Subgroups — #22569291

Order 8576.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#16863527	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863525	C₂	1	2	yes	maximal, complemented
5	#16863524	C₂	1	2	yes	maximal, complemented
6	#16863523	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863452	C₂	1	2	yes	maximal
8	#16863451	C₂	1	2	yes	maximal
9	#16863448	C₂	1	1	yes	maximal, characteristic
10	#16863447	C₂	1	1	yes	maximal, characteristic
11	#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₆₇	C₄	1	1	yes	characteristic
13	Q₈×Dic₆₇	C₄	1	1	yes	characteristic
14	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₃₄	C₂²	1	2	yes	—
16	#14524008	C₄	1	2	yes	—
17	#14524008	C₄	1	1	yes	characteristic
18	C₄⋊C₄×D₆₇	C₄	1	2	yes	—
19	C₄⋊C₄×D₆₇	C₄	1	1	yes	characteristic
20	C₄²×D₆₇	C₂²	1	1	yes	characteristic
21	C₄²×D₆₇	C₂²	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₆₇⋊C₈	C₂²	1	2	yes	—
25	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₆₇⋊C₈	C₂²	1	2	yes	—
27	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
31	#14523954	C₂²	1	2	yes	—
32	#14523953	C₂²	1	2	yes	—
33	#14523953	C₂²	1	2	yes	—
34	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₆₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
39	#97×C₆₇	C₂²	1	2	yes	—
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	Q₈×D₆₇	—	2	4	no	—
44	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
54	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₆₇	C₂³	1	2	yes	—
63	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₆₇	—	2	4	no	—
84	C₄×D₆₇	—	2	1	no	—
85	C₄×D₆₇	—	2	1	no	—
86	C₄×D₆₇	—	4	1	no	—
87	C₄×D₆₇	—	4	2	no	—
88	C₄×D₆₇	M₁₆	1	4	yes	—
89	Dic₁₃₄	—	2	2	no	—
90	Dic₁₃₄	—	2	4	no	—
91	C₈×C₆₇	—	2	2	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₆₇⋊C₈	—	2	1	no	—
95	C₆₇⋊C₈	—	2	2	no	—
96	C₆₇⋊C₈	—	2	1	no	—
97	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₆₇	—	2	1	no	—
99	C₂×D₆₇	—	2	1	no	—
100	C₂×D₆₇	C₂×M₁₆	1	2	yes	—
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
105	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₆₇	—	2	1	no	—
108	Dic₆₇	—	2	2	no	—
109	Dic₆₇	—	2	1	no	—
110	Dic₆₇	—	2	2	no	—
111	Dic₆₇	C₂×M₁₆	1	2	yes	—
112	C₂×C₆₇	#519	1	1	yes	characteristic
113	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₆₇	#568	1	1	yes	characteristic
115	D₆₇	—	2	2	no	—
116	#2983	—	67	1	no	maximal, Sylow
117	C₆₇	#2983	1	1	yes	characteristic, complemented, Sylow
118	C₂×C₄×Q₈	—	67	1	no	—
119	#446	—	67	1	no	—
120	#446	—	67	1	no	—
121	#446	—	67	2	no	—
122	#446	—	67	2	no	—
123	#446	—	67	1	no	—
124	#446	D₆₇	1	1	yes	characteristic, complemented
125	#432	—	67	2	no	—
126	#432	—	67	2	no	—
127	#431	—	67	1	no	—
128	#431	—	67	1	no	—
129	C₂²×Q₈	—	67	1	no	—
130	C₄×Q₈	—	67	1	no	—
131	C₄×Q₈	—	67	2	no	—
