Group #30765148 — subgroups

Subgroups — #30765148

Order 17152.

430 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	2	yes	maximal, complemented
2	#22558133	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
6	C₄×C₆₇⋊C₁₆	—	2	2	no	—
7	C₂³×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
9	C₈×D₄×C₆₇	—	2	2	no	—
10	C₈×D₄×C₆₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₆₇⋊C₁₆	—	2	4	no	—
15	C₂²×D₄×C₆₇	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₇	—	2	2	no	—
18	C₂²×C₈×C₆₇	—	2	2	no	—
19	C₂²×C₈×C₆₇	—	4	4	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	C₄×D₄×C₆₇	—	2	2	no	—
24	C₄×D₄×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₂²⋊C₈×C₆₇	—	2	4	no	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₆₇⋊C₁₆	—	4	4	no	—
33	C₂⁴×C₆₇	C₂×C₈	1	2	yes	—
34	C₂×D₄×C₆₇	—	2	2	no	—
35	C₂×D₄×C₆₇	—	2	2	no	—
36	C₂×D₄×C₆₇	C₁₆	1	4	yes	complemented
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	—	4	2	no	—
40	C₂²×C₄×C₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	4	1	no	—
48	C₂×C₈×C₆₇	—	4	4	no	—
49	C₂×C₈×C₆₇	—	4	4	no	—
50	C₂×C₈×C₆₇	—	4	2	no	—
51	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
54	C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂²⋊C₄×C₆₇	—	2	4	no	—
56	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₆₇	M₁₆	1	2	yes	—
58	C₂³×C₆₇	—	2	2	no	—
59	C₂³×C₆₇	—	2	2	no	—
60	C₂³×C₆₇	—	4	2	no	—
61	C₂³×C₆₇	—	4	2	no	—
62	C₂³×C₆₇	C₂×C₁₆	1	4	yes	—
63	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
64	D₄×C₆₇	—	2	8	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	4	1	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	2	1	no	—
81	C₈×C₆₇	—	4	2	no	—
82	C₈×C₆₇	#100	1	2	yes	—
83	C₂²×C₆₇	—	2	1	no	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	4	4	no	—
87	C₂²×C₆₇	—	4	4	no	—
88	C₂²×C₆₇	—	4	2	no	—
89	C₂²×C₆₇	—	4	2	no	—
90	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
93	C₄×C₆₇	—	2	2	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	4	2	no	—
96	C₄×C₆₇	C₂²⋊C₁₆	1	4	yes	—
97	C₄×C₆₇	#410	1	1	yes	characteristic
98	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
99	#11487	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	4	4	no	—
103	C₂×C₆₇	#1460	1	1	yes	characteristic
104	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
105	C₂×C₆₇	#2115	1	1	yes	characteristic
106	C₂×C₈×D₄	D₆₇	1	1	yes	characteristic
107	#2162	—	67	2	no	—
108	#1314	—	67	4	no	—
109	C₆₇	#11487	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₈	Dic₆₇	1	2	yes	—
111	C₂×C₄×D₄	Dic₆₇	1	1	yes	characteristic
112	C₈×D₄	—	2	2	no	—
113	C₈×D₄	—	2	2	no	—
114	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
115	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	134	2	no	—
118	C₂²⋊C₁₆	—	67	4	no	—
119	C₄×C₁₆	—	134	2	no	—
120	C₂²×D₄	C₆₇⋊C₈	1	1	yes	characteristic
121	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	4	4	no	—
125	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₄×D₄	—	2	2	no	—
129	C₄×D₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
132	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
133	C₂×C₁₆	—	134	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
137	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
138	C₂⁴	C₂×C₆₇⋊C₈	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₆₇⋊C₁₆	1	4	yes	complemented
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
147	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
162	C₄²	#11761612	1	2	yes	—
163	C₁₆	—	268	4	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	—	4	2	no	—
168	C₂³	C₂×C₆₇⋊C₁₆	1	4	yes	—
169	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
180	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
181	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
182	C₂×C₄	#14523983	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	1	no	—
187	C₈	—	4	2	no	—
188	C₈	#14523960	1	2	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
197	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
198	C₂²	C₂×#14523983	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	C₂²⋊(C₆₇⋊C₁₆)	1	4	yes	—
203	C₄	#16863370	1	1	yes	characteristic
204	C₄	C₂×#14523960	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	4	4	no	—
208	C₂	#22558530	1	1	yes	characteristic
209	C₂	C₂×C₂²⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
210	C₂	#22561830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562008

C₂

yes

maximal, complemented

#22558133

C₂

yes

maximal, complemented

C₂×C₈×D₄×C₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²⋊(C₆₇⋊C₁₆)

