Subgroups — #22568634

Order 8576.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524049	C₂	1	2	yes	maximal
2	C₄×#11761628	C₂	1	1	yes	maximal, characteristic
3	#16863353	C₂	1	4	yes	maximal, complemented
4	#16863351	C₂	1	2	yes	maximal, complemented
5	C₄×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863348	C₂	1	2	yes	maximal, complemented
7	#97×D₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11761628	C₂²	1	1	yes	characteristic
10	#14524049	C₄	1	16	yes	complemented
11	C₂×#11761612	C₂²	1	2	yes	—
12	C₂×#11761611	C₂²	1	4	yes	—
13	C₂×C₈×D₆₇	C₂²	1	4	yes	—
14	C₄×Dic₁₃₄	C₄	1	2	yes	—
15	C₄×D₂₆₈	C₄	1	2	yes	—
16	C₄×C₂²⋊D₆₇	C₄	1	4	yes	—
17	#14524002	C₂²	1	1	yes	characteristic
18	#14524000	C₂²	1	2	yes	—
19	C₄²×D₆₇	C₂²	1	2	yes	—
20	#14523954	C₂²	1	2	yes	—
21	#14523953	C₂²	1	4	yes	—
22	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
23	C₈×Dic₆₇	C₂²	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#97×C₆₇	C₂²	1	1	yes	characteristic
27	#97×C₆₇	C₂²	1	1	yes	characteristic
28	#97×C₆₇	C₂²	1	2	yes	—
29	#11761628	C₂×C₄	1	8	yes	—
30	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₆₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₆₇	C₂³	1	2	yes	—
35	#11761612	C₂×C₄	1	8	yes	complemented
36	#11761611	C₂×C₄	1	16	yes	complemented
37	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₈×D₆₇	C₂×C₄	1	16	yes	complemented
39	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
40	Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
42	(C₂×D₆₇)⋊C₄	C₂×C₄	1	4	yes	—
43	C₄×Dic₆₇	C₂×C₄	1	4	yes	—
44	C₄×Dic₆₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
48	C₂×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₆₇	—	4	2	no	—
56	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₆₇	C₄²	1	8	yes	—
62	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
63	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
64	D₂₆₈	C₄²	1	4	yes	—
65	C₄×D₆₇	—	4	4	no	—
66	C₄×D₆₇	C₂²×C₄	1	8	yes	—
67	Dic₁₃₄	C₄²	1	4	yes	—
68	C₈×C₆₇	C₂²×C₄	1	4	yes	—
69	C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂²×C₆₇	—	2	2	no	—
72	C₂²×C₆₇	C₂×C₄²	1	2	yes	—
73	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₆₇	—	2	4	no	—
75	C₂×D₆₇	C₂×C₄²	1	4	yes	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	2	2	no	—
78	C₄×C₆₇	C₂×C₄²	1	2	yes	—
79	C₄×C₆₇	C₂³×C₄	1	2	yes	—
80	Dic₆₇	—	2	4	no	—
81	Dic₆₇	C₂×C₄²	1	4	yes	—
82	C₂×C₆₇	—	2	2	no	—
83	C₂×C₆₇	C₂²×C₄²	1	1	yes	characteristic
84	C₂×C₆₇	#568	1	2	yes	—
85	D₆₇	—	2	4	no	—
86	#2875	—	67	1	no	maximal, Sylow
87	C₆₇	#2875	1	1	yes	characteristic, complemented, Sylow
88	C₂×#131	—	67	2	no	—
89	C₄×#41	—	67	1	no	—
90	#405	—	67	2	no	—
91	#405	—	67	4	no	—
92	C₄×M₁₆	—	67	1	no	—
