Group #30689773 — subgroups

Subgroups — C₆₇×#10522 #30689773

Order 17152.

402 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₆₇×#3249	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1994	C₂	1	2	yes	maximal
3	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1550	C₂	1	2	yes	maximal
5	C₆₇×#1292	C₂	1	1	yes	maximal, characteristic
6	#533×C₆₇	C₂²	1	2	yes	—
7	#477×C₆₇	—	2	2	no	—
8	#467×C₆₇	—	2	2	no	—
9	#462×C₆₇	—	2	2	no	—
10	#423×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	#339×C₆₇	—	2	2	no	—
16	#339×C₆₇	—	2	2	no	—
17	#333×C₆₇	—	2	2	no	—
18	#332×C₆₇	—	2	2	no	—
19	C₂×Q₁₆×C₆₇	—	2	2	no	—
20	C₂×M₁₆×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	—	2	1	no	—
22	C₂²×C₈×C₆₇	D₄	1	2	yes	—
23	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
25	#125×C₆₇	—	2	2	no	—
26	C₂²⋊Q₈×C₆₇	—	2	2	no	—
27	C₂²⋊Q₈×C₆₇	—	2	2	no	—
28	C₄×Q₈×C₆₇	—	2	2	no	—
29	C₄×Q₈×C₆₇	—	2	2	no	—
30	#117×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	—	2	1	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
36	#107×C₆₇	—	2	2	no	—
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	4	4	no	—
42	#103×C₆₇	—	2	4	no	—
43	C₂×Q₈×C₆₇	—	2	2	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
50	Q₁₆×C₆₇	—	8	2	no	—
51	M₁₆×C₆₇	—	4	2	no	—
52	M₁₆×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	4	2	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	—	4	1	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₂²⋊C₄×C₆₇	—	4	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	2	1	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	—	4	2	no	—
71	C₄²×C₆₇	Q₁₆	1	2	yes	complemented
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₂³×C₆₇	#121	1	1	yes	characteristic
75	Q₈×C₆₇	—	4	2	no	—
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	8	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	4	1	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	2	yes	—
91	C₂×C₄×C₆₇	#123	1	2	yes	—
92	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#137	1	2	yes	—
96	C₈×C₆₇	—	4	2	no	—
97	C₈×C₆₇	—	8	1	no	—
98	C₈×C₆₇	—	8	2	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	#394	1	1	yes	characteristic
102	C₂²×C₆₇	#467	1	1	yes	characteristic
103	C₂²×C₆₇	#470	1	1	yes	characteristic
104	C₄×C₆₇	—	2	4	no	—
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	—	8	2	no	—
109	C₄×C₆₇	—	8	2	no	—
110	C₄×C₆₇	#453	1	1	yes	characteristic
111	C₄×C₆₇	#456	1	1	yes	characteristic
112	C₄×C₆₇	#462	1	2	yes	—
113	C₄×C₆₇	#464	1	2	yes	—
114	#10522	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	—	2	1	no	—
117	C₂×C₆₇	#2039	1	1	yes	characteristic
118	C₂×C₆₇	#2050	1	1	yes	characteristic
119	C₂×C₆₇	#2056	1	1	yes	characteristic
120	#3249	C₂×C₆₇	1	1	yes	characteristic
121	#1994	C₂×C₆₇	1	2	yes	—
122	#1842	C₂×C₆₇	1	1	yes	characteristic
123	#1550	C₂×C₆₇	1	2	yes	—
124	#1292	C₂×C₆₇	1	1	yes	characteristic
125	C₆₇	#10522	1	1	yes	characteristic, direct product factor, Sylow
126	#533	C₂²×C₆₇	1	2	yes	—
127	#477	—	2	2	no	—
128	#467	—	2	2	no	—
129	#462	—	2	2	no	—
130	#423	C₂²×C₆₇	1	2	yes	—
131	C₂×C₄⋊C₈	—	2	2	no	—
132	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
134	C₄³	C₂²×C₆₇	1	1	yes	characteristic
135	#339	—	2	2	no	—
136	#339	—	2	2	no	—
137	#333	—	2	2	no	—
138	#332	—	2	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	D₄×C₆₇	1	2	yes	—
143	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
144	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
145	#125	—	2	2	no	—
146	C₂²⋊Q₈	—	2	2	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	4	4	no	—
162	#103	—	2	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
169	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
170	Q₁₆	—	8	2	no	—
171	M₁₆	—	4	2	no	—
172	M₁₆	—	4	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	Q₁₆×C₆₇	1	2	yes	complemented
192	C₄²	C₂×D₄×C₆₇	1	2	yes	—
193	C₄²	#41×C₆₇	1	2	yes	—
194	C₂³	#121×C₆₇	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	Q₈	—	8	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
210	C₂×C₄	#121×C₆₇	1	2	yes	—
211	C₂×C₄	#123×C₆₇	1	2	yes	—
212	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
214	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
215	C₂×C₄	#137×C₆₇	1	2	yes	—
216	C₈	—	4	2	no	—
217	C₈	—	8	1	no	—
218	C₈	—	8	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#394×C₆₇	1	1	yes	characteristic
222	C₂²	#467×C₆₇	1	1	yes	characteristic
223	C₂²	#470×C₆₇	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	#453×C₆₇	1	1	yes	characteristic
231	C₄	#456×C₆₇	1	1	yes	characteristic
232	C₄	#462×C₆₇	1	2	yes	—
233	C₄	#464×C₆₇	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	C₆₇×#2039	1	1	yes	characteristic
237	C₂	C₆₇×#2050	1	1	yes	characteristic
238	C₂	C₆₇×#2056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#3249

