Subgroups — #30772177

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	4	no	—
21	C₄×C₆₇⋊C₈	—	2	2	no	—
22	C₈×Dic₆₇	—	2	4	no	—
23	C₈×Dic₆₇	—	2	2	no	—
24	C₈×Dic₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	C₂×C₁₆×C₆₇	—	2	4	no	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	Q₈	1	2	yes	—
32	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
33	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
34	C₂×C₆₇⋊C₈	—	2	2	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
37	C₄×Dic₆₇	—	2	2	no	—
38	C₄×Dic₆₇	—	2	2	no	—
39	C₄×Dic₆₇	—	2	2	no	—
40	C₄×Dic₆₇	—	2	4	no	—
41	C₄×Dic₆₇	C₁₆	1	4	yes	complemented
42	C₆₇⋊C₁₆	—	4	2	no	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₆₇	#41	1	2	yes	—
51	C₂×C₈×C₆₇	#41	1	2	yes	—
52	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₄²×C₆₇	M₁₆	1	2	yes	—
54	C₁₆×C₆₇	—	4	2	no	—
55	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#117	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₆₇	—	2	2	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
70	C₂×Dic₆₇	M₃₂	1	4	yes	—
71	C₈×C₆₇	—	2	2	no	—
72	C₈×C₆₇	#124	1	1	yes	characteristic
73	C₈×C₆₇	#124	1	1	yes	characteristic
74	C₈×C₆₇	#125	1	2	yes	—
75	C₆₇⋊C₈	—	2	4	no	—
76	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂²×C₆₇	#431	1	2	yes	—
78	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
79	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
80	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	C₄⋊C₁₆	1	4	yes	—
83	C₄×C₆₇	#382	1	1	yes	characteristic
84	C₄×C₆₇	#423	1	1	yes	characteristic
85	C₄×C₆₇	#432	1	2	yes	—
86	Dic₆₇	—	2	4	no	—
87	Dic₆₇	—	2	4	no	—
88	#12879	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	#1831	1	1	yes	characteristic
90	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
91	C₂×C₆₇	#2150	1	1	yes	characteristic
92	C₂×C₆₇	#2162	1	2	yes	—
93	C₂×C₆₇	#2163	1	2	yes	—
94	C₂×C₄⋊C₁₆	—	67	1	no	—
95	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
96	C₄²×C₈	—	67	1	no	—
97	#1374	—	67	4	no	—
98	C₆₇	#12879	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₁₆	—	67	2	no	—
100	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
101	C₂×C₄×C₈	—	67	2	no	—
102	C₂×C₄×C₈	—	67	2	no	—
103	C₂×C₄×C₈	—	67	1	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₄³	—	67	1	no	—
106	C₄⋊C₁₆	—	2	2	no	—
107	C₄⋊C₁₆	—	134	2	no	—
108	C₂²×C₈	—	67	2	no	—
109	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
111	C₂×C₄²	—	67	1	no	—
112	C₂×C₄²	—	67	2	no	—
113	C₂×C₄²	—	67	1	no	—
114	C₂×C₄²	—	67	2	no	—
115	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	134	2	no	—
119	C₂×C₁₆	—	134	4	no	—
120	C₄×C₈	—	134	4	no	—
121	C₄×C₈	—	134	2	no	—
122	C₄×C₈	—	134	2	no	—
123	C₄×C₈	—	134	2	no	—
124	C₄×C₈	Dic₁₃₄	1	2	yes	—
125	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
126	C₂²×C₄	—	67	2	no	—
127	C₂²×C₄	—	67	2	no	—
128	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	67	4	no	—
133	C₂×C₈	—	134	2	no	—
134	C₂×C₈	—	134	2	no	—
135	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
138	C₂×C₈	#11761627	1	2	yes	—
139	C₂×C₈	#11761629	1	2	yes	—
140	C₄²	—	67	4	no	—
141	C₄²	—	134	4	no	—
142	C₄²	—	134	2	no	—
143	C₄²	—	134	2	no	—
144	C₄²	—	134	2	no	—
145	C₄²	Dic₆₇⋊C₄	1	2	yes	—
146	C₄²	#11761611	1	2	yes	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	268	2	no	—
149	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	67	4	no	—
152	C₂×C₄	—	67	4	no	—
153	C₂×C₄	—	134	2	no	—
154	C₂×C₄	—	134	2	no	—
155	C₂×C₄	—	134	2	no	—
156	C₂×C₄	—	134	2	no	—
157	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
158	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
159	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
160	C₂×C₄	#14523982	1	2	yes	—
161	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#14524001	1	2	yes	—
163	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
164	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	134	4	no	—
167	C₈	#14524030	1	1	yes	characteristic
168	C₈	#14524032	1	1	yes	characteristic
169	C₈	#14524034	1	2	yes	—
170	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
171	C₂²	#16863448	1	2	yes	—
172	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
173	C₂²	M₃₂×D₆₇	1	2	yes	—
174	C₂²	C₂×#14523982	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	134	4	no	—
177	C₄	—	134	4	no	—
178	C₄	Dic₆₇⋊C₁₆	1	4	yes	—
179	C₄	#16863261	1	1	yes	characteristic
180	C₄	#16863419	1	1	yes	characteristic
181	C₄	#16863452	1	2	yes	—
182	C₂	#22560325	1	1	yes	characteristic
183	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic
184	C₂	#22561964	1	1	yes	characteristic
185	C₂	#22562007	1	2	yes	—
186	C₂	#22562011	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#12879
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
41. C₄×Dic₆₇	C₁₆	semidirect
98. C₆₇	#12879	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₄×C₈×Dic₆₇	2
#22558286	2
C₂×C₄⋊C₁₆×C₆₇	2
#12879	67