Subgroups — #22569332

Order 8576.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11761627	C₂	1	1	yes	maximal, characteristic
2	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863523	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863467	C₂	1	2	yes	maximal, complemented
5	#16863464	C₂	1	2	yes	maximal, complemented
6	D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863450	C₂	1	2	yes	maximal, complemented
8	#16863447	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761627	C₄	1	1	yes	characteristic
13	C₂×#11761611	C₂²	1	2	yes	—
14	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
16	#14524008	C₄	1	1	yes	characteristic
17	D₄×Dic₆₇	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
19	(C₂²⋊D₆₇)⋊C₄	C₄	1	2	yes	—
20	#14524001	C₄	1	1	yes	characteristic
21	#14523999	C₄	1	2	yes	—
22	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
23	C₄²×D₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₆₇⋊C₈	C₂²	1	2	yes	—
26	Dic₆₇⋊C₈	C₂²	1	2	yes	—
27	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₈	C₂²	1	2	yes	—
29	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
33	#14523953	C₂²	1	2	yes	—
34	#14523953	C₂²	1	2	yes	—
35	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
37	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#11761627	—	2	4	no	—
42	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
49	#11761611	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
54	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
55	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
58	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
59	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₆₇	C₂³	1	2	yes	—
64	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₈×C₆₇	—	4	2	no	—
68	C₂×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
76	D₄×C₆₇	M₁₆	1	4	yes	complemented
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₆₇	—	2	4	no	—
84	C₂×Dic₆₇	—	2	4	no	—
85	C₂×Dic₆₇	—	2	4	no	—
86	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
90	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄×D₆₇	—	2	2	no	—
92	C₄×D₆₇	—	2	1	no	—
93	C₄×D₆₇	—	2	1	no	—
94	C₄×D₆₇	—	2	2	no	—
95	Dic₁₃₄	—	2	2	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₆₇⋊C₈	—	2	1	no	—
100	C₆₇⋊C₈	—	2	2	no	—
101	C₆₇⋊C₈	—	2	1	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	C₂×M₁₆	1	4	yes	—
105	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₆₇	—	2	1	no	—
107	C₂×D₆₇	—	2	1	no	—
108	C₂×D₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
112	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₆₇	—	2	1	no	—
115	Dic₆₇	—	2	2	no	—
116	Dic₆₇	—	2	1	no	—
117	Dic₆₇	—	2	2	no	—
118	Dic₆₇	#131	1	2	yes	—
119	C₂×C₆₇	—	2	4	no	—
120	C₂×C₆₇	#520	1	1	yes	characteristic
121	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
123	D₆₇	—	4	1	no	—
124	#2990	—	67	1	no	maximal, Sylow
125	C₆₇	#2990	1	1	yes	characteristic, complemented, Sylow
126	C₄×#41	—	67	1	no	—
127	#446	—	67	1	no	—
128	#446	—	67	1	no	—
129	#435	—	67	2	no	—
130	#435	—	67	2	no	—
131	C₈×D₄	—	67	1	no	—
132	C₈×D₄	D₆₇	1	1	yes	characteristic, complemented
133	#432	—	67	2	no	—
134	#431	—	67	1	no	—
135	#423	—	67	1	no	—
136	C₂×C₄⋊C₈	—	67	2	no	—
137	C₂×#41	—	67	1	no	—
138	C₂×M₁₆	—	67	2	no	—
