Group #30748753 — subgroups

Subgroups — #30748753

Order 17152.

398 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 137 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	#22560205	C₂	1	1	yes	maximal, characteristic
3	#22558964	C₂	1	2	yes	maximal
4	#22558083	C₂	1	2	yes	maximal
5	C₆₇×#1801	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863420	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×#14523962	—	2	1	no	—
12	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863165	—	2	1	no	—
15	#16863146	—	2	2	no	—
16	#16863146	—	2	2	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	#423×C₆₇	C₂²	1	2	yes	—
19	C₂×#101×C₆₇	—	2	1	no	—
20	#343×C₆₇	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	#14523962	—	4	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
33	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
34	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	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	#101×C₆₇	—	4	2	no	—
45	#11761612	—	4	2	no	—
46	#11761612	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	4	1	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	—
58	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	M₁₆×C₆₇	—	4	2	no	—
65	M₁₆×C₆₇	—	4	2	no	—
66	C₂×C₈×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₄²×C₆₇	#41	1	2	yes	—
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	Q₈×C₆₇	—	2	4	no	—
75	Q₈×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	#106	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₈×C₆₇	—	8	2	no	—
92	C₆₇⋊C₈	—	2	1	no	—
93	C₆₇⋊C₈	—	2	2	no	—
94	C₆₇⋊C₈	—	4	2	no	—
95	C₆₇⋊C₈	—	8	2	no	—
96	C₆₇⋊C₈	#101	1	2	yes	complemented
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	#380	1	1	yes	characteristic
100	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
101	C₂²×C₆₇	#427	1	1	yes	characteristic
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	4	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	2	no	—
106	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
107	C₄×C₆₇	#413	1	1	yes	characteristic
108	C₄×C₆₇	#465	1	2	yes	—
109	C₄×C₆₇	#465	1	2	yes	—
110	#8615	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	#1801	1	1	yes	characteristic
114	C₂×C₆₇	#1807	1	1	yes	characteristic
115	C₂×C₆₇	#1813	1	1	yes	characteristic
116	C₂×C₈×Q₈	—	67	1	no	—
117	#1801	—	67	1	no	—
118	#1801	D₆₇	1	1	yes	characteristic
119	#1566	—	67	2	no	—
120	#1292	—	67	2	no	—
121	C₆₇	#8615	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
123	C₈×Q₈	—	134	2	no	—
124	C₈×Q₈	—	134	2	no	—
125	#423	—	67	2	no	—
126	#423	C₂×D₆₇	1	2	yes	—
127	C₂×C₄⋊C₈	—	67	2	no	—
128	C₂×C₄⋊C₈	—	67	1	no	—
129	C₂×#101	—	2	1	no	—
130	C₂×#101	—	134	1	no	—
131	C₂×C₄×C₈	—	67	1	no	—
132	C₂×C₄×C₈	—	67	2	no	—
133	#343	—	2	1	no	—
134	#343	—	134	1	no	—
135	#334	—	134	2	no	—
136	#334	—	134	2	no	—
137	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	134	2	no	—
140	C₂²×C₈	—	67	1	no	—
141	C₂²×C₈	—	67	2	no	—
142	C₂²×C₈	—	67	1	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₆₇	1	2	yes	—
148	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	134	4	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₄⋊C₈	—	134	2	no	—
154	C₄⋊C₈	—	134	2	no	—
155	#101	—	4	2	no	—
156	#101	—	268	2	no	—
157	C₄×C₈	—	67	2	no	—
158	C₄×C₈	—	67	2	no	—
159	C₄×C₈	—	134	2	no	—
160	C₄×C₈	—	134	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	#11761615	1	2	yes	—
164	C₂×Q₈	#11761616	1	2	yes	—
165	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
166	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
170	M₁₆	—	4	2	no	—
171	M₁₆	—	4	2	no	—
172	M₁₆	—	268	2	no	—
173	M₁₆	—	268	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	268	2	no	—
176	C₂×C₈	—	268	2	no	—
177	C₂×C₈	—	268	1	no	—
178	C₂×C₈	—	67	1	no	—
179	C₂×C₈	—	67	1	no	—
180	C₂×C₈	—	67	2	no	—
181	C₂×C₈	—	134	2	no	—
182	C₂×C₈	—	134	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	D₄×D₆₇	1	2	yes	—
188	C₄²	#11761627	1	2	yes	—
189	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
190	Q₈	—	2	4	no	—
191	Q₈	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#14523974	1	4	yes	—
200	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
201	C₂×C₄	#14523999	1	1	yes	characteristic
202	C₂×C₄	#14524015	1	2	yes	—
203	C₂×C₄	#14524020	1	2	yes	—
204	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
205	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
206	C₂×C₄	#14524063	1	2	yes	—
207	C₈	—	8	2	no	—
208	C₈	—	268	2	no	—
209	C₈	—	536	2	no	—
210	C₈	—	67	2	no	—
211	C₈	—	134	1	no	—
212	C₈	—	134	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16863254	1	1	yes	characteristic
216	C₂²	C₂×#14523974	1	1	yes	characteristic
217	C₂²	#16863432	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	#101×D₆₇	1	1	yes	characteristic
223	C₄	#16863379	1	1	yes	characteristic
224	C₄	#16863623	1	2	yes	—
225	C₄	#16863627	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	#22560203	1	1	yes	characteristic
229	C₂	#22560232	1	1	yes	characteristic
230	C₂	#22560257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#22560205

