Group #30746396 — subgroups

Subgroups — #101×C₆₇⋊C₈ #30746396

Order 17152.

454 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 193 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	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22558533	C₂	1	1	yes	maximal, characteristic
3	M₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal
4	#22558060	C₂	1	2	yes	maximal
5	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863449	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863122	C₄	1	4	yes	—
14	#16863117	—	2	2	no	—
15	C₈×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#101×C₆₇	C₄	1	2	yes	—
18	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
19	#343×C₆₇	C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#101×C₆₇	C₈	1	8	yes	complemented
41	#97×C₆₇	—	2	2	no	—
42	C₄×C₈×C₆₇	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	4	1	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
50	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	2	2	no	—
53	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
55	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	4	no	—
60	M₁₆×C₆₇	C₂×C₈	1	8	yes	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	C₂×D₄	1	2	yes	—
67	C₄²×C₆₇	#41	1	2	yes	—
68	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₆₇	—	2	8	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	1	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	4	1	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₆₇	#131	1	2	yes	—
84	C₈×C₆₇	—	2	8	no	—
85	C₆₇⋊C₈	—	2	1	no	—
86	C₆₇⋊C₈	—	2	2	no	—
87	C₆₇⋊C₈	—	4	4	no	—
88	C₆₇⋊C₈	—	4	2	no	—
89	C₆₇⋊C₈	#101	1	2	yes	direct product factor
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	—	2	1	no	—
92	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₆₇	#405	1	1	yes	characteristic
95	C₄×C₆₇	—	2	4	no	—
96	C₄×C₆₇	—	2	1	no	—
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	4	2	no	—
99	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
100	C₄×C₆₇	#413	1	1	yes	characteristic
101	C₄×C₆₇	C₈×D₄	1	4	yes	—
102	C₈×#101	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	1	no	—
104	C₂×C₆₇	—	2	1	no	—
105	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₆₇	C₄×#101	1	1	yes	characteristic
107	C₂×C₆₇	#1757	1	1	yes	characteristic
108	C₂×C₈×Q₈	—	67	1	no	—
109	C₄×#101	D₆₇	1	1	yes	characteristic
110	#1461	—	67	1	no	—
111	C₈×M₁₆	—	67	2	no	—
112	#1283	—	67	2	no	—
113	C₆₇	C₈×#101	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
115	C₈×Q₈	—	134	2	no	—
116	C₈×Q₈	—	134	2	no	—
117	#431	—	67	2	no	—
118	C₂×C₄⋊C₈	—	67	2	no	—
119	C₂×C₄⋊C₈	—	67	1	no	—
120	C₂×#101	Dic₆₇	1	2	yes	—
121	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
122	C₂×C₄×C₈	—	67	1	no	—
123	C₂×C₄×C₈	—	67	2	no	—
124	#343	Dic₆₇	1	2	yes	—
125	#324	—	67	4	no	—
126	#322	—	134	2	no	—
127	C₈²	—	134	2	no	—
128	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
129	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
130	C₂²×C₈	—	67	1	no	—
131	C₂²×C₈	—	67	2	no	—
132	C₂²×C₈	—	67	1	no	—
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
137	C₂×C₄²	C₄×D₆₇	1	2	yes	—
138	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
139	C₄⋊C₈	—	134	4	no	—
140	C₄⋊C₈	—	134	2	no	—
141	C₄⋊C₈	—	134	2	no	—
142	#101	C₆₇⋊C₈	1	8	yes	direct product factor
143	C₂²⋊C₈	—	67	4	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	67	2	no	—
147	C₄×C₈	—	67	2	no	—
148	C₄×C₈	—	134	2	no	—
149	C₄×C₈	—	134	2	no	—
150	C₄×C₈	—	134	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
154	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
155	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	C₂×C₆₇⋊C₈	1	8	yes	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	268	2	no	—
163	C₂×C₈	—	268	1	no	—
164	C₂×C₈	—	67	1	no	—
165	C₂×C₈	—	67	1	no	—
166	C₂×C₈	—	67	2	no	—
167	C₂×C₈	—	134	2	no	—
168	C₂×C₈	—	134	4	no	—
169	C₂×C₈	—	134	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	D₄×D₆₇	1	2	yes	—
175	C₄²	#11761627	1	2	yes	—
176	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
186	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
187	C₂×C₄	#14523999	1	1	yes	characteristic
188	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
189	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
190	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
191	C₂×C₄	#14524050	1	2	yes	—
192	C₈	—	2	8	no	—
193	C₈	—	268	4	no	—
194	C₈	—	268	2	no	—
195	C₈	—	67	2	no	—
196	C₈	—	134	1	no	—
197	C₈	—	134	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
201	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
202	C₂²	#16863352	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	—	4	2	no	—
207	C₄	#101×D₆₇	1	1	yes	characteristic
208	C₄	#16863379	1	1	yes	characteristic
209	C₄	D₄×C₆₇⋊C₈	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	C₂²⋊C₄×C₆₇⋊C₈	1	1	yes	characteristic
213	C₂	#101×Dic₆₇	1	1	yes	characteristic
214	C₂	#22560012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#22558533

C₂

yes

maximal, characteristic

M₁₆×C₆₇⋊C₈

C₂

yes

maximal

#22558060

C₂

yes

maximal

C₄×#101×C₆₇

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

—

Q₈×C₆₇⋊C₈

—

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863122

C₄

yes

—

#16863117

—

C₈×C₆₇⋊C₈

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₂×#101×C₆₇

C₄

yes

—

C₄×M₁₆×C₆₇

C₂²

yes

—

#343×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂²×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#101×C₆₇

