Subgroups — #30764865

Order 17152.

290 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558128	C₂	1	2	yes	maximal, complemented
4	#22558125	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	—
9	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₆₇	C₄	1	2	yes	—
17	C₂²⋊C₁₆×C₆₇	C₄	1	2	yes	—
18	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₆₇⋊C₈	C₈	1	2	yes	—
20	C₂²×C₆₇⋊C₈	C₈	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
23	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₆₇⋊C₈	—	2	4	no	—
37	C₂×C₆₇⋊C₈	—	2	4	no	—
38	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
39	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
40	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
41	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
42	C₆₇⋊C₁₆	—	2	4	no	—
43	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₄²	1	1	yes	characteristic
53	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₆₇	—	2	4	no	—
57	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
65	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	C₈×C₆₇	—	2	1	no	—
69	C₈×C₆₇	—	2	1	no	—
70	C₈×C₆₇	#131	1	2	yes	—
71	C₈×C₆₇	#131	1	2	yes	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	—	2	2	no	—
74	C₆₇⋊C₈	—	2	2	no	—
75	C₆₇⋊C₈	M₃₂	1	4	yes	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	#346	1	2	yes	—
79	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	#346	1	2	yes	—
83	C₄×C₆₇	#405	1	1	yes	characteristic
84	C₄×C₆₇	#405	1	1	yes	characteristic
85	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
86	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
87	#11411	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	2	no	—
89	C₂×C₆₇	#1450	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
91	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
92	#2162	—	67	1	no	—
93	#2162	D₆₇	1	1	yes	characteristic
94	C₂×C₈²	—	67	1	no	—
95	#1312	—	67	2	no	—
96	#1311	—	67	2	no	—
97	C₆₇	#11411	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄×C₈	—	67	1	no	—
99	C₂×C₄×C₈	—	67	1	no	—
100	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	C₄⋊C₁₆	—	67	2	no	—
102	C₄⋊C₁₆	Dic₆₇	1	2	yes	—
103	C₂²⋊C₁₆	—	67	2	no	—
104	C₂²⋊C₁₆	Dic₆₇	1	2	yes	—
105	C₄×C₁₆	—	67	2	no	—
106	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
107	C₈²	—	67	1	no	—
108	C₈²	—	67	1	no	—
109	C₈²	—	67	1	no	—
110	C₈²	—	67	1	no	—
111	C₂²×C₈	—	67	2	no	—
112	C₂²×C₈	—	67	2	no	—
113	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₁₆	—	67	4	no	—
117	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
118	C₄×C₈	—	67	2	no	—
119	C₄×C₈	—	67	2	no	—
120	C₄×C₈	—	67	1	no	—
121	C₄×C₈	—	67	1	no	—
122	C₄×C₈	—	67	1	no	—
123	C₄×C₈	—	67	1	no	—
124	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
125	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	67	2	no	—
134	C₂×C₈	—	67	2	no	—
135	C₂×C₈	—	67	2	no	—
136	C₂×C₈	—	67	2	no	—
137	C₂×C₈	—	134	4	no	—
138	C₂×C₈	—	134	4	no	—
139	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
140	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
143	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
144	C₄²	C₈×D₆₇	1	1	yes	characteristic
145	C₄²	C₈×D₆₇	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
147	C₁₆	—	2	4	no	—
148	C₁₆	—	134	4	no	—
149	C₂³	C₈×Dic₆₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
154	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
155	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
156	C₂×C₄	#14523982	1	4	yes	—
157	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	67	4	no	—
163	C₈	—	134	2	no	—
164	C₈	—	134	2	no	—
165	C₈	—	134	2	no	—
166	C₈	#14524050	1	2	yes	—
167	C₈	#14524051	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16863171	1	2	yes	—
171	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#16863171	1	2	yes	—
175	C₄	#16863352	1	1	yes	characteristic
176	C₄	#16863354	1	1	yes	characteristic
177	C₄	M₃₂×D₆₇	1	2	yes	—
178	C₄	C₂×#14523982	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#22558493	1	1	yes	characteristic
181	C₂	C₂×#16863171	1	1	yes	characteristic
182	C₂	M₃₂×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#2162	C₂
prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2162	D₆₇
hypocenter	C₆₇	#11411
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #22558128	C₂	semidirect
4. #22558125	C₂	semidirect
97. C₆₇	#11411	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
C₂×C₈×C₆₇⋊C₈	2
#22558128	2
#22558125	2
C₆₇×#2162	2
#11411	67