Group #30749700 — subgroups

Subgroups — #30749700

Order 17152.

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

normal subgroups only

This group has 410 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	#22569290	C₂	1	1	yes	maximal, characteristic
2	#22561146	C₂	1	1	yes	maximal, characteristic
3	#22558961	C₂	1	2	yes	maximal
4	#22558047	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	—	2	2	no	—
9	#16863449	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#16863151	—	2	2	no	—
13	#16863149	—	2	2	no	—
14	#16863147	—	2	2	no	—
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#101×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#101×C₆₇	C₄	1	1	yes	characteristic
18	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
19	#343×C₆₇	C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	—	2	1	no	—
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	4	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#101×C₆₇	—	2	2	no	—
41	#101×C₆₇	C₈	1	4	yes	complemented
42	#97×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	4	1	no	—
48	C₂×C₆₇⋊C₈	—	4	1	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	1	no	—
51	C₂×Q₈×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	2	2	no	—
53	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
54	C₂×Q₈×C₆₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	M₁₆×C₆₇	—	2	2	no	—
61	M₁₆×C₆₇	—	2	2	no	—
62	M₁₆×C₆₇	—	2	2	no	—
63	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	C₂×D₄	1	2	yes	—
71	C₄²×C₆₇	#41	1	2	yes	—
72	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈×C₆₇	—	2	4	no	—
74	Q₈×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	4	1	no	—
81	C₂×C₄×C₆₇	—	4	2	no	—
82	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
83	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#131	1	2	yes	—
90	C₈×C₆₇	—	2	4	no	—
91	C₈×C₆₇	—	4	2	no	—
92	C₆₇⋊C₈	—	4	2	no	—
93	C₆₇⋊C₈	—	4	2	no	—
94	C₆₇⋊C₈	—	4	2	no	—
95	C₆₇⋊C₈	—	8	1	no	—
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	#385	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₆₇	#424	1	1	yes	characteristic
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	C₈×D₄	1	2	yes	—
106	C₄×C₆₇	#435	1	2	yes	—
107	C₄×C₆₇	#454	1	1	yes	characteristic
108	C₄×C₆₇	#456	1	1	yes	characteristic
109	#8836	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	—	2	1	no	—
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	#1850	1	1	yes	characteristic
113	C₂×C₆₇	#1857	1	1	yes	characteristic
114	C₂×C₆₇	#1858	1	1	yes	characteristic
115	#2982	—	67	1	no	—
116	#1984	—	67	1	no	—
117	C₄×#101	D₆₇	1	1	yes	characteristic
118	#1565	—	67	2	no	—
119	#1278	—	67	2	no	—
120	C₆₇	#8836	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
122	C₈×Q₈	—	134	2	no	—
123	C₈×Q₈	—	134	2	no	—
124	#431	—	67	2	no	—
125	#431	—	134	2	no	—
126	C₂×C₄⋊C₈	—	67	1	no	—
127	C₂×#101	Dic₆₇	1	1	yes	characteristic
128	C₂×#101	Dic₆₇	1	1	yes	characteristic
129	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
130	C₂×C₄×C₈	—	67	1	no	—
131	#343	Dic₆₇	1	2	yes	—
132	#336	—	134	2	no	—
133	#335	—	134	2	no	—
134	#334	—	134	2	no	—
135	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
136	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
137	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
138	C₂²×C₈	—	134	1	no	—
139	C₂²×C₈	—	134	1	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂×C₄²	D₂₆₈	1	2	yes	—
145	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
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	#101	—	2	2	no	—
152	#101	C₆₇⋊C₈	1	4	yes	complemented
153	C₂²⋊C₈	—	134	2	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	134	4	no	—
157	C₄×C₈	—	134	2	no	—
158	C₄×C₈	—	134	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
162	C₂×Q₈	#11761612	1	2	yes	—
163	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	2	2	no	—
170	M₁₆	—	2	2	no	—
171	M₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	268	1	no	—
175	C₂×C₈	—	268	1	no	—
176	C₂×C₈	—	268	2	no	—
177	C₂×C₈	—	268	1	no	—
178	C₂×C₈	—	134	2	no	—
179	C₂×C₈	—	134	2	no	—
180	C₂×C₈	—	134	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	D₄×D₆₇	1	2	yes	—
186	C₄²	#11761627	1	2	yes	—
187	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	4	yes	—
198	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
199	C₂×C₄	#14524013	1	1	yes	characteristic
200	C₂×C₄	#14524020	1	2	yes	—
201	C₂×C₄	#14524027	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
204	C₂×C₄	#14524050	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	4	2	no	—
207	C₈	—	268	2	no	—
208	C₈	—	268	2	no	—
209	C₈	—	268	2	no	—
210	C₈	—	536	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16863271	1	1	yes	characteristic
214	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
215	C₂²	#16863423	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	2	no	—
220	C₄	D₄×C₆₇⋊C₈	1	2	yes	—
221	C₄	#16863468	1	2	yes	—
222	C₄	#16863571	1	1	yes	characteristic
223	C₄	#16863581	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	#22560389	1	1	yes	characteristic
227	C₂	#22560426	1	1	yes	characteristic
228	C₂	#22560431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569290

