Group #30758844 — subgroups

Subgroups — #30758844

Order 17152.

504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22565238	C₂	1	1	yes	maximal, characteristic
2	#22561158	C₂	1	2	yes	maximal
3	#22558092	C₂	1	1	yes	maximal, characteristic
4	#22558049	C₂	1	2	yes	maximal
5	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524042	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863316	—	2	2	no	—
9	#16863292	—	2	2	no	—
10	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	#16863158	—	2	2	no	—
12	#16863158	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#103×C₆₇	—	2	2	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	#14524042	—	4	4	no	—
20	#14524042	—	4	4	no	—
21	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
22	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
23	C₂×Dic₆₇⋊C₄	—	2	4	no	—
24	C₂×C₄×Dic₆₇	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₆₇⋊#95	—	2	2	no	—
28	C₆₇⋊#95	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	1	no	—
33	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	complemented
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	#103×C₆₇	—	4	4	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₂×Dic₁₃₄	—	4	8	no	—
41	C₂×Dic₁₃₄	—	8	2	no	—
42	C₂²×Dic₆₇	—	2	2	no	—
43	C₂²×Dic₆₇	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₄⋊Dic₆₇	—	2	2	no	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	SD₁₆	1	4	yes	—
50	C₄⋊Dic₆₇	Q₁₆	1	4	yes	—
51	Dic₆₇⋊C₄	—	4	8	no	—
52	C₄×Dic₆₇	—	4	2	no	—
53	C₄×Dic₆₇	—	4	2	no	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	4	2	no	—
57	C₂×Q₈×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	4	1	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	4	1	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₆₇	#41	1	2	yes	—
72	C₄²×C₆₇	#41	1	1	yes	characteristic
73	C₂³×C₆₇	#120	1	1	yes	characteristic
74	Q₈×C₆₇	—	4	4	no	—
75	Q₈×C₆₇	—	8	2	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	—	8	1	no	—
82	C₂×C₄×C₆₇	—	8	1	no	—
83	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#124	1	2	yes	—
86	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₆₇	#137	1	2	yes	—
91	C₂×Dic₆₇	—	2	4	no	—
92	C₂×Dic₆₇	—	2	4	no	—
93	C₂×Dic₆₇	—	4	2	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	C₂×Dic₆₇	—	4	2	no	—
97	Dic₁₃₄	—	8	8	no	—
98	C₈×C₆₇	—	4	2	no	—
99	C₆₇⋊C₈	—	8	2	no	—
100	C₂²×C₆₇	#393	1	1	yes	characteristic
101	C₂²×C₆₇	#450	1	2	yes	—
102	C₂²×C₆₇	#451	1	2	yes	—
103	C₂²×C₆₇	#453	1	1	yes	characteristic
104	C₂²×C₆₇	#454	1	1	yes	characteristic
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	8	2	no	—
108	C₄×C₆₇	—	8	2	no	—
109	C₄×C₆₇	#475	1	2	yes	—
110	C₄×C₆₇	#477	1	2	yes	—
111	C₄×C₆₇	#487	1	2	yes	—
112	C₄×C₆₇	#489	1	2	yes	—
113	Dic₆₇	—	4	4	no	—
114	Dic₆₇	—	4	4	no	—
115	#10349	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	#1629	1	1	yes	characteristic
117	C₂×C₆₇	#1633	1	1	yes	characteristic
118	C₂×C₆₇	#1637	1	2	yes	—
119	C₂×C₆₇	#2024	1	2	yes	—
120	C₂×C₆₇	#2027	1	1	yes	characteristic
121	#2551	—	67	1	no	—