139	C₂²×C₈	—	67	2	no	—
140	C₂²×C₈	C₂×D₆₇	1	2	yes	—
141	C₄×Q₈	—	67	1	no	—
142	C₄×Q₈	—	67	1	no	—
143	C₄×D₄	—	67	1	no	—
144	C₄×D₄	—	67	2	no	—
145	C₄×D₄	—	67	2	no	—
146	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
147	#117	—	67	1	no	—
148	#117	—	67	2	no	—
149	C₂×C₄²	—	67	2	no	—
150	C₂×C₄²	—	67	1	no	—
151	C₄⋊C₈	—	67	1	no	—
152	C₄⋊C₈	—	67	2	no	—
153	C₄⋊C₈	—	67	2	no	—
154	C₄⋊C₈	—	67	2	no	—
155	C₄⋊C₈	—	67	1	no	—
156	C₄⋊C₈	—	67	1	no	—
157	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂²⋊C₈	—	67	2	no	—
159	C₂²⋊C₈	—	67	1	no	—
160	C₂²⋊C₈	—	67	1	no	—
161	C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
162	#97	—	67	2	no	—
163	#97	—	67	2	no	—
164	C₄×C₈	—	67	1	no	—
165	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	#41	—	134	4	no	—
167	C₂×Q₈	—	67	1	no	—
168	C₂×D₄	—	67	2	no	—
169	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
170	C₂²×C₄	—	67	2	no	—
171	C₂²×C₄	—	67	2	no	—
172	C₂²×C₄	—	67	1	no	—
173	C₂²×C₄	—	67	1	no	—
174	C₂²×C₄	—	67	1	no	—
175	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
176	M₁₆	—	268	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	268	2	no	—
179	C₂×C₈	—	67	1	no	—
180	C₂×C₈	—	67	1	no	—
181	C₂×C₈	—	67	2	no	—
182	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₆₇	1	2	yes	—
185	C₄⋊C₄	—	67	2	no	—
186	C₄⋊C₄	—	67	2	no	—
187	C₄⋊C₄	—	67	1	no	—
188	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	67	2	no	—
190	C₂²⋊C₄	—	67	2	no	—
191	C₂²⋊C₄	C₄×D₆₇	1	2	yes	—
192	C₄²	—	67	2	no	—
193	C₄²	—	67	1	no	—
194	C₄²	—	67	1	no	—
195	C₄²	—	67	2	no	—
196	C₄²	—	67	2	no	—
197	C₄²	—	67	1	no	—
198	C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	C₂³	—	67	1	no	—
200	C₂³	C₂×C₄×D₆₇	1	2	yes	—
201	Q₈	—	134	2	no	—
202	D₄	—	134	4	no	—
203	D₄	#11761611	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	67	1	no	—
207	C₂×C₄	—	67	1	no	—
208	C₂×C₄	—	67	1	no	—
209	C₂×C₄	—	67	2	no	—
210	C₂×C₄	—	67	2	no	—
211	C₂×C₄	—	134	2	no	—
212	C₂×C₄	—	134	4	no	—
213	C₂×C₄	—	134	4	no	—
214	C₂×C₄	—	134	2	no	—
215	C₂×C₄	—	134	1	no	—
216	C₂×C₄	—	134	1	no	—
217	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	134	1	no	—
225	C₈	—	134	2	no	—
226	C₈	—	134	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	134	1	no	—
230	C₂²	—	134	1	no	—
231	C₂²	—	134	1	no	—
232	C₂²	C₂×#11761611	1	4	yes	—
233	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	67	2	no	—
237	C₄	—	134	1	no	—
238	C₄	—	134	1	no	—
239	C₄	—	134	2	no	—
240	C₄	—	134	2	no	—
241	C₄	C₂×#11761611	1	2	yes	—
242	C₄	#14524088	1	1	yes	characteristic
243	C₄	#14524090	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	268	1	no	—
246	C₂	#16863875	1	1	yes	characteristic
247	C₂	C₂²×#11761611	1	1	yes	characteristic
248	C₂	#131×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₈×D₄×C₆₇	C₂
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#2990
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #16863526	C₂	semidirect
3. #16863523	C₂	semidirect
4. #16863467	C₂	semidirect
5. #16863464	C₂	semidirect
6. D₄×C₆₇⋊C₈	C₂	semidirect
7. #16863450	C₂	semidirect
8. #16863447	C₂	semidirect
9. #16863418	C₂	semidirect
10. C₂×Dic₆₇⋊C₈	C₂	semidirect
11. C₈×D₄×C₆₇	C₂	semidirect
76. D₄×C₆₇	M₁₆	semidirect
125. C₆₇	#2990	semidirect
132. C₈×D₄	D₆₇	semidirect
203. D₄	#11761611	semidirect