Group #30754937 — subgroups

Subgroups — #30754937

Order 17152.

486 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 151 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	#22568587	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863143	C₂	1	2	yes	maximal
4	C₂×#16863141	C₂	1	2	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	C₂×#14524043	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523954	—	2	1	no	—
10	#16863302	—	2	1	no	—
11	#16863302	—	2	1	no	—
12	#16863302	—	2	2	no	—
13	#16863262	C₄	1	4	yes	—
14	#16863151	—	2	1	no	—
15	#16863143	—	2	4	no	—
16	#16863141	—	2	4	no	—
17	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂²×Dic₁₃₄	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	—	2	2	no	—
25	#14524043	—	2	2	no	—
26	#14524043	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	1	no	—
28	C₂×C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
33	#14523954	—	4	2	no	—
34	C₆₇⋊#95	—	2	4	no	—
35	C₆₇⋊#95	—	2	4	no	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	—	4	4	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	—	4	1	no	—
47	C₂×Dic₁₃₄	—	4	1	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂×Dic₁₃₄	—	4	1	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	1	no	—
53	C₂²×Dic₆₇	—	2	2	no	—
54	C₂²×Dic₆₇	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	1	no	—
62	C₄⋊Dic₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	4	2	no	—
69	C₂×C₈×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	1	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	1	no	—
76	C₄²×C₆₇	C₂×D₄	1	2	yes	—
77	C₄²×C₆₇	#41	1	2	yes	—
78	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	#101	1	4	yes	—
86	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₆₇	#121	1	2	yes	—
88	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#137	1	2	yes	—
94	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
95	C₂×Dic₆₇	—	4	2	no	—
96	C₂×Dic₆₇	—	4	2	no	—
97	C₂×Dic₆₇	—	4	2	no	—
98	C₂×Dic₆₇	—	4	4	no	—
99	C₂×Dic₆₇	—	4	2	no	—
100	Dic₁₃₄	—	8	2	no	—
101	Dic₁₃₄	—	8	2	no	—
102	Dic₁₃₄	—	8	2	no	—
103	Dic₁₃₄	—	8	1	no	—
104	Dic₁₃₄	—	8	1	no	—
105	C₈×C₆₇	—	8	2	no	—
106	C₆₇⋊C₈	—	4	4	no	—
107	C₂²×C₆₇	#390	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#101	1	2	yes	—
109	C₂²×C₆₇	#416	1	1	yes	characteristic
110	C₂²×C₆₇	#417	1	2	yes	—
111	C₂²×C₆₇	#417	1	1	yes	characteristic
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	4	2	no	—
115	C₄×C₆₇	#463	1	2	yes	—
116	C₄×C₆₇	#464	1	2	yes	—
117	C₄×C₆₇	#489	1	2	yes	—
118	C₄×C₆₇	#496	1	1	yes	characteristic
119	C₄×C₆₇	#497	1	1	yes	characteristic
120	Dic₆₇	—	8	2	no	—
121	Dic₆₇	—	8	2	no	—
122	#9708	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	#1552	1	2	yes	—
124	C₂×C₆₇	#1561	1	2	yes	—
125	C₂×C₆₇	#1959	1	1	yes	characteristic
126	C₂×C₆₇	#1966	1	1	yes	characteristic
127	C₂×C₆₇	#1983	1	1	yes	characteristic
128	#2862	—	67	1	no	—
129	#1842	D₆₇	1	1	yes	characteristic
130	#1767	—	67	1	no	—
131	C₂×#332	—	67	2	no	—
132	C₂×#332	—	67	2	no	—