C₈

yes

complemented

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄²

yes

characteristic

Q₈×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

C₄×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

direct product factor

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₇

#405

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#101

yes

characteristic

100

C₄×C₆₇

#413

yes

characteristic

101

C₄×C₆₇

C₈×D₄

yes

—

102

C₈×#101

—

maximal, Sylow

103

C₂×C₆₇

—

104

C₂×C₆₇

—

105

C₂×C₆₇

C₈×C₂²⋊C₄

yes

characteristic

106

C₂×C₆₇

C₄×#101

yes

characteristic

107

C₂×C₆₇

#1757

yes

characteristic

108

C₂×C₈×Q₈

—

109

C₄×#101

D₆₇

yes

characteristic

110

#1461

—

111

C₈×M₁₆

—

112

#1283

—

113

C₆₇

C₈×#101

yes

characteristic, complemented, Sylow

114

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

115

C₈×Q₈

—

134

—

116

C₈×Q₈

—

134

—

117

#431

—

118

C₂×C₄⋊C₈

—

119

C₂×C₄⋊C₈

—

120

C₂×#101

Dic₆₇

yes

—

121

C₄×M₁₆

C₂×D₆₇

yes

—

122

C₂×C₄×C₈

—

123

C₂×C₄×C₈

—

124

#343

Dic₆₇

yes

—

125

#324

—

126

#322

—

134

—

127

C₈²

—

134

—

128

C₂²×Q₈

C₂×Dic₆₇

yes

characteristic

129

C₂×M₁₆

C₂×Dic₆₇

yes

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₄×Q₈

—

134

C₄×Q₈

—

135

C₂×C₄⋊C₄

C₄×D₆₇

yes

—

136

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

characteristic

137

C₂×C₄²

C₄×D₆₇

yes

—

138

C₂×C₄²

C₂²×D₆₇

yes

characteristic

139

C₄⋊C₈

—

134

—

140

C₄⋊C₈

—

134

—

141

C₄⋊C₈

—

134

—

142

#101

C₆₇⋊C₈

yes

direct product factor

143

C₂²⋊C₈

—

144

#97

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

134

—

149

C₄×C₈

—

134

—

150

C₄×C₈

—

134

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

C₂×C₆₇⋊C₈

yes

—

154

C₂²×C₄

C₄×Dic₆₇

yes

—

155

C₂²×C₄

C₄×Dic₆₇

yes

—

156

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

157

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

158

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

159

M₁₆

—

160

M₁₆

C₂×C₆₇⋊C₈

yes

—

161

C₂×C₈

—

162

C₂×C₈

—

268

—

163

C₂×C₈

—

268

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

134

—

168

C₂×C₈

—

134

—

169

C₂×C₈

—

134

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₄²

D₄×D₆₇

yes

—

175

C₄²

#11761627

yes

—

176

C₂³

C₂×C₄×Dic₆₇

yes

characteristic

177

Q₈

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₄×C₆₇⋊C₈

yes

—

186

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

187

C₂×C₄

#14523999

yes

characteristic

188

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

189

C₂×C₄

D₄×Dic₆₇

yes

—

190

C₂×C₄

C₂²×C₆₇⋊C₈

yes

—

191

C₂×C₄

#14524050

yes

—

192

C₈

—

193

C₈

—

268

—

194

C₈

—

268

—

195

C₈

—

196

C₈

—

134

—

197

C₈

—

134

—

198

C₂²

—

199

C₂²

—

200

C₂²

C₂²⋊C₄×Dic₆₇

yes

characteristic

201

C₂²

C₂×C₄×C₆₇⋊C₈

yes

characteristic

202

C₂²

#16863352

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

#101×D₆₇

yes

characteristic

208

C₄

#16863379

yes

characteristic

209

C₄

D₄×C₆₇⋊C₈

yes

—

210

C₂

—

211

C₂

—

212

C₂

C₂²⋊C₄×C₆₇⋊C₈

yes

characteristic

213

C₂

#101×Dic₆₇

yes

characteristic

214

C₂

#22560012

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#101×C₆₇	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#101	D₆₇
hypocenter	C₆₇	C₈×#101
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₆₇

commutator

C₂²×C₆₇

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#101×C₆₇

C₂

socle

C₂²×C₆₇

C₄×C₂²⋊C₄

prime radical

C₂³×C₆₇

C₂×C₄²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

N	H ≅ G/N	type
40. #101×C₆₇	C₈	semidirect
89. C₆₇⋊C₈	#101	direct (N × H)
113. C₆₇	C₈×#101	semidirect
142. #101	C₆₇⋊C₈	direct (N × H)

H ≅ G/N

type

40. #101×C₆₇

C₈

semidirect

89. C₆₇⋊C₈

#101

direct (N × H)

113. C₆₇

C₈×#101

semidirect

142. #101

C₆₇⋊C₈

direct (N × H)

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
#22558533	2
M₁₆×C₆₇⋊C₈	2
#22558060	2
C₄×#101×C₆₇	2
C₈×#101	67

subgroup

index

C₂×Q₈×C₆₇⋊C₈

#22558533

M₁₆×C₆₇⋊C₈

#22558060

C₄×#101×C₆₇

C₈×#101

Subgroups — #101×C₆₇⋊C₈ #30746396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×C67⋊C8 #30746396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #101×C₆₇⋊C₈ #30746396