C₂

yes

maximal, characteristic

#22561146

C₂

yes

maximal, characteristic

#22558961

C₂

yes

maximal

#22558047

C₂

yes

maximal

C₄×#101×C₆₇

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

—

Q₈×C₆₇⋊C₈

—

#16863449

—

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₂×#101×C₆₇

C₄

yes

characteristic

C₂×#101×C₆₇

C₄

yes

characteristic

C₄×M₁₆×C₆₇

C₂²

yes

—

#343×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#101×C₆₇

—

#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

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

—

C₂×Q₈×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

—

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×C₄⋊C₄

yes

characteristic

Q₈×C₆₇

—

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

C₄×D₄

yes

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#385

yes

characteristic

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

100

C₂²×C₆₇

#424

yes

characteristic

101

C₄×C₆₇

—

102

C₄×C₆₇

—

103

C₄×C₆₇

—

104

C₄×C₆₇

—

105

C₄×C₆₇

C₈×D₄

yes

—

106

C₄×C₆₇

#435

yes

—

107

C₄×C₆₇

#454

yes

characteristic

108

C₄×C₆₇

yes

109

—

110

C₂×C₆₇

—

111

C₂×C₆₇

—

112

C₂×C₆₇

#1850

yes

characteristic

113

C₂×C₆₇

#1857

yes

characteristic

114

C₂×C₆₇

#1858

yes

characteristic

115

#2982

—

116

#1984

—

117

C₄×#101

D₆₇

yes

characteristic

118

#1565

—

119

#1278

—

120

C₆₇

#8836

yes

characteristic, complemented, Sylow

121

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

122

C₈×Q₈

—

134

—

123

C₈×Q₈

—

134

—

124

#431

—

125

#431

—

134

—

126

C₂×C₄⋊C₈

—

127

C₂×#101

Dic₆₇

yes

characteristic

128

C₂×#101

Dic₆₇

yes

characteristic

129

C₄×M₁₆

C₂×D₆₇

yes

—

130

C₂×C₄×C₈

—

131

#343

Dic₆₇

yes

—

132

#336

—

134

—

133

#335

—

134

—

134

#334

—

134

—

135

C₂²×Q₈

C₂×Dic₆₇

yes

characteristic

136

C₂×M₁₆

C₂×Dic₆₇

yes

—

137

C₂×M₁₆

C₂×Dic₆₇

yes

—

138

C₂²×C₈

—

134

—

139

C₂²×C₈

—

134

—

140

C₄×Q₈

—

141

C₄×Q₈

—

142

C₂×C₄⋊C₄

Dic₁₃₄

yes

—

143

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

characteristic

144

C₂×C₄²

D₂₆₈

yes

—

145

C₂×C₄²

C₂²×D₆₇

yes

characteristic

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

134

—

149

C₄⋊C₈

—

134

—

150

C₄⋊C₈

—

134

—

151

#101

—

152

#101

C₆₇⋊C₈

yes

complemented

153

C₂²⋊C₈

—

134

—

154

#97

—

155

C₄×C₈

—

156

C₄×C₈

—

134

—

157

C₄×C₈

—

134

—

158

C₄×C₈

—

134

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

C₂×C₆₇⋊C₈

yes

—

162

C₂×Q₈

#11761612

yes

—

163

C₂²×C₄

C₄⋊Dic₆₇

yes

—

164

C₂²×C₄

C₄⋊Dic₆₇

yes

—

165

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

166

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

167

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

168

M₁₆

—

169

M₁₆

—

170

M₁₆

—

171

M₁₆

C₂×C₆₇⋊C₈

yes

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

268

—

175

C₂×C₈

—

268

—

176

C₂×C₈

—

268

—

177

C₂×C₈

—

268

—

178

C₂×C₈

—

134

—

179

C₂×C₈

—

134

—

180

C₂×C₈

—

134

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

D₄×D₆₇

yes

—

186

C₄²

#11761627

yes

—

187

C₂³

C₂×C₄⋊Dic₆₇

yes

characteristic

188

Q₈

—

189

Q₈

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

C₄⋊(C₆₇⋊C₈)

yes

—

198

C₂×C₄

D₄×Dic₆₇

yes

—

199

C₂×C₄

#14524013

yes

characteristic

200

C₂×C₄

#14524020

yes

—

201

C₂×C₄

#14524027

yes

characteristic

202

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

203

C₂×C₄

C₂×#11761612

yes

characteristic

204

C₂×C₄

#14524050

yes

—

205

C₈

—

206

C₈

—

207

C₈

—

268

—

208

C₈

—

268

—

209

C₈

—

268

—

210

C₈

—

536

—

211

C₂²

—

212

C₂²

—

213

C₂²

#16863271

yes

characteristic

214

C₂²

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

215

C₂²

#16863423

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

D₄×C₆₇⋊C₈

yes

—

221

C₄

#16863468

yes

—

222

C₄

#16863571

yes

characteristic

223

C₄

#16863581

yes

characteristic

224

C₂

—

225

C₂

—

226

C₂

#22560389

yes

characteristic

227

C₂

#22560426

yes

characteristic

228

C₂

#22560431

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524013
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#101×C₆₇	C₂
socle	C₂²×C₆₇	#385
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#101	D₆₇
hypocenter	C₆₇	#8836
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

#14524013

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#101×C₆₇

C₂

socle

C₂²×C₆₇

#385

prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

N	H ≅ G/N	type
41. #101×C₆₇	C₈	semidirect
120. C₆₇	#8836	semidirect
152. #101	C₆₇⋊C₈	semidirect

H ≅ G/N

type

41. #101×C₆₇

C₈

semidirect

120. C₆₇

#8836

semidirect

152. #101

C₆₇⋊C₈

semidirect

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

subgroup	index
#22569290	2
#22561146	2
#22558961	2
#22558047	2
C₄×#101×C₆₇	2
#8836	67

subgroup

index

C₄×#101×C₆₇

Subgroups — #30749700

Conjugacy classes of subgroups

Complements

Maximal subgroups