133	C₆₇	#9708	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄⋊Q₈	—	67	1	no	—
135	C₂×C₄⋊Q₈	—	67	1	no	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	—	67	2	no	—
138	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
139	C₂×#97	—	134	1	no	—
140	#391	—	134	1	no	—
141	#391	—	134	1	no	—
142	#391	—	134	2	no	—
143	#382	—	67	4	no	—
144	C₄³	C₂×D₆₇	1	1	yes	characteristic
145	#336	—	134	1	no	—
146	#332	—	134	4	no	—
147	#332	—	134	4	no	—
148	C₂²×Q₈	—	134	1	no	—
149	C₂²×Q₈	—	134	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	134	2	no	—
152	C₄⋊Q₈	—	134	2	no	—
153	C₄⋊Q₈	—	134	2	no	—
154	C₄⋊Q₈	—	134	2	no	—
155	C₄⋊Q₈	—	134	2	no	—
156	C₂×C₄⋊C₄	—	67	2	no	—
157	C₂×C₄⋊C₄	—	67	2	no	—
158	C₂×C₄⋊C₄	—	134	1	no	—
159	C₂×C₄⋊C₄	—	134	1	no	—
160	C₂×C₄²	C₄×D₆₇	1	2	yes	—
161	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
162	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	4	4	no	—
168	C₄⋊C₈	—	67	4	no	—
169	C₄⋊C₈	—	134	2	no	—
170	#97	—	268	2	no	—
171	#95	—	134	4	no	—
172	#95	—	134	4	no	—
173	C₂×Q₈	—	268	2	no	—
174	C₂×Q₈	—	268	1	no	—
175	C₂×Q₈	—	268	1	no	—
176	C₂×Q₈	—	268	2	no	—
177	C₂×Q₈	—	268	2	no	—
178	C₂×Q₈	—	268	1	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	268	1	no	—
181	C₂²×C₄	—	134	2	no	—
182	C₂²×C₄	—	134	2	no	—
183	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
184	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	—	268	2	no	—
191	C₂×C₈	—	268	2	no	—
192	C₂×C₈	—	134	4	no	—
193	C₄⋊C₄	—	268	2	no	—
194	C₄⋊C₄	—	268	1	no	—
195	C₄⋊C₄	—	268	1	no	—
196	C₄⋊C₄	—	134	4	no	—
197	C₄⋊C₄	—	134	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	D₄×D₆₇	1	2	yes	—
205	C₄²	#11761629	1	2	yes	—
206	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
207	Q₈	—	536	2	no	—
208	Q₈	—	536	1	no	—
209	Q₈	—	536	1	no	—
210	Q₈	—	536	2	no	—
211	Q₈	—	536	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	268	2	no	—
219	C₂×C₄	—	268	2	no	—
220	C₂×C₄	—	268	2	no	—
221	C₂×C₄	—	268	4	no	—
222	C₂×C₄	—	268	2	no	—
223	C₂×C₄	#14523961	1	4	yes	—
224	C₂×C₄	#14524005	1	2	yes	—
225	C₂×C₄	#14524018	1	2	yes	—
226	C₂×C₄	#14524032	1	1	yes	characteristic
227	C₂×C₄	#14524039	1	1	yes	characteristic
228	C₂×C₄	#14524065	1	2	yes	—
229	C₂×C₄	#14524069	1	2	yes	—
230	C₂×C₄	#14524071	1	1	yes	characteristic
231	C₂×C₄	#14524074	1	2	yes	—
232	C₂×C₄	#14524076	1	1	yes	characteristic
233	C₈	—	8	2	no	—
234	C₈	—	268	4	no	—
235	C₂²	#16863297	1	1	yes	characteristic
236	C₂²	C₂×#14523961	1	2	yes	—
237	C₂²	#16863393	1	1	yes	characteristic
238	C₂²	#16863395	1	2	yes	—
239	C₂²	#16863398	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	536	2	no	—
244	C₄	—	536	2	no	—
245	C₄	#16863612	1	2	yes	—
246	C₄	#16863619	1	2	yes	—
247	C₄	#16863758	1	2	yes	—
248	C₄	#16863783	1	1	yes	characteristic
249	C₄	#16863788	1	1	yes	characteristic
250	C₂	#22558907	1	2	yes	—
251	C₂	#22558946	1	2	yes	—
252	C₂	#22561023	1	1	yes	characteristic
253	C₂	#22561052	1	1	yes	characteristic
254	C₂	#22561139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568587

