Group #30749412 — subgroups

Subgroups — #30749412

Order 17152.

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

normal subgroups only

This group has 422 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	#22560205	C₂	1	1	yes	maximal, characteristic
2	#22558533	C₂	1	1	yes	maximal, characteristic
3	#22558112	C₂	1	2	yes	maximal
4	#22558067	C₂	1	2	yes	maximal
5	C₆₇×#2351	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523962	—	2	1	no	—
9	#16863165	—	2	1	no	—
10	#16863122	—	2	2	no	—
11	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#389×C₆₇	—	2	2	no	—
14	#389×C₆₇	—	2	1	no	—
15	#382×C₆₇	C₄	1	2	yes	—
16	#382×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	#14523962	—	4	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₆₇	—	2	2	no	—
28	C₄×Q₈×C₆₇	—	2	2	no	—
29	C₄×Q₈×C₆₇	—	4	4	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#95×C₆₇	—	2	2	no	—
41	#95×C₆₇	—	2	1	no	—
42	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
43	#95×C₆₇	C₂×C₄	1	2	yes	—
44	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
45	#11761612	—	4	2	no	—
46	#11761612	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×Q₈×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	—
52	C₂×Q₈×C₆₇	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	—	4	2	no	—
71	C₄²×C₆₇	#41	1	2	yes	—
72	C₄²×C₆₇	#41	1	2	yes	—
73	C₂³×C₆₇	C₂×C₂²⋊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	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	4	1	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	4	1	no	—
83	C₂×C₄×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	1	no	—
91	C₂×C₄×C₆₇	#103	1	4	yes	—
92	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#123	1	2	yes	—
95	C₂×C₄×C₆₇	#123	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₆₇⋊C₈	—	8	2	no	—
100	C₆₇⋊C₈	—	8	2	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	—	2	1	no	—
103	C₂²×C₆₇	#381	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
105	C₂²×C₆₇	#418	1	1	yes	characteristic
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	—	8	1	no	—
114	C₄×C₆₇	#481	1	2	yes	—
115	C₄×C₆₇	#484	1	2	yes	—
116	#8767	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	1	no	—
118	C₂×C₆₇	—	2	1	no	—
119	C₂×C₆₇	#1824	1	1	yes	characteristic
120	C₂×C₆₇	#1827	1	1	yes	characteristic
121	C₂×C₆₇	#1830	1	1	yes	characteristic
122	#2351	D₆₇	1	1	yes	characteristic
123	#1801	—	67	1	no	—
124	#1461	—	67	1	no	—
125	#1303	—	67	2	no	—
126	#1286	—	67	2	no	—
127	C₆₇	#8767	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄×Q₈	—	2	1	no	—
129	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
130	#431	—	67	2	no	—
131	#423	—	67	2	no	—
132	C₂×#101	—	134	1	no	—
133	#389	—	2	1	no	—
134	#389	—	2	2	no	—
135	#382	Dic₆₇	1	2	yes	—
136	#382	C₂×D₆₇	1	1	yes	characteristic
137	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
138	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
139	#343	—	134	1	no	—
140	#324	—	134	2	no	—
141	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂×M₁₆	—	134	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	4	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₄²	—	2	2	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
156	C₄⋊C₈	—	134	2	no	—
157	C₄⋊C₈	—	134	2	no	—
158	C₄⋊C₈	—	134	2	no	—
159	#101	—	268	2	no	—
160	C₂²⋊C₈	—	134	2	no	—
161	C₄×C₈	—	134	2	no	—
162	#95	—	2	2	no	—
163	#95	—	2	1	no	—
164	#95	C₂×Dic₆₇	1	1	yes	characteristic
165	#95	C₂×Dic₆₇	1	2	yes	—
166	#95	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	#11761615	1	2	yes	—
170	C₂×Q₈	#11761618	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	M₁₆	—	268	2	no	—
180	M₁₆	—	268	2	no	—
181	C₂×C₈	—	268	2	no	—
182	C₂×C₈	—	268	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	4	no	—
192	C₄²	—	4	2	no	—
193	C₄²	#11761628	1	2	yes	—
194	C₄²	#11761628	1	2	yes	—
195	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	#14523967	1	4	yes	—
214	C₂×C₄	#14524002	1	1	yes	characteristic
215	C₂×C₄	#14524002	1	1	yes	characteristic
216	C₂×C₄	#14524026	1	2	yes	—
217	C₂×C₄	#14524026	1	2	yes	—
218	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
219	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
220	C₂×C₄	#14524069	1	2	yes	—
221	C₈	—	536	2	no	—
222	C₈	—	536	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	#16863259	1	1	yes	characteristic
226	C₂²	C₂×#14523967	1	1	yes	characteristic
227	C₂²	#16863401	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	8	1	no	—
236	C₄	#16863714	1	2	yes	—
237	C₄	#16863731	1	2	yes	—
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	#22560299	1	1	yes	characteristic
241	C₂	#22560313	1	1	yes	characteristic
242	C₂	#22560323	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560205