93	C₄×M₁₆	—	67	2	no	—
94	C₂×#97	—	67	2	no	—
95	C₂×#97	D₆₇	1	1	yes	characteristic, complemented
96	C₂×#41	—	67	1	no	—
97	#131	—	67	16	no	—
98	C₂×M₁₆	—	67	2	no	—
99	C₂×M₁₆	—	67	4	no	—
100	C₂²×C₈	—	67	4	no	—
101	C₂²×C₈	C₂×D₆₇	1	2	yes	—
102	C₄×Q₈	—	67	2	no	—
103	C₄×D₄	—	67	4	no	—
104	C₄×D₄	—	67	2	no	—
105	#117	—	67	1	no	—
106	#117	—	67	2	no	—
107	C₂×C₄²	—	67	2	no	—
108	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
109	#97	—	67	4	no	—
110	#97	—	67	2	no	—
111	#97	C₂×D₆₇	1	1	yes	characteristic
112	#97	C₂×D₆₇	1	1	yes	characteristic
113	#97	C₂×D₆₇	1	2	yes	—
114	C₄×C₈	—	67	4	no	—
115	C₄×C₈	—	67	2	no	—
116	#41	—	67	8	no	—
117	C₂×Q₈	—	67	1	no	—
118	C₂×D₄	—	67	2	no	—
119	C₂×D₄	—	67	1	no	—
120	C₂²×C₄	—	67	4	no	—
121	C₂²×C₄	—	67	2	no	—
122	C₂²×C₄	C₄×D₆₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
124	M₁₆	—	67	8	no	—
125	M₁₆	—	67	16	no	—
126	C₂×C₈	—	67	4	no	—
127	C₂×C₈	—	67	16	no	—
128	C₂×C₈	C₄×D₆₇	1	8	yes	complemented
129	C₂×C₈	C₂²×D₆₇	1	2	yes	—
130	C₂×C₈	C₂²×D₆₇	1	2	yes	—
131	C₄⋊C₄	—	67	4	no	—
132	C₄⋊C₄	—	67	2	no	—
133	C₂²⋊C₄	—	67	2	no	—
134	C₂²⋊C₄	—	67	4	no	—
135	C₄²	—	67	4	no	—
136	C₄²	—	67	2	no	—
137	C₄²	C₄×D₆₇	1	2	yes	—
138	C₄²	C₂²×D₆₇	1	1	yes	characteristic
139	C₄²	C₂²×D₆₇	1	1	yes	characteristic
140	C₂³	—	67	2	no	—
141	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
142	Q₈	—	67	4	no	—
143	D₄	—	67	8	no	—
144	D₄	—	67	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	268	4	no	—
147	C₂×C₄	—	67	2	no	—
148	C₂×C₄	—	67	8	no	—
149	C₂×C₄	—	67	4	no	—
150	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₆₇	1	4	yes	—
154	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
155	C₈	—	67	8	no	—
156	C₈	C₂×C₄×D₆₇	1	4	yes	—
157	C₈	C₂×C₄×D₆₇	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	67	4	no	—
160	C₂²	—	134	4	no	—
161	C₂²	C₄²×D₆₇	1	2	yes	—
162	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	67	4	no	—
166	C₄	—	134	4	no	—
167	C₄	C₄²×D₆₇	1	2	yes	—
168	C₄	C₂²×C₄×D₆₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	134	4	no	—
171	C₂	C₂×C₄²×D₆₇	1	1	yes	characteristic
172	C₂	#16864162	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂²×C₄²
Fitting	C₂×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	#2875
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #16863353	C₂	semidirect
4. #16863351	C₂	semidirect
5. C₄×#11761612	C₂	semidirect
6. #16863348	C₂	semidirect
7. #97×D₆₇	C₂	semidirect
8. C₂×#97×C₆₇	C₂	semidirect
10. #14524049	C₄	semidirect
35. #11761612	C₂×C₄	semidirect
36. #11761611	C₂×C₄	semidirect
38. C₈×D₆₇	C₂×C₄	semidirect
47. C₂×C₈×C₆₇	C₂×C₄	semidirect
87. C₆₇	#2875	semidirect
95. C₂×#97	D₆₇	semidirect
128. C₂×C₈	C₄×D₆₇	semidirect