C₂

yes

maximal, characteristic

#22560062

C₂

yes

maximal, characteristic

C₂×#16863143

C₂

yes

maximal

C₂×#16863141

C₂

yes

maximal

C₆₇×#1842

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

characteristic

C₂×#14524043

C₂²

yes

characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₄

yes

—

#16863151

—

#16863143

—

#16863141

—

C₂×C₄⋊C₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×Dic₁₃₄

—

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523954

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#101

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

#137

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

100

Dic₁₃₄

—

101

Dic₁₃₄

—

102

Dic₁₃₄

—

103

Dic₁₃₄

—

104

Dic₁₃₄

—

105

C₈×C₆₇

—

106

C₆₇⋊C₈

—

107

C₂²×C₆₇

#390

yes

characteristic

108

C₂²×C₆₇

C₂×#101

yes

—

109

C₂²×C₆₇

#416

yes

characteristic

110

C₂²×C₆₇

#417

yes

—

111

C₂²×C₆₇

#417

yes

characteristic

112

C₄×C₆₇

—

113

C₄×C₆₇

—

114

C₄×C₆₇

—

115

C₄×C₆₇

#463

yes

—

116

C₄×C₆₇

#464

yes

—

117

C₄×C₆₇

#489

yes

—

118

C₄×C₆₇

#496

yes

characteristic

119

C₄×C₆₇

#497

yes

characteristic

120

Dic₆₇

—

121

Dic₆₇

—

122

#9708

—

maximal, Sylow

123

C₂×C₆₇

#1552

yes

—

124

C₂×C₆₇

#1561

yes

—

125

C₂×C₆₇

#1959

yes

characteristic

126

C₂×C₆₇

#1966

yes

characteristic

127

C₂×C₆₇

#1983

yes

characteristic

128

#2862

—

129

#1842

D₆₇

yes

characteristic

130

#1767

—

131

C₂×#332

—

132

C₂×#332

—

133

C₆₇

#9708

yes

characteristic, complemented, Sylow

134

C₂×C₄⋊Q₈

—

135

C₂×C₄⋊Q₈

—

136

C₂×C₄⋊C₈

—

137

C₂×C₄⋊C₈

—

138

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

139

C₂×#97

—

134

—

140

#391

—

134

—

141

#391

—

134

—

142

#391

—

134

—

143

#382

—

144

C₄³

C₂×D₆₇

yes

characteristic

145

#336

—

134

—

146

#332

—

134

—

147

#332

—

134

—

148

C₂²×Q₈

—

134

—

149

C₂²×Q₈

—

134

—

150

C₂²×C₈

—

151

C₂²×C₈

—

134

—

152

C₄⋊Q₈

—

134

—

153

C₄⋊Q₈

—

134

—

154

C₄⋊Q₈

—

134

—

155

C₄⋊Q₈

—

134

—

156

C₂×C₄⋊C₄

—

157

C₂×C₄⋊C₄

—

158

C₂×C₄⋊C₄

—

134

—

159

C₂×C₄⋊C₄

—

134

—

160

C₂×C₄²

C₄×D₆₇

yes

—

161

C₂×C₄²

D₂₆₈

yes

characteristic

162

C₂×C₄²

D₂₆₈

yes

characteristic

163

C₂×C₄²

C₂²⋊D₆₇

yes

—

164

C₂×C₄²

C₂²×D₆₇

yes

characteristic

165

C₄⋊C₈

—

166

C₄⋊C₈

—

167

C₄⋊C₈

—

168

C₄⋊C₈

—

169

C₄⋊C₈

—

134

—

170

#97

—

268

—

171

#95

—

134

—

172

#95

—

134

—

173

C₂×Q₈

—

268

—

174

C₂×Q₈

—

268

—

175

C₂×Q₈

—

268

—

176

C₂×Q₈

—

268

—

177

C₂×Q₈

—

268

—

178

C₂×Q₈

—

268

—

179

C₂×Q₈

—

268

—

180

C₂×Q₈

—

268

—

181

C₂²×C₄

—

134

—

182

C₂²×C₄

—

134

—

183

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

184

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

185

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

186

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

187

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

—

268

—

191

C₂×C₈

—

268

—

192

C₂×C₈

—

134

—

193

C₄⋊C₄

—

268

—

194

C₄⋊C₄

—

268

—

195

C₄⋊C₄

—

268

—

196

C₄⋊C₄

—

134

—

197

C₄⋊C₄

—

134

—

198

C₄²

—

199

C₄²

—

200

C₄²

—

201

C₄²

—

202

C₄²

—

203

C₄²

—

204

C₄²

D₄×D₆₇

yes

—

205

C₄²

#11761629

yes

—

206

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

207

Q₈

—

536

—

208

Q₈

—

536

—

209

Q₈

—

536

—

210

Q₈

—

536

—

211

Q₈

—

536

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

268

—

219

C₂×C₄

—

268

—

220

C₂×C₄

—

268

—

221

C₂×C₄

—

268

—

222

C₂×C₄

—

268

—

223

C₂×C₄

#14523961

yes

—

224

C₂×C₄

#14524005

yes

—

225

C₂×C₄

#14524018

yes

—

226

C₂×C₄

#14524032

yes

characteristic

227

C₂×C₄

#14524039

yes

characteristic

228

C₂×C₄

#14524065

yes

—

229

C₂×C₄

#14524069

yes

—

230

C₂×C₄

#14524071

yes

characteristic

231

C₂×C₄

#14524074

yes

—

232

C₂×C₄

#14524076

yes

characteristic

233

C₈

—

234

C₈

—

268

—

235

C₂²

#16863297

yes

characteristic

236

C₂²

C₂×#14523961

yes

—

237

C₂²

#16863393

yes

characteristic

238

C₂²

#16863395

yes

—

239

C₂²

#16863398

yes

characteristic

240

C₄

—

241

C₄

—

242

C₄

—

243

C₄

—

536

—

244

C₄

—

536

—

245

C₄

#16863612

yes

—

246

C₄

#16863619

yes

—

247

C₄

#16863758

yes

—

248

C₄

#16863783

yes

characteristic

249

C₄

#16863788

yes

characteristic

250

C₂

#22558907

yes

—

251

C₂

#22558946

yes

—

252

C₂

#22561023

yes

characteristic

253

C₂

#22561052

yes

characteristic

254

C₂

#22561139

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#9708
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1842

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
133. C₆₇	#9708	semidirect

H ≅ G/N

type

133. C₆₇

#9708

semidirect

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

subgroup	index
#22568587	2
#22560062	2
C₂×#16863143	2
C₂×#16863141	2
C₆₇×#1842	2
#9708	67

subgroup

index

C₆₇×#1842

Subgroups — #30754937

Conjugacy classes of subgroups

Complements

Maximal subgroups