132	C₄×Q₈	—	67	1	no	—
133	C₄×Q₈	—	67	2	no	—
134	C₄×Q₈	—	67	1	no	—
135	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	67	1	no	—
137	C₂×C₄⋊C₄	—	67	2	no	—
138	C₂×C₄²	—	67	1	no	—
139	C₂×C₄²	—	67	2	no	—
140	C₄⋊C₈	—	67	1	no	—
141	C₄⋊C₈	—	67	1	no	—
142	C₄⋊C₈	—	67	2	no	—
143	C₄⋊C₈	—	67	2	no	—
144	C₄⋊C₈	—	67	1	no	—
145	C₄⋊C₈	—	67	2	no	—
146	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
147	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
148	C₂²⋊C₈	—	67	2	no	—
149	C₂²⋊C₈	—	67	1	no	—
150	C₂²⋊C₈	—	67	1	no	—
151	#97	—	67	2	no	—
152	#97	—	67	2	no	—
153	#97	—	67	2	no	—
154	#97	C₂×D₆₇	1	2	yes	—
155	C₄×C₈	—	67	2	no	—
156	C₄×C₈	—	67	1	no	—
157	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂×Q₈	—	67	2	no	—
159	C₂×Q₈	—	67	1	no	—
160	C₂×Q₈	—	134	4	no	—
161	C₂×Q₈	C₄×D₆₇	1	1	yes	characteristic
162	C₂²×C₄	—	67	1	no	—
163	C₂²×C₄	—	67	2	no	—
164	C₂²×C₄	—	67	1	no	—
165	C₂²×C₄	—	67	2	no	—
166	C₂²×C₄	—	67	1	no	—
167	C₂×C₈	—	67	1	no	—
168	C₂×C₈	—	67	2	no	—
169	C₂×C₈	—	67	1	no	—
170	C₂×C₈	C₂²×D₆₇	1	2	yes	—
171	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	C₄⋊C₄	—	67	2	no	—
174	C₄⋊C₄	—	67	2	no	—
175	C₄⋊C₄	—	67	2	no	—
176	C₄⋊C₄	—	67	2	no	—
177	C₄⋊C₄	—	67	1	no	—
178	C₄⋊C₄	C₄×D₆₇	1	2	yes	—
179	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
180	C₄²	—	67	1	no	—
181	C₄²	—	67	2	no	—
182	C₄²	—	67	1	no	—
183	C₄²	—	67	2	no	—
184	C₄²	—	67	1	no	—
185	C₄²	—	67	2	no	—
186	C₄²	C₂²×D₆₇	1	1	yes	characteristic
187	C₄²	C₂²×D₆₇	1	2	yes	—
188	C₂³	—	67	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	134	4	no	—
191	Q₈	—	134	2	no	—
192	C₂×C₄	—	268	1	no	—
193	C₂×C₄	—	268	2	no	—
194	C₂×C₄	—	67	1	no	—
195	C₂×C₄	—	67	4	no	—
196	C₂×C₄	—	67	1	no	—
197	C₂×C₄	—	67	1	no	—
198	C₂×C₄	—	67	2	no	—
199	C₂×C₄	—	67	2	no	—
200	C₂×C₄	—	134	4	no	—
201	C₂×C₄	—	134	1	no	—
202	C₂×C₄	—	134	1	no	—
203	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	134	1	no	—
212	C₈	—	134	2	no	—
213	C₈	—	134	1	no	—
214	C₂²	—	67	2	no	—
215	C₂²	—	134	1	no	—
216	C₂²	—	134	1	no	—
217	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	67	2	no	—
222	C₄	—	134	1	no	—
223	C₄	—	134	2	no	—
224	C₄	—	134	1	no	—
225	C₄	—	134	2	no	—
226	C₄	M₁₆×D₆₇	1	2	yes	—
227	C₄	#14524090	1	1	yes	characteristic
228	C₄	#14524091	1	1	yes	characteristic
229	C₂	—	134	2	no	—
230	C₂	#16863872	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₆₇	1	1	yes	characteristic
232	C₂	#16864161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#446×C₆₇	C₂
hypercenter	#446	D₆₇
hypocenter	C₆₇	#2983
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #16863527	C₂	semidirect
3. #16863526	C₂	semidirect
4. #16863525	C₂	semidirect
5. #16863524	C₂	semidirect
6. #16863523	C₂	semidirect
11. #446×C₆₇	C₂	semidirect
117. C₆₇	#2983	semidirect
124. #446	D₆₇	semidirect