Subgroups — #22562607

Order 8576.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863264	C₂	1	2	yes	maximal, complemented
2	#16863262	C₂	1	2	yes	maximal, complemented
3	#16863259	C₂	1	2	yes	maximal
4	#16863256	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₆₇	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊Dic₆₇	C₂	1	2	yes	maximal
7	#16863242	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₆₇	C₂²	1	4	yes	—
11	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
12	C₆₇⋊#95	C₂²	1	4	yes	—
13	C₆₇⋊#95	C₂²	1	4	yes	—
14	C₆₇⋊#95	C₂²	1	4	yes	—
15	C₆₇⋊#95	C₂²	1	4	yes	—
16	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₄	1	8	yes	complemented
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₆₇	C₂³	1	8	yes	—
25	C₄⋊Dic₆₇	—	2	8	no	—
26	C₂²⋊Dic₆₇	—	2	8	no	—
27	C₄×Dic₆₇	—	2	8	no	—
28	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	—	2	4	no	—
30	C₂²×C₄×C₆₇	—	2	4	no	—
31	C₂²×C₄×C₆₇	—	2	4	no	—
32	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₄²×C₆₇	C₂×C₄	1	4	yes	—
42	C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₄²×C₆₇	C₂×C₄	1	4	yes	—
44	C₄²×C₆₇	C₂×C₄	1	4	yes	—
45	C₂³×C₆₇	—	2	2	no	—
46	C₂³×C₆₇	—	2	2	no	—
47	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₆₇	#41	1	4	yes	—
49	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₆₇	—	2	8	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	8	no	—
56	C₂×C₄×C₆₇	—	2	8	no	—
57	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₆₇	#41	1	4	yes	—
64	C₂×C₄×C₆₇	#41	1	4	yes	—
65	C₂×Dic₆₇	—	2	8	no	—
66	C₂×Dic₆₇	—	2	8	no	—
67	C₂×Dic₆₇	—	2	8	no	—
68	C₂²×C₆₇	—	2	2	no	—
69	C₂²×C₆₇	—	2	2	no	—
70	C₂²×C₆₇	—	2	8	no	—
71	C₂²×C₆₇	#117	1	4	yes	—
72	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂×#41	1	2	yes	—
74	C₂²×C₆₇	C₂×#41	1	2	yes	—
75	C₂²×C₆₇	C₂×#41	1	2	yes	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	2	2	no	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	#117	1	4	yes	—
81	C₄×C₆₇	#117	1	4	yes	—
82	Dic₆₇	—	4	8	no	—
83	C₂×C₆₇	—	2	4	no	—
84	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
85	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
86	C₂×C₆₇	C₂×#117	1	1	yes	characteristic
87	C₂×C₆₇	#529	1	4	yes	—
88	#2283	—	67	1	no	maximal, Sylow
89	C₆₇	#2283	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄²	D₆₇	1	1	yes	characteristic
91	#383	—	67	2	no	—
92	#382	—	67	2	no	—
93	#381	—	67	2	no	—
94	#380	—	67	2	no	—
95	C₄×C₄⋊C₄	—	67	2	no	—
96	C₄×C₂²⋊C₄	—	67	2	no	—
97	#376	—	67	2	no	—
98	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
99	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
100	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	67	4	no	—
102	C₂×C₂²⋊C₄	—	67	4	no	—
103	C₂×C₄²	—	67	4	no	—
104	C₂×C₄²	Dic₆₇	1	8	yes	complemented
105	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
109	#95	—	67	4	no	—
110	#95	—	67	4	no	—
111	#95	—	67	4	no	—
112	#95	—	67	4	no	—
113	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	67	8	no	—
118	C₂²×C₄	C₂×Dic₆₇	1	4	yes	—
119	C₂²×C₄	C₂×Dic₆₇	1	4	yes	—
120	C₂²×C₄	C₂×Dic₆₇	1	4	yes	—
121	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊C₄	—	134	8	no	—
128	C₂²⋊C₄	—	134	8	no	—
129	C₄²	—	134	8	no	—
130	C₄²	C₂×Dic₆₇	1	4	yes	—
131	C₄²	C₂×Dic₆₇	1	4	yes	—
132	C₄²	C₂×Dic₆₇	1	4	yes	—
133	C₄²	C₂×Dic₆₇	1	4	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	C₂²×Dic₆₇	1	2	yes	—
137	C₂³	#11761628	1	4	yes	—
138	C₂³	C₂³×D₆₇	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	134	8	no	—
147	C₂×C₄	—	134	8	no	—
148	C₂×C₄	—	134	8	no	—
149	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
150	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
151	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
152	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
153	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
154	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
155	C₂×C₄	#11761628	1	4	yes	—
156	C₂×C₄	#11761628	1	4	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	8	no	—
160	C₂²	#14524002	1	4	yes	—
161	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
162	C₂²	C₂×#11761628	1	2	yes	—
163	C₂²	C₂×#11761628	1	2	yes	—
164	C₂²	C₂×#11761628	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	268	8	no	—
170	C₄	#14524002	1	4	yes	—
171	C₄	#14524002	1	4	yes	—
172	C₂	—	2	4	no	—
173	C₂	C₂×#14524002	1	1	yes	characteristic
174	C₂	C₂×#14524002	1	1	yes	characteristic
175	C₂	C₂×#14524002	1	1	yes	characteristic
176	C₂	#16863930	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Frattini	C₂³	C₂³×D₆₇
Fitting	C₂²×C₄²×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂²×C₄²	D₆₇
hypocenter	C₆₇	#2283
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #16863264	C₂	semidirect
2. #16863262	C₂	semidirect
5. C₄×C₄⋊Dic₆₇	C₂	semidirect
7. #16863242	C₂	semidirect
19. C₂×C₄²×C₆₇	C₄	semidirect
89. C₆₇	#2283	semidirect
104. C₂×C₄²	Dic₆₇	semidirect