Subgroups — #22559902

Order 8576.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863150	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₆₇	C₄	1	2	yes	—
6	C₂×#11761612	—	2	2	no	—
7	C₂×#11761612	—	2	2	no	—
8	C₂×#11761612	C₄	1	4	yes	complemented
9	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₆₇	—	2	2	no	—
11	C₂×C₄×Dic₆₇	—	2	2	no	—
12	C₂×C₄×Dic₆₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₆₇	—	2	2	no	—
16	C₂×M₁₆×C₆₇	—	2	2	no	—
17	C₂×M₁₆×C₆₇	C₄	1	4	yes	complemented
18	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂²×Dic₆₇	—	2	4	no	—
20	C₂²×Dic₆₇	—	2	8	no	—
21	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
22	#11761612	—	2	8	no	—
23	C₂×C₆₇⋊C₈	—	2	2	no	—
24	C₂×C₆₇⋊C₈	—	2	2	no	—
25	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
26	C₄×Dic₆₇	—	2	4	no	—
27	C₄×Dic₆₇	—	2	2	no	—
28	C₄×Dic₆₇	—	2	2	no	—
29	C₂⁴×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₆₇	D₄	1	2	yes	—
34	C₂²×C₄×C₆₇	D₄	1	2	yes	—
35	C₂²×C₄×C₆₇	D₄	1	2	yes	—
36	C₂²×C₄×C₆₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₆₇	—	2	8	no	—
39	C₂×C₈×C₆₇	—	2	2	no	—
40	C₂×C₈×C₆₇	—	2	2	no	—
41	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
42	C₂³×C₆₇	—	2	4	no	—
43	C₂³×C₆₇	C₄²	1	2	yes	—
44	C₂³×C₆₇	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₄²	1	2	yes	—
48	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₆₇	—	2	16	no	—
61	C₂×Dic₆₇	—	2	4	no	—
62	C₂×Dic₆₇	—	2	4	no	—
63	C₂×Dic₆₇	M₁₆	1	8	yes	complemented
64	C₈×C₆₇	—	2	4	no	—
65	C₆₇⋊C₈	—	2	4	no	—
66	C₂²×C₆₇	—	2	4	no	—
67	C₂²×C₆₇	—	2	4	no	—
68	C₂²×C₆₇	—	2	4	no	—
69	C₂²×C₆₇	#95	1	4	yes	—
70	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂×M₁₆	1	4	yes	—
74	C₄×C₆₇	#95	1	4	yes	—
75	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Dic₆₇	—	2	8	no	—
80	C₂×C₆₇	—	2	4	no	—
81	C₂×C₆₇	C₂×#95	1	1	yes	characteristic
82	C₂×C₆₇	C₄×M₁₆	1	2	yes	—
83	C₂×C₆₇	#407	1	2	yes	—
84	C₂×C₆₇	#423	1	2	yes	—
85	#1728	—	67	1	no	maximal, Sylow
86	C₆₇	#1728	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	67	1	no	—
88	C₂²×M₁₆	D₆₇	1	1	yes	characteristic
89	C₂²×C₄²	—	67	1	no	—
90	#336	—	67	4	no	—
91	C₂³×C₄	—	67	2	no	—
92	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	2	2	no	—
95	C₂×M₁₆	—	67	4	no	—
96	C₂×M₁₆	—	134	2	no	—
97	C₂×M₁₆	—	134	2	no	—
98	C₂×M₁₆	Dic₆₇	1	4	yes	complemented
99	C₂²×C₈	—	67	2	no	—
100	C₂²×C₈	C₂×D₆₇	1	2	yes	—
101	C₂×C₄²	—	67	2	no	—
102	C₂×C₄²	—	67	2	no	—
103	C₂×C₄²	—	134	2	no	—
104	C₂×C₄²	—	134	2	no	—
105	C₂⁴	C₄×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	—	67	4	no	—
107	C₂²×C₄	—	134	8	no	—
108	C₂²×C₄	—	134	4	no	—
109	C₂²×C₄	Dic₁₃₄	1	2	yes	—
110	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₆₇	1	2	yes	—
112	C₂²×C₄	D₂₆₈	1	2	yes	—
113	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
116	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	134	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	67	4	no	—
122	C₂×C₈	—	134	2	no	—
123	C₂×C₈	—	134	2	no	—
124	C₂×C₈	C₂×Dic₆₇	1	4	yes	complemented
125	C₄²	—	134	4	no	—
126	C₄²	—	134	2	no	—
127	C₄²	—	134	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	C₄×Dic₆₇	1	2	yes	—
130	C₂³	(C₂×D₆₇)⋊C₄	1	2	yes	—
131	C₂³	Dic₆₇⋊C₄	1	2	yes	—
132	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
133	C₂×C₄	—	67	8	no	—
134	C₂×C₄	—	134	16	no	—
135	C₂×C₄	—	134	4	no	—
136	C₂×C₄	—	134	4	no	—
137	C₂×C₄	C₄×Dic₆₇	1	2	yes	—
138	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
140	C₂×C₄	C₂²⋊Dic₆₇	1	4	yes	—
141	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
142	C₂×C₄	Dic₆₇⋊C₄	1	4	yes	—
143	C₂×C₄	C₄⋊Dic₆₇	1	4	yes	—
144	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	134	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₆₇⋊#95	1	4	yes	—
156	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
157	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
159	C₂²	M₁₆×D₆₇	1	4	yes	—
160	C₄	—	134	8	no	—
161	C₄	C₆₇⋊#95	1	4	yes	—
162	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
164	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
165	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
166	C₂	—	2	4	no	—
167	C₂	C₂×C₆₇⋊#95	1	1	yes	characteristic
168	C₂	M₁₆×Dic₆₇	1	2	yes	—
169	C₂	#16863359	1	2	yes	—
170	C₂	#16863419	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄²
Fitting	C₂²×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	C₂²×M₁₆	D₆₇
hypocenter	C₆₇	#1728
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16863150	C₂	semidirect
8. C₂×#11761612	C₄	semidirect
17. C₂×M₁₆×C₆₇	C₄	semidirect
25. C₂×C₆₇⋊C₈	C₂×C₄	semidirect
41. C₂×C₈×C₆₇	C₂×C₄	semidirect
63. C₂×Dic₆₇	M₁₆	semidirect
86. C₆₇	#1728	semidirect
98. C₂×M₁₆	Dic₆₇	semidirect
124. C₂×C₈	C₂×Dic₆₇	semidirect