Group #30773281 — subgroups

Subgroups — #30773281

Order 17152.

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

normal subgroups only

This group has 490 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	#22573312	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561925	C₂	1	2	yes	maximal, complemented
3	#22560255	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558310	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3058	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	4	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863434	C₂²	1	2	yes	—
10	#16863370	—	2	1	no	—
11	#16863197	—	2	4	no	—
12	#16863153	—	2	1	no	—
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#535×C₆₇	—	2	1	no	—
16	#469×C₆₇	C₄	1	4	yes	complemented
17	#466×C₆₇	—	2	2	no	—
18	#427×C₆₇	C₂²	1	1	yes	characteristic
19	#418×C₆₇	—	2	1	no	—
20	#14524002	—	2	2	no	—
21	#14523983	—	2	4	no	—
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	4	no	—
25	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	complemented
26	#14523977	—	2	2	no	—
27	#14523975	—	2	2	no	—
28	#14523958	—	4	2	no	—
29	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂×D₈×C₆₇	—	2	2	no	—
32	C₂×D₈×C₆₇	—	2	1	no	—
33	C₂×D₈×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
35	#131×C₆₇	—	2	2	no	—
36	#131×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#127×C₆₇	—	4	1	no	—
42	#124×C₆₇	—	4	1	no	—
43	#121×C₆₇	—	2	4	no	—
44	#120×C₆₇	—	4	2	no	—
45	#117×C₆₇	—	2	1	no	—
46	#107×C₆₇	—	2	1	no	—
47	#106×C₆₇	C₂×C₄	1	2	yes	—
48	#102×C₆₇	—	2	4	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₂²⋊Dic₆₇	—	4	2	no	—
51	C₄×Dic₆₇	—	4	2	no	—
52	C₆₇⋊C₁₆	—	2	4	no	—
53	C₂⁴×C₆₇	—	4	1	no	—
54	#41×C₆₇	—	2	2	no	—
55	#41×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
57	C₂×D₄×C₆₇	—	2	2	no	—
58	C₂×D₄×C₆₇	—	4	4	no	—
59	C₂×D₄×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	4	1	no	—
61	C₂×D₄×C₆₇	—	4	2	no	—
62	C₂×D₄×C₆₇	D₈	1	2	yes	—
63	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	D₈×C₆₇	—	2	4	no	—
67	D₈×C₆₇	—	4	1	no	—
68	D₈×C₆₇	—	4	1	no	—
69	M₁₆×C₆₇	—	2	2	no	—
70	M₁₆×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	#41	1	2	yes	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₂²⋊C₄×C₆₇	—	4	1	no	—
80	C₂²⋊C₄×C₆₇	—	4	4	no	—
81	C₄²×C₆₇	—	4	1	no	—
82	C₂³×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	8	2	no	—
84	C₂³×C₆₇	—	16	2	no	—
85	C₂³×C₆₇	#102	1	2	yes	—
86	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₆₇	—	2	2	no	—
88	D₄×C₆₇	—	2	4	no	—
89	D₄×C₆₇	—	2	2	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	4	2	no	—
92	D₄×C₆₇	—	8	2	no	—
93	D₄×C₆₇	—	16	4	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	#102	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#135	1	2	yes	—
102	C₂×Dic₆₇	—	8	2	no	—
103	C₈×C₆₇	—	2	2	no	—
104	C₈×C₆₇	C₄×D₄	1	2	yes	—
105	C₈×C₆₇	#120	1	1	yes	characteristic
106	C₈×C₆₇	#123	1	1	yes	characteristic
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	4	1	no	—
109	C₂²×C₆₇	—	4	2	no	—
110	C₂²×C₆₇	—	8	4	no	—
111	C₂²×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	—	16	4	no	—
113	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
114	C₂²×C₆₇	#466	1	2	yes	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	8	2	no	—
117	C₄×C₆₇	#386	1	1	yes	characteristic
118	C₄×C₆₇	#416	1	1	yes	characteristic
119	C₄×C₆₇	#465	1	2	yes	—
120	Dic₆₇	—	16	2	no	—
121	#13074	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	2	no	—
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	4	2	no	—
125	C₂×C₆₇	—	16	2	no	—
126	C₂×C₆₇	#1893	1	1	yes	characteristic
127	#3407	—	67	1	no	—
128	#3058	D₆₇	1	1	yes	characteristic
129	#2140	—	67	2	no	—
130	#1812	—	67	1	no	—
131	#1384	—	67	2	no	—
132	C₆₇	#13074	1	1	yes	characteristic, complemented, Sylow
133	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
134	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
135	#535	—	2	1	no	—
136	#504	—	134	4	no	—
137	C₂×M₃₂	—	67	2	no	—
138	C₂×M₃₂	—	134	2	no	—
139	#469	Dic₆₇	1	4	yes	complemented
140	#466	—	2	2	no	—
141	#427	—	67	2	no	—
142	#427	C₂×D₆₇	1	1	yes	characteristic
143	#418	—	2	1	no	—
144	#410	—	134	1	no	—
145	#357	—	134	4	no	—
146	H(3,ℤ/4)	—	134	1	no	—
147	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂²×D₄	—	2	1	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
156	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	#127	—	4	1	no	—
160	#124	—	4	1	no	—
161	#121	—	2	4	no	—
162	#120	—	4	2	no	—
163	#117	—	2	1	no	—
164	#117	—	134	2	no	—
165	M₃₂	—	134	4	no	—
166	M₃₂	—	134	2	no	—
167	M₃₂	—	134	2	no	—
168	C₂×C₁₆	—	67	4	no	—
169	C₂×C₁₆	—	134	4	no	—
170	#107	—	2	1	no	—
171	#107	—	134	2	no	—
172	#106	—	134	2	no	—
173	#106	C₂×Dic₆₇	1	2	yes	—
174	#102	—	2	4	no	—
175	#99	—	268	2	no	—
176	C₂⁴	—	4	1	no	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	#11761616	1	1	yes	characteristic
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	#11761613	1	2	yes	—
186	C₂×D₄	#11761616	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	D₈	—	2	4	no	—
190	D₈	—	4	1	no	—
191	D₈	—	4	1	no	—
192	M₁₆	—	2	2	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	D₄×D₆₇	1	2	yes	—
199	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂×C₈	#11761627	1	2	yes	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	268	2	no	—
203	C₂²⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	4	4	no	—
205	C₂²⋊C₄	—	268	2	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	268	2	no	—
208	C₁₆	—	134	4	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	—	16	2	no	—
212	C₂³	#14523964	1	2	yes	—
213	C₂³	C₂×#11761613	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	D₄	—	2	4	no	—
216	D₄	—	2	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	8	2	no	—
220	D₄	—	16	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	536	2	no	—
225	C₂×C₄	#14523964	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	#14524015	1	2	yes	—
228	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
229	C₂×C₄	#14524063	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	D₄×Dic₆₇	1	2	yes	—
232	C₈	#14524011	1	1	yes	characteristic
233	C₈	#14524024	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	8	4	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	16	4	no	—
240	C₂²	C₂×#14523964	1	1	yes	characteristic
241	C₂²	#16863630	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	1072	2	no	—
245	C₄	#16863276	1	1	yes	characteristic
246	C₄	#16863392	1	1	yes	characteristic
247	C₄	#16863623	1	2	yes	—
248	C₂	—	2	1	no	—
249	C₂	—	2	2	no	—
250	C₂	—	4	2	no	—
251	C₂	—	16	2	no	—
252	C₂	#22560638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22573312