122	#1987	—	67	2	no	—
123	#1956	D₆₇	1	1	yes	characteristic
124	#1296	—	67	1	no	—
125	#1279	—	67	2	no	—
126	C₆₇	#10349	1	1	yes	characteristic, complemented, Sylow
127	C₂×C₄⋊Q₈	—	134	2	no	—
128	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	67	2	no	—
130	C₂×#103	—	2	2	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	#395	—	134	2	no	—
133	#389	—	134	2	no	—
134	#384	C₂×D₆₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	67	2	no	—
136	#340	—	134	2	no	—
137	#340	—	134	2	no	—
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	—	134	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	134	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	1	no	—
144	C₄⋊Q₈	—	268	4	no	—
145	C₄⋊Q₈	—	268	4	no	—
146	C₄⋊Q₈	Dic₁₃₄	1	2	yes	complemented
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	67	2	no	—
150	C₂×C₄⋊C₄	—	67	2	no	—
151	C₂×C₄⋊C₄	—	134	4	no	—
152	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
153	C₂×C₄²	—	134	2	no	—
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	C₄⋊C₈	—	134	2	no	—
156	C₄⋊C₈	—	134	2	no	—
157	#103	—	4	4	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	134	2	no	—
160	#95	—	134	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	268	8	no	—
166	C₂×Q₈	—	536	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	134	2	no	—
170	C₂²×C₄	—	134	2	no	—
171	C₂²×C₄	D₄×D₆₇	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	268	2	no	—
177	C₂×C₈	—	268	2	no	—
178	C₂×C₈	—	268	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	268	8	no	—
185	C₄⋊C₄	—	67	4	no	—
186	C₄⋊C₄	—	67	4	no	—
187	C₄⋊C₄	—	134	2	no	—
188	C₄⋊C₄	—	134	2	no	—
189	C₄²	—	268	2	no	—
190	C₄²	—	268	2	no	—
191	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
192	C₄²	#11761627	1	2	yes	—
193	C₄²	#11761628	1	1	yes	characteristic
194	C₂³	#14524011	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	Q₈	—	8	2	no	—
197	Q₈	—	536	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	268	2	no	—
206	C₂×C₄	—	268	2	no	—
207	C₂×C₄	—	268	2	no	—
208	C₂×C₄	—	268	2	no	—
209	C₂×C₄	—	134	4	no	—
210	C₂×C₄	—	134	4	no	—
211	C₂×C₄	#14524020	1	2	yes	—
212	C₂×C₄	#14524022	1	1	yes	characteristic
213	C₂×C₄	#14524029	1	2	yes	—
214	C₂×C₄	#14524030	1	1	yes	characteristic
215	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
216	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
217	C₂×C₄	#14524070	1	2	yes	—
218	C₂×C₄	#14524075	1	2	yes	—
219	C₈	—	4	2	no	—
220	C₈	—	536	2	no	—
221	C₂²	#16863308	1	1	yes	characteristic
222	C₂²	#16863545	1	2	yes	—
223	C₂²	#16863550	1	2	yes	—
224	C₂²	#16863563	1	1	yes	characteristic
225	C₂²	#16863568	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	—	268	4	no	—
231	C₄	—	268	4	no	—
232	C₄	#16863679	1	2	yes	—
233	C₄	#16863689	1	2	yes	—
234	C₄	#16863745	1	2	yes	—
235	C₄	#16863757	1	2	yes	—
236	C₂	#22559323	1	1	yes	characteristic
237	C₂	#22559347	1	1	yes	characteristic
238	C₂	#22559371	1	2	yes	—
239	C₂	#22561348	1	2	yes	—
240	C₂	#22561365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22565238