C₂

yes

maximal, characteristic

#22558964

C₂

yes

maximal

#22558083

C₂

yes

maximal

C₆₇×#1801

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

—

Q₈×C₆₇⋊C₈

—

#16863420

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×#14523962

—

C₂×C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

—

C₂×#101×C₆₇

—

#343×C₆₇

—

C₂×#11761612

—

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)

—

C₄⋊(C₆₇⋊C₈)

—

#14523962

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂²×Q₈×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂×C₄⋊C₄×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#101×C₆₇

—

#11761612

—

#11761612

—

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

SD₁₆

yes

—

C₂×Q₈×C₆₇

SD₁₆

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

—

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

#106

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

100

C₂²×C₆₇

C₂×#106

yes

characteristic

101

C₂²×C₆₇

#427

yes

characteristic

102

C₄×C₆₇

—

103

C₄×C₆₇

—

104

C₄×C₆₇

—

105

C₄×C₆₇

—

106

C₄×C₆₇

C₂×#101

yes

characteristic

107

C₄×C₆₇

#413

yes

characteristic

108

C₄×C₆₇

#465

yes

—

109

C₄×C₆₇

#465

yes

—

110

#8615

—

maximal, Sylow

111

C₂×C₆₇

—

112

C₂×C₆₇

—

113

C₂×C₆₇

#1801

yes

characteristic

114

C₂×C₆₇

#1807

yes

characteristic

115

C₂×C₆₇

#1813

yes

characteristic

116

C₂×C₈×Q₈

—

117

#1801

—

118

#1801

D₆₇

yes

characteristic

119

#1566

—

120

#1292

—

121

C₆₇

#8615

yes

characteristic, complemented, Sylow

122

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

123

C₈×Q₈

—

134

—

124

C₈×Q₈

—

134

—

125

#423

—

126

#423

C₂×D₆₇

yes

—

127

C₂×C₄⋊C₈

—

128

C₂×C₄⋊C₈

—

129

C₂×#101

—

130

C₂×#101

—

134

—

131

C₂×C₄×C₈

—

132

C₂×C₄×C₈

—

133

#343

—

134

#343

—

134

—

135

#334

—

134

—

136

#334

—

134

—

137

C₂²×Q₈

C₂²⋊D₆₇

yes

characteristic

138

C₂×M₁₆

—

139

C₂×M₁₆

—

134

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₂²×C₈

—

143

C₄×Q₈

—

144

C₄×Q₈

—

145

C₂×C₄⋊C₄

Dic₁₃₄

yes

—

146

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

147

C₂×C₄²

C₄×D₆₇

yes

—

148

C₂×C₄²

C₂²×D₆₇

yes

characteristic

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

134

—

152

C₄⋊C₈

—

134

—

153

C₄⋊C₈

—

134

—

154

C₄⋊C₈

—

134

—

155

#101

—

156

#101

—

268

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

134

—

160

C₄×C₈

—

134

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

#11761615

yes

—

164

C₂×Q₈

#11761616

yes

—

165

C₂²×C₄

Dic₆₇⋊C₄

yes

—

166

C₂²×C₄

Dic₆₇⋊C₄

yes

—

167

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

168

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

169

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

170

M₁₆

—

171

M₁₆

—

172

M₁₆

—

268

—

173

M₁₆

—

268

—

174

C₂×C₈

—

175

C₂×C₈

—

268

—

176

C₂×C₈

—

268

—

177

C₂×C₈

—

268

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

134

—

182

C₂×C₈

—

134

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄²

—

186

C₄²

—

187

C₄²

D₄×D₆₇

yes

—

188

C₄²

#11761627

yes

—

189

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

#14523974

yes

—

200

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

201

C₂×C₄

#14523999

yes

characteristic

202

C₂×C₄

#14524015

yes

—

203

C₂×C₄

#14524020

yes

—

204

C₂×C₄

C₂×#11761615

yes

characteristic

205

C₂×C₄

C₂×#11761616

yes

characteristic

206

C₂×C₄

#14524063

yes

—

207

C₈

—

208

C₈

—

268

—

209

C₈

—

536

—

210

C₈

—

211

C₈

—

134

—

212

C₈

—

134

—

213

C₂²

—

214

C₂²

—

215

C₂²

#16863254

yes

characteristic

216

C₂²

C₂×#14523974

yes

characteristic

217

C₂²

#16863432

yes

characteristic

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

#101×D₆₇

yes

characteristic

223

C₄

#16863379

yes

characteristic

224

C₄

#16863623

yes

—

225

C₄

#16863627

yes

—

226

C₂

—

227

C₂

—

228

C₂

#22560203

yes

characteristic

229

C₂

#22560232

yes

characteristic

230

C₂

#22560257

yes

characteristic

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1801	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1801	D₆₇
hypocenter	C₆₇	#8615
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1801

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1801

C₂

socle

C₂²×C₆₇

#380

prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1801

N	H ≅ G/N	type
96. C₆₇⋊C₈	#101	semidirect
121. C₆₇	#8615	semidirect

H ≅ G/N

type

96. C₆₇⋊C₈

#101

semidirect

121. C₆₇

#8615

semidirect

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
#22560205	2
#22558964	2
#22558083	2
C₆₇×#1801	2
#8615	67

subgroup

index

C₂×Q₈×C₆₇⋊C₈

C₆₇×#1801

Subgroups — #30748753

Conjugacy classes of subgroups

Complements

Maximal subgroups