C₂

yes

maximal, characteristic, complemented

#22561925

C₂

yes

maximal, complemented

#22560255

C₂

yes

maximal, characteristic, complemented

#22558310

C₂

yes

maximal, complemented

C₆₇×#3058

C₂

yes

maximal, characteristic

#16863813

—

C₂×#14523983

—

C₂×#14523983

C₂²

yes

—

#16863434

C₂²

yes

—

#16863370

—

#16863197

—

#16863153

—

C₂²×D₈×C₆₇

C₂²

yes

characteristic

C₂×#131×C₆₇

C₂²

yes

characteristic

#535×C₆₇

—

#469×C₆₇

C₄

yes

complemented

#466×C₆₇

—

#427×C₆₇

C₂²

yes

characteristic

#418×C₆₇

—

—

—

—

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

—

—

C₂×#41×C₆₇

D₄

yes

characteristic

C₂²×D₄×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

C₂×C₄

yes

—

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#127×C₆₇

—

#124×C₆₇

—

#121×C₆₇

—

#120×C₆₇

—

#117×C₆₇

—

#107×C₆₇

—

#106×C₆₇

C₂×C₄

yes

—

#102×C₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

SD₁₆

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

D₈

yes

—

C₂×D₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#102

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

100

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

101

C₂×C₄×C₆₇

#135

yes

—

102

C₂×Dic₆₇

—

103

C₈×C₆₇

—

104

C₈×C₆₇

C₄×D₄

yes

—

105

C₈×C₆₇

#120

yes

characteristic

106

C₈×C₆₇

#123

yes

characteristic

107

C₂²×C₆₇

—

108

C₂²×C₆₇

—

109

C₂²×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

—

113

C₂²×C₆₇

C₂×#102

yes

characteristic

114

C₂²×C₆₇

#466

yes

—

115

C₄×C₆₇

—

116

C₄×C₆₇

—

117

C₄×C₆₇

#386

yes

characteristic

118

C₄×C₆₇

#416

yes

characteristic

119

C₄×C₆₇

#465

yes

—

120

Dic₆₇

—

121

#13074

—

maximal, Sylow

122

C₂×C₆₇

—

123

C₂×C₆₇

—

124

C₂×C₆₇

—

125

C₂×C₆₇

—

126

C₂×C₆₇

#1893

yes

characteristic

127

#3407

—

128

#3058

D₆₇

yes

characteristic

129

#2140

—

130

#1812

—

131

#1384

—

132

C₆₇

#13074

yes

characteristic, complemented, Sylow

133

C₂²×D₈

C₂×D₆₇

yes

characteristic

134

C₂×#131

C₂×D₆₇

yes

characteristic

135

#535

—

136

#504

—

134

—

137

C₂×M₃₂

—

138

C₂×M₃₂

—

134

—

139

#469

Dic₆₇

yes

complemented

140

#466

—

141

#427

—

142

#427

C₂×D₆₇

yes

characteristic

143

#418

—

144

#410

—

134

—

145

#357

—

134

—

146

H(3,ℤ/4)