C₂

yes

maximal, characteristic

#22561158

C₂

yes

maximal

#22558092

C₂

yes

maximal, characteristic

#22558049

C₂

yes

maximal

C₆₇×#1956

C₂

yes

maximal, characteristic

C₂×#14524042

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863316

—

#16863292

—

C₄×C₄⋊Dic₆₇

C₂²

yes

—

#16863158

—

#16863158

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×#103×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

—

#14524042

—

#14524042

—

C₂×C₄⋊Dic₆₇

D₄

yes

—

C₂×C₄⋊Dic₆₇

D₄

yes

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

Q₈

yes

complemented

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#103×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

SD₁₆

yes

—

C₄⋊Dic₆₇

Q₁₆

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

100

C₂²×C₆₇

#393

yes

characteristic

101

C₂²×C₆₇

#450

yes

—

102

C₂²×C₆₇

#451

yes

—

103

C₂²×C₆₇

#453

yes

characteristic

104

C₂²×C₆₇

#454

yes

characteristic

105

C₄×C₆₇

—

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

#475

yes

—

110

C₄×C₆₇

#477

yes

—

111

C₄×C₆₇

#487

yes

—

112

C₄×C₆₇

#489

yes

—

113

Dic₆₇

—

114

Dic₆₇

—

115

#10349

—

maximal, Sylow

116

C₂×C₆₇

#1629

yes

characteristic

117

C₂×C₆₇

#1633

yes

characteristic

118

C₂×C₆₇

#1637

yes

—

119

C₂×C₆₇

#2024

yes

—

120

C₂×C₆₇

#2027

yes

characteristic

121

#2551

—

122

#1987

—

123

#1956

D₆₇

yes

characteristic

124

#1296

—

125

#1279

—

126

C₆₇

#10349

yes

characteristic, complemented, Sylow

127

C₂×C₄⋊Q₈

—

134

—

128

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

129

C₂×C₄⋊C₈

—

130

C₂×#103

—

131

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

132

#395

—

134

—

133

#389

—

134

—

134

#384

C₂×D₆₇

yes

characteristic

135

C₄×C₄⋊C₄

—

136

#340

—

134

—

137

#340

—

134

—

138

C₂²×Q₈

—

139

C₂²×Q₈

—

134

—

140

C₂²×C₈

—

141

C₂²×C₈

—

134

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

268

—

145

C₄⋊Q₈

—

268

—

146

C₄⋊Q₈

Dic₁₃₄

yes

complemented

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

134

—

152

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

153

C₂×C₄²

—

134

—

154

C₂×C₄²

C₂²×D₆₇

yes

characteristic

155

C₄⋊C₈

—

134

—

156

C₄⋊C₈

—

134

—

157

#103

—

158

C₄×C₈

—

159

#95

—

134

—

160

#95

—

134

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

268

—

166

C₂×Q₈

—

536

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

134

—

170

C₂²×C₄

—

134

—

171

C₂²×C₄

D₄×D₆₇

yes

—

172

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

268

—

177

C₂×C₈

—

268

—

178

C₂×C₈

—

268

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

268

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

134

—

188

C₄⋊C₄

—

134

—

189

C₄²

—

268

—

190

C₄²

—

268

—

191

C₄²

C₂×Dic₁₃₄

yes

characteristic

192

C₄²

#11761627

yes

—

193

C₄²

#11761628

yes

characteristic

194

C₂³

#14524011

yes

characteristic

195

Q₈

—

196

Q₈

—

197

Q₈

—

536

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

268

—

206

C₂×C₄

—

268

—

207

C₂×C₄

—

268

—

208

C₂×C₄

—

268

—

209

C₂×C₄

—

134

—

210

C₂×C₄

—

134

—

211

C₂×C₄

#14524020

yes

—

212

C₂×C₄

#14524022

yes

characteristic

213

C₂×C₄

#14524029

yes

—

214

C₂×C₄

#14524030

yes

characteristic

215

C₂×C₄

SD₁₆×D₆₇

yes

—

216

C₂×C₄

Q₁₆×D₆₇

yes

—

217

C₂×C₄

#14524070

yes

—

218

C₂×C₄

#14524075

yes

—

219

C₈

—

220

C₈

—

536

—

221

C₂²

#16863308

yes

characteristic

222

C₂²

#16863545

yes

—

223

C₂²

#16863550

yes

—

224

C₂²

#16863563

yes

characteristic

225

C₂²

#16863568

yes

characteristic

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

268

—

231

C₄

—

268

—

232

C₄

#16863679

yes

—

233

C₄

#16863689

yes

—

234

C₄

#16863745

yes

—

235

C₄

#16863757

yes

—

236

C₂

#22559323

yes

characteristic

237

C₂

#22559347

yes

characteristic

238

C₂

#22559371

yes

—

239

C₂

#22561348

yes

—

240

C₂

#22561365

yes

characteristic

	subgroup	G/H
center	C₂³	#14524011
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1956	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#1956	D₆₇
hypocenter	C₆₇	#10349
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14524011

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1956

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
33. C₄⋊Q₈×C₆₇	Q₈	semidirect
126. C₆₇	#10349	semidirect
146. C₄⋊Q₈	Dic₁₃₄	semidirect

H ≅ G/N

type

33. C₄⋊Q₈×C₆₇

Q₈

semidirect

126. C₆₇

#10349

semidirect

146. C₄⋊Q₈

Dic₁₃₄

semidirect

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

subgroup	index
#22565238	2
#22561158	2
#22558092	2
#22558049	2
C₆₇×#1956	2
#10349	67

subgroup

index

C₆₇×#1956

Subgroups — #30758844

Conjugacy classes of subgroups

Complements

Maximal subgroups