C₂

yes

maximal, characteristic

C₆₇×#1994

C₂

yes

maximal

C₆₇×#1842

C₂

yes

maximal, characteristic

C₆₇×#1550

C₂

yes

maximal

C₆₇×#1292

C₂

yes

maximal, characteristic

#533×C₆₇

C₂²

yes

—

#477×C₆₇

—

#467×C₆₇

—

#462×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄⋊C₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

#339×C₆₇

—

#333×C₆₇

—

#332×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

D₄

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#103×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

Q₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

complemented

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#121

yes

characteristic

Q₈×C₆₇

—

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

#120

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

—

101

C₂²×C₆₇

#394

yes

characteristic

102

C₂²×C₆₇

#467

yes

characteristic

103

C₂²×C₆₇

#470

yes

characteristic

104

C₄×C₆₇

—

105

C₄×C₆₇

—

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

#453

yes

characteristic

111

C₄×C₆₇

#456

yes

characteristic

112

C₄×C₆₇

#462

yes

—

113

C₄×C₆₇

#464

yes

—

114

#10522

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

115

C₂×C₆₇

—

116

C₂×C₆₇

—

117

C₂×C₆₇

#2039

yes

characteristic

118

C₂×C₆₇

#2050

yes

characteristic

119

C₂×C₆₇

#2056

yes

characteristic

120

#3249

C₂×C₆₇

yes

characteristic

121

#1994

C₂×C₆₇

yes

—

122

#1842

C₂×C₆₇

yes

characteristic

123

#1550

C₂×C₆₇

yes

—

124

#1292

C₂×C₆₇

yes

characteristic

125

C₆₇

#10522

yes

characteristic, direct product factor, Sylow

126

#533

C₂²×C₆₇

yes

—

127

#477

—

128

#467

—

129

#462

—

130

#423

C₂²×C₆₇

yes

—

131

C₂×C₄⋊C₈

—

132

C₂×C₄⋊C₈

C₂²×C₆₇

yes

characteristic

133

C₂×C₄⋊C₈

C₂²×C₆₇

yes

characteristic

134

C₄³

C₂²×C₆₇

yes

characteristic

135

#339

—

136

#339

—

137

#333

—

138

#332

—

139

C₂×Q₁₆

—

140

C₂×M₁₆

—

141

C₂²×C₈

—

142

C₂²×C₈

D₄×C₆₇

yes

—

143

C₄⋊Q₈

D₄×C₆₇

yes

—

144

C₄⋊Q₈

D₄×C₆₇

yes

—

145

#125

—

146

C₂²⋊Q₈

—

147

C₂²⋊Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

#117

—

151

C₂×C₄²

—

152

C₂×C₄²

—

153

C₂×C₄²

D₄×C₆₇

yes

characteristic

154

C₂×C₄²

D₄×C₆₇

yes

characteristic

155

C₂×C₄²

C₂³×C₆₇

yes

characteristic

156

#107

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

C₄⋊C₈

—

162

#103

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

168

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

169

C₂²×C₄

#41×C₆₇

yes

characteristic

170

Q₁₆

—

171

M₁₆

—

172

M₁₆

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₂²⋊C₄

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

191

C₄²

Q₁₆×C₆₇

yes

complemented

192

C₄²

C₂×D₄×C₆₇

yes

—

193

C₄²

#41×C₆₇

yes

—

194

C₂³

#121×C₆₇

yes

characteristic

195

Q₈

—

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

#120×C₆₇

yes

characteristic

210

C₂×C₄

#121×C₆₇

yes

—

211

C₂×C₄

#123×C₆₇

yes

—

212

C₂×C₄

#124×C₆₇

yes

characteristic

213

C₂×C₄

C₂×Q₁₆×C₆₇

yes

characteristic

214

C₂×C₄

#135×C₆₇

yes

characteristic

215

C₂×C₄

#137×C₆₇

yes

—

216

C₈

—

217

C₈

—

218

C₈

—

219

C₂²

—

220

C₂²

—

221

C₂²

#394×C₆₇

yes

characteristic

222

C₂²

#467×C₆₇

yes

characteristic

223

C₂²

#470×C₆₇

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

#453×C₆₇

yes

characteristic

231

C₄

#456×C₆₇

yes

characteristic

232

C₄

#462×C₆₇

yes

—

233

C₄

#464×C₆₇

yes

—

234

C₂

—

235

C₂

—

236

C₂

C₆₇×#2039

yes

characteristic

237

C₂

C₆₇×#2050

yes

characteristic

238

C₂

C₆₇×#2056

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂³×C₆₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

prime radical

C₂³×C₆₇

#121

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
71. C₄²×C₆₇	Q₁₆	semidirect
114. #10522	C₆₇	direct (N × H)
125. C₆₇	#10522	direct (N × H)
191. C₄²	Q₁₆×C₆₇	semidirect

H ≅ G/N

type

71. C₄²×C₆₇

Q₁₆

semidirect

114. #10522

C₆₇

direct (N × H)

125. C₆₇

#10522

direct (N × H)

191. C₄²

Q₁₆×C₆₇

semidirect

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

subgroup	index
C₆₇×#3249	2
C₆₇×#1994	2
C₆₇×#1842	2
C₆₇×#1550	2
C₆₇×#1292	2
#10522	67

subgroup

index

C₆₇×#3249

C₆₇×#1994

C₆₇×#1842

C₆₇×#1550

C₆₇×#1292

#10522

Subgroups — C₆₇×#10522 #30689773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#10522 #30689773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10522 #30689773