—

134

—

147

C₂×#41

C₂²⋊D₆₇

yes

characteristic

148

C₂²×D₄

—

149

C₂×D₈

—

150

C₂×D₈

—

151

C₂×D₈

—

152

C₂×D₈

C₂×Dic₆₇

yes

—

153

#131

—

154

#131

—

155

C₂×M₁₆

C₂×Dic₆₇

yes

—

156

C₂×M₁₆

C₂²⋊D₆₇

yes

characteristic

157

C₂²×C₈

C₂²⋊D₆₇

yes

—

158

C₂²×C₈

C₂²×D₆₇

yes

characteristic

159

#127

—

160

#124

—

161

#121

—

162

#120

—

163

#117

—

164

#117

—

134

—

165

M₃₂

—

134

—

166

M₃₂

—

134

—

167

M₃₂

—

134

—

168

C₂×C₁₆

—

169

C₂×C₁₆

—

134

—

170

#107

—

171

#107

—

134

—

172

#106

—

134

—

173

#106

C₂×Dic₆₇

yes

—

174

#102

—

175

#99

—

268

—

176

C₂⁴

—

177

#41

—

178

#41

—

179

C₂×Q₈

#11761616

yes

characteristic

180

C₂×D₄

—

181

C₂×D₄

—

182

C₂×D₄

—

183

C₂×D₄

—

184

C₂×D₄

—

185

C₂×D₄

#11761613

yes

—

186

C₂×D₄

#11761616

yes

characteristic

187

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

188

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

189

D₈

—

190

D₈

—

191

D₈

—

192

M₁₆

—

193

M₁₆

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

C₂²⋊Dic₆₇

yes

—

197

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

198

C₂×C₈

D₄×D₆₇

yes

—

199

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

200

C₂×C₈

#11761627

yes

—

201

C₄⋊C₄

—

202

C₄⋊C₄

—

268

—

203

C₂²⋊C₄

—

204

C₂²⋊C₄

—

205

C₂²⋊C₄

—

268

—

206

C₄²

—

207

C₄²

—

268

—

208

C₁₆

—

134

—

209

C₂³

—

210

C₂³

—

211

C₂³

—

212

C₂³

#14523964

yes

—

213

C₂³

C₂×#11761613

yes

characteristic

214

Q₈

—

215

D₄

—

216

D₄

—

217

D₄

—

218

D₄

—

219

D₄

—

220

D₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

536

—

225

C₂×C₄

#14523964

yes

—

226

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

227

C₂×C₄

#14524015

yes

—

228

C₂×C₄

C₂×#11761616

yes

characteristic

229

C₂×C₄

#14524063

yes

—

230

C₈

—

231

C₈

D₄×Dic₆₇

yes

—

232

C₈

#14524011

yes

characteristic

233

C₈

#14524024

yes

characteristic

234

C₂²

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

C₂×#14523964

yes

characteristic

241

C₂²

#16863630

yes

—

242

C₄

—

243

C₄

—

244

C₄

—

1072

—

245

C₄

#16863276

yes

characteristic

246

C₄

#16863392

yes

characteristic

247

C₄

#16863623

yes

—

248

C₂

—

249

C₂

—

250

C₂

—

251

C₂

—

252

C₂

#22560638

yes

characteristic

	subgroup	G/H
center	C₂	#22560638
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₆₇×#3058	C₂
socle	C₂×C₆₇	#1893
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3058	D₆₇
hypocenter	C₆₇	#13074
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3058

subgroup

G/H

center

C₂

#22560638

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting, prime radical

C₆₇×#3058

C₂

socle

C₂×C₆₇

#1893

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3058

N	H ≅ G/N	type
1. #22573312	C₂	semidirect
2. #22561925	C₂	semidirect
3. #22560255	C₂	semidirect
4. #22558310	C₂	semidirect
16. #469×C₆₇	C₄	semidirect
25. C₂×C₆₇⋊C₁₆	D₄	semidirect
132. C₆₇	#13074	semidirect
139. #469	Dic₆₇	semidirect

H ≅ G/N

type

1. #22573312

C₂

semidirect

2. #22561925

C₂

semidirect

3. #22560255

C₂

semidirect

4. #22558310

C₂

semidirect

16. #469×C₆₇

C₄

semidirect

25. C₂×C₆₇⋊C₁₆

D₄

semidirect

132. C₆₇

#13074

semidirect

139. #469

Dic₆₇

semidirect

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

subgroup	index
#22573312	2
#22561925	2
#22560255	2
#22558310	2
C₆₇×#3058	2
#13074	67

subgroup

index

C₆₇×#3058

Subgroups — #30773281

Conjugacy classes of subgroups

Complements

Maximal subgroups