C₂

yes

maximal, characteristic

#22558533

C₂

yes

maximal, characteristic

#22558112

C₂

yes

maximal

#22558067

C₂

yes

maximal

C₆₇×#2351

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863420

C₂²

yes

—

C₂×#14523962

—

#16863165

—

#16863122

—

C₂×C₄×Q₈×C₆₇

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

#389×C₆₇

—

#389×C₆₇

—

#382×C₆₇

C₄

yes

—

#382×C₆₇

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₆₇

C₄

yes

—

C₄×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523962

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

D₄

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

C₂×C₄

yes

characteristic

#95×C₆₇

C₂×C₄

yes

—

#95×C₆₇

C₂×C₄

yes

characteristic

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

SD₁₆

yes

—

C₂×Q₈×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

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

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₆₇⋊C₈

—

100

C₆₇⋊C₈

—

101

C₂²×C₆₇

—

102

C₂²×C₆₇

—

103

C₂²×C₆₇

#381

yes

characteristic

104

C₂²×C₆₇

C₂×#103

yes

characteristic

105

C₂²×C₆₇

#418

yes

characteristic

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

—

113

C₄×C₆₇

—

114

C₄×C₆₇

#481

yes

—

115

C₄×C₆₇

#484

yes

—

116

#8767

—

maximal, Sylow

117

C₂×C₆₇

—

118

C₂×C₆₇

—

119

C₂×C₆₇

#1824

yes

characteristic

120

C₂×C₆₇

#1827

yes

characteristic

121

C₂×C₆₇

#1830

yes

characteristic

122

#2351

D₆₇

yes

characteristic

123

#1801

—

124

#1461

—

125

#1303

—

126

#1286

—

127

C₆₇

#8767

yes

characteristic, complemented, Sylow

128

C₂×C₄×Q₈

—

129

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

130

#431

—

131

#423

—

132

C₂×#101

—

134

—

133

#389

—

134

#389

—

135

#382

Dic₆₇

yes

—

136

#382

C₂×D₆₇

yes

characteristic

137

C₄×C₄⋊C₄

Dic₆₇

yes

—

138

C₄×C₄⋊C₄

C₂×D₆₇

yes

characteristic

139

#343

—

134

—

140

#324

—

134

—

141

C₂²×Q₈

C₂²⋊D₆₇

yes

characteristic

142

C₂×M₁₆

—

134

—

143

C₄×Q₈

—

144

C₄×Q₈

—

145

C₄×Q₈

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

151

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

152

C₂×C₄²

—

153

C₂×C₄²

—

154

C₂×C₄²

C₂×Dic₆₇

yes

—

155

C₂×C₄²

C₂²×D₆₇

yes

characteristic

156

C₄⋊C₈

—

134

—

157

C₄⋊C₈

—

134

—

158

C₄⋊C₈

—

134

—

159

#101

—

268

—

160

C₂²⋊C₈

—

134

—

161

C₄×C₈

—

134

—

162

#95

—

163

#95

—

164

#95

C₂×Dic₆₇

yes

characteristic

165

#95

C₂×Dic₆₇

yes

—

166

#95

C₂×Dic₆₇

yes

characteristic

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

#11761615

yes

—

170

C₂×Q₈

#11761618

yes

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

175

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

176

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

177

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

178

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

179

M₁₆

—

268

—

180

M₁₆

—

268

—

181

C₂×C₈

—

268

—

182

C₂×C₈

—

268

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

#11761628

yes

—

194

C₄²

#11761628

yes

—

195

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

196

Q₈

—

197

Q₈

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

#14523967

yes

—

214

C₂×C₄

#14524002

yes

characteristic

215

C₂×C₄

#14524002

yes

characteristic

216

C₂×C₄

#14524026

yes

—

217

C₂×C₄

#14524026

yes

—

218

C₂×C₄

C₂×#11761615

yes

characteristic

219

C₂×C₄

C₂×#11761618

yes

characteristic

220

C₂×C₄

#14524069

yes

—

221

C₈

—

536

—

222

C₈

—

536

—

223

C₂²

—

224

C₂²

—

225

C₂²

#16863259

yes

characteristic

226

C₂²

C₂×#14523967

yes

characteristic

227

C₂²

#16863401

yes

characteristic

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

#16863714

yes

—

237

C₄

#16863731

yes

—

238

C₂

—

239

C₂

—

240

C₂

#22560299

yes

characteristic

241

C₂

#22560313

yes

characteristic

242

C₂

#22560323

yes

characteristic

	subgroup	G/H
center	C₂²	#16863259
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2351	C₂
socle	C₂²×C₆₇	#381
prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2351	D₆₇
hypocenter	C₆₇	#8767
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2351

subgroup

G/H

center

C₂²

#16863259

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2351

C₂

socle

C₂²×C₆₇

#381

prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2351

N	H ≅ G/N	type
127. C₆₇	#8767	semidirect

H ≅ G/N

type

127. C₆₇

#8767

semidirect

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

subgroup	index
#22560205	2
#22558533	2
#22558112	2
#22558067	2
C₆₇×#2351	2
#8767	67

subgroup

index

C₆₇×#2351

Subgroups — #30749412

Conjugacy classes of subgroups

Complements

Maximal subgroups