C₂²

yes

—

C₄×C₆₇⋊C₁₆

—

C₂³×C₈×C₆₇

C₄

yes

—

C₂×C₄×D₄×C₆₇

C₄

yes

characteristic

C₈×D₄×C₆₇

—

C₈×D₄×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₄

yes

characteristic

C₂×C₂²⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂²×D₄×C₆₇

C₈

yes

characteristic

C₂³×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

C₈

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

C₂×C₈

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₁₆

yes

complemented

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×C₁₆

yes

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

—

C₂×C₄×C₆₇

M₃₂

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#100

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₇

C₂²×C₁₆

yes

characteristic

C₂²×C₆₇

C₂×M₃₂

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊C₁₆

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

100

C₂×C₆₇

—

101

C₂×C₆₇

—

102

C₂×C₆₇

—

103

C₂×C₆₇

#1460

yes

characteristic

104

C₂×C₆₇

C₂×C₂²⋊C₁₆

yes

characteristic

105

C₂×C₆₇

#2115

yes

characteristic

106

C₂×C₈×D₄

D₆₇

yes

characteristic

107

#2162

—

108

#1314

—

109

C₆₇

#11487

yes

characteristic, complemented, Sylow

110

C₂³×C₈

Dic₆₇

yes

—

111

C₂×C₄×D₄

Dic₆₇

yes

characteristic

112

C₈×D₄

—

113

C₈×D₄

—

114

C₂×C₄⋊C₈

Dic₆₇

yes

characteristic

115

C₂×C₂²⋊C₈

C₂×D₆₇

yes

—

116

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

117

C₄⋊C₁₆

—

134

—

118

C₂²⋊C₁₆

—

119

C₄×C₁₆

—

134

—

120

C₂²×D₄

C₆₇⋊C₈

yes

characteristic

121

C₂³×C₄

C₂×Dic₆₇

yes

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

126

C₂²×C₈

C₂×Dic₆₇

yes

—

127

C₂²×C₈

C₂²×D₆₇

yes

characteristic

128

C₄×D₄

—

129

C₄×D₄

—

130

C₂×C₄⋊C₄

C₆₇⋊C₈

yes

characteristic

131

C₂×C₂²⋊C₄

C₆₇⋊C₈

yes

—

132

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

133

C₂×C₁₆

—

134

—

134

C₄⋊C₈

—

135

C₂²⋊C₈

—

136

C₄×C₈

C₂²⋊D₆₇

yes

—

137

C₄×C₈

C₂²⋊D₆₇

yes

—

138

C₂⁴

C₂×C₆₇⋊C₈

yes

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

C₆₇⋊C₁₆

yes

complemented

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

147

C₂²×C₄

C₂×C₆₇⋊C₈

yes

characteristic

148

C₂²×C₄

C₂×C₆₇⋊C₈

yes

characteristic

149

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

157

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

158

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²⋊Dic₆₇

yes

—

162

C₄²

#11761612

yes

—

163

C₁₆

—

268

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

C₂×C₆₇⋊C₁₆

yes

—

169

C₂³

C₂²×C₆₇⋊C₈

yes

characteristic

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

180

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

181

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

182

C₂×C₄

#14523983

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

184

C₂×C₄

C₂×#11761612

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

#14523960

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂×C₂²⋊(C₆₇⋊C₈)

yes

characteristic

197

C₂²

C₂²×C₆₇⋊C₁₆

yes

characteristic

198

C₂²

C₂×#14523983

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

C₂²⋊(C₆₇⋊C₁₆)

yes

—

203

C₄

#16863370

yes

characteristic

204

C₄

C₂×#14523960

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

#22558530

yes

characteristic

209

C₂

C₂×C₂²⋊(C₆₇⋊C₁₆)

yes

characteristic

210

C₂

#22561830

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₈×D₄×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₆₇	C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₈×D₄	D₆₇
hypocenter	C₆₇	#11487
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₆₇

commutator

C₂²×C₆₇

C₂²×C₁₆

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×C₈×D₄×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

C₂²×D₄×C₆₇

C₈

cosocle

C₂²×C₈×C₆₇

C₂³

hypercenter

C₂×C₈×D₄

D₆₇

hypocenter

C₆₇

#11487

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

N	H ≅ G/N	type
1. #22562008	C₂	semidirect
2. #22558133	C₂	semidirect
36. C₂×D₄×C₆₇	C₁₆	semidirect
109. C₆₇	#11487	semidirect
141. C₂×D₄	C₆₇⋊C₁₆	semidirect

H ≅ G/N

type

1. #22562008

C₂

semidirect

2. #22558133

C₂

semidirect

36. C₂×D₄×C₆₇

C₁₆

semidirect

109. C₆₇

#11487

semidirect

141. C₂×D₄

C₆₇⋊C₁₆

semidirect

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

subgroup	index
#22562008	2
#22558133	2
C₂×C₈×D₄×C₆₇	2
#11487	67

subgroup

index

#22562008

#22558133

C₂×C₈×D₄×C₆₇

#11487

Subgroups — #30765148

Conjugacy classes of subgroups

Complements

Maximal subgroups