	no	—
165	C₂×C₄⋊C₄	—	67	1	no	—
166	C₂×C₄⋊C₄	—	67	1	no	—
167	C₂×C₄⋊C₄	—	67	2	no	—
168	C₂×C₄⋊C₄	—	67	2	no	—
169	C₂×C₄⋊C₄	—	67	2	no	—
170	C₂×C₄⋊C₄	—	67	2	no	—
171	C₂×C₄⋊C₄	—	67	1	no	—
172	C₂×C₄⋊C₄	—	67	2	no	—
173	C₂×C₄⋊C₄	—	67	1	no	—
174	C₂×C₄⋊C₄	—	134	4	no	—
175	C₂×C₂²⋊C₄	—	67	2	no	—
176	C₂×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
180	#95	—	67	2	no	—
181	#95	—	67	2	no	—
182	#95	—	67	2	no	—
183	#95	C₂×D₆₇	1	1	yes	characteristic
184	#95	C₂×D₆₇	1	1	yes	characteristic
185	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
186	C₂×Q₈	—	268	2	no	—
187	C₂×Q₈	—	134	4	no	—
188	C₂×Q₈	—	134	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	268	2	no	—
192	C₂²×C₄	—	67	2	no	—
193	C₂²×C₄	—	67	2	no	—
194	C₂²×C₄	—	67	2	no	—
195	C₂²×C₄	—	67	2	no	—
196	C₂²×C₄	—	134	4	no	—
197	C₂²×C₄	—	134	4	no	—
198	C₂²×C₄	Dic₁₃₄	1	4	yes	complemented
199	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
205	C₄⋊C₄	—	268	2	no	—
206	C₄⋊C₄	—	67	4	no	—
207	C₄⋊C₄	—	134	4	no	—
208	C₄⋊C₄	—	134	4	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₄⋊C₄	—	134	4	no	—
212	C₄⋊C₄	—	134	4	no	—
213	C₄⋊C₄	—	134	4	no	—
214	C₄⋊C₄	—	134	2	no	—
215	C₂²⋊C₄	—	134	4	no	—
216	C₂²⋊C₄	Dic₁₃₄	1	4	yes	complemented
217	C₂²⋊C₄	D₂₆₈	1	4	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	C₂×D₂₆₈	1	2	yes	—
225	C₂³	C₂×Dic₁₃₄	1	2	yes	—
226	C₂³	C₂×Dic₁₃₄	1	2	yes	—
227	C₂³	C₂³×D₆₇	1	1	yes	characteristic
228	Q₈	—	268	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	268	8	no	—
241	C₂×C₄	—	67	4	no	—
242	C₂×C₄	—	134	2	no	—
243	C₂×C₄	—	134	2	no	—
244	C₂×C₄	—	134	2	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	—	134	2	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	—	134	2	no	—
251	C₂×C₄	—	134	2	no	—
252	C₂×C₄	—	134	2	no	—
253	C₂×C₄	D₄×D₆₇	1	2	yes	—
254	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
255	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
256	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
257	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
258	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
259	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
260	C₂×C₄	#11761627	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#14524043	1	4	yes	complemented
268	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
269	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
270	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
271	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
272	C₂²	C₂×#11761627	1	1	yes	characteristic
273	C₂²	#14524089	1	1	yes	characteristic
274	C₂²	#14524090	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	268	2	no	—
281	C₄	—	268	2	no	—
282	C₄	—	268	2	no	—
283	C₄	—	134	4	no	—
284	C₄	#14524020	1	4	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	C₂×#14524020	1	1	yes	characteristic
287	C₂	C₂×#14524043	1	1	yes	characteristic
288	C₂	#16864024	1	1	yes	characteristic
289	C₂	D₄×D₂₆₈	1	1	yes	characteristic
290	C₂	#16864050	1	1	yes	characteristic
291	C₂	D₄×Dic₁₃₄	1	1	yes	characteristic
292	C₂	#16864103	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₆₇	C₂
hypercenter	C₄×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	#2440
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14524043	C₂	semidirect
4. #16863328	C₂	semidirect
5. #16863317	C₂	semidirect
6. #16863294	C₂	semidirect
7. #16863294	C₂	semidirect
9. #16863266	C₂	semidirect
62. C₂²×C₄×C₆₇	Q₈	semidirect
70. C₂²⋊C₄×C₆₇	Q₈	semidirect
120. C₂²×C₆₇	C₄⋊Q₈	semidirect
133. C₄×C₆₇	C₂²⋊Q₈	semidirect
147. C₆₇	#2440	semidirect
198. C₂²×C₄	Dic₁₃₄	semidirect
216. C₂²⋊C₄	Dic₁₃₄	semidirect
267. C₂²	#14524043	semidirect
284. C₄	#14524020	semidirect