Group #30755027 — subgroups

Subgroups — #30755027

Order 17152.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
2	#22558804	C₂	1	2	yes	maximal
3	#22558630	C₂	1	2	yes	maximal
4	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	1	yes	characteristic
7	C₂×#14523979	—	2	1	no	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	C₄×#11761612	C₂²	1	1	yes	characteristic
10	#16863161	—	2	1	no	—
11	#16863138	—	2	4	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
13	#16863118	—	2	1	no	—
14	C₈×C₆₇⋊C₈	—	2	1	no	—
15	#533×C₆₇	C₂²	1	2	yes	—
16	#500×C₆₇	C₄	1	1	yes	characteristic
17	#494×C₆₇	C₄	1	1	yes	characteristic
18	#487×C₆₇	C₄	1	2	yes	—
19	#467×C₆₇	—	2	1	no	—
20	#467×C₆₇	—	2	1	no	—
21	C₄×Q₁₆×C₆₇	—	2	1	no	—
22	C₄×Q₁₆×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	1	no	—
26	#14523979	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
29	#14523954	—	2	2	no	—
30	#14523954	D₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	1	no	—
33	C₄×C₆₇⋊C₈	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×Q₁₆×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
40	#125×C₆₇	—	2	2	no	—
41	C₂²⋊Q₈×C₆₇	—	2	2	no	—
42	C₂²⋊Q₈×C₆₇	—	2	2	no	—
43	C₄×Q₈×C₆₇	—	2	2	no	—
44	C₄×Q₈×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	#107×C₆₇	—	2	1	no	—
48	#107×C₆₇	—	2	1	no	—
49	#103×C₆₇	—	2	2	no	—
50	#103×C₆₇	—	2	2	no	—
51	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	#11761612	—	2	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	4	2	no	—
67	Q₁₆×C₆₇	—	4	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₂²⋊C₄×C₆₇	—	4	2	no	—
81	C₄²×C₆₇	—	4	2	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₆₇	#124	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	2	no	—
88	Q₈×C₆₇	—	4	2	no	—
89	Q₈×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
102	C₈×C₆₇	—	2	1	no	—
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	#104	1	2	yes	—
106	C₆₇⋊C₈	—	2	4	no	—
107	C₆₇⋊C₈	—	4	4	no	—
108	C₆₇⋊C₈	—	4	2	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	#390	1	1	yes	characteristic
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	8	2	no	—
117	C₄×C₆₇	—	8	2	no	—
118	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
119	C₄×C₆₇	#421	1	1	yes	characteristic
120	C₄×C₆₇	#443	1	2	yes	—
121	C₄×C₆₇	#473	1	2	yes	—
122	#9725	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	4	1	no	—
124	C₂×C₆₇	#1974	1	1	yes	characteristic
125	C₂×C₆₇	#1979	1	1	yes	characteristic
126	C₂×C₆₇	#1980	1	1	yes	characteristic
127	#3170	D₆₇	1	1	yes	characteristic
128	C₄×#108	—	67	1	no	—
129	#1526	—	67	2	no	—
130	#1485	—	67	2	no	—
131	C₈×M₁₆	—	67	1	no	—
132	C₆₇	#9725	1	1	yes	characteristic, complemented, Sylow
133	#533	C₂×D₆₇	1	2	yes	—
134	#500	Dic₆₇	1	1	yes	characteristic
135	#494	Dic₆₇	1	1	yes	characteristic
136	#487	Dic₆₇	1	2	yes	—
137	#467	—	2	1	no	—
138	#467	—	2	1	no	—
139	C₄×Q₁₆	—	2	1	no	—
140	C₄×Q₁₆	—	2	1	no	—
141	#431	—	67	1	no	—
142	C₂×#108	—	134	1	no	—
143	C₄×M₁₆	—	67	1	no	—
144	C₄×M₁₆	—	67	2	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
146	#341	—	134	1	no	—
147	#330	—	134	4	no	—
148	Q₈⋊C₈	—	134	4	no	—
149	#322	—	134	1	no	—
150	C₈²	—	134	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×M₁₆	—	134	2	no	—
154	C₂×M₁₆	—	134	1	no	—
155	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
157	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
158	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
159	#125	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	#117	—	2	2	no	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	#108	—	134	4	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	C₄⋊C₈	—	67	2	no	—
170	#103	—	2	2	no	—
171	#103	—	2	2	no	—
172	C₂²⋊C₈	—	134	1	no	—
173	#97	—	67	2	no	—
174	#97	—	134	2	no	—
175	C₄×C₈	—	134	2	no	—
176	C₄×C₈	—	134	1	no	—
177	C₄×C₈	—	134	1	no	—
178	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂²×C₄	#11761627	1	1	yes	characteristic
186	C₂²×C₄	#11761627	1	1	yes	characteristic
187	Q₁₆	—	4	2	no	—
188	Q₁₆	—	4	2	no	—
189	M₁₆	—	268	2	no	—
190	M₁₆	—	134	4	no	—
191	M₁₆	—	134	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	268	2	no	—
195	C₂×C₈	—	134	2	no	—
196	C₂×C₈	—	134	2	no	—
197	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₄²	D₄×D₆₇	1	1	yes	characteristic
211	C₄²	D₄×D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂³	#14524030	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
224	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
226	C₂×C₄	#14524014	1	1	yes	characteristic
227	C₂×C₄	#14524014	1	1	yes	characteristic
228	C₂×C₄	#14524039	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	268	4	no	—
233	C₈	—	268	2	no	—
234	C₈	—	134	4	no	—
235	C₈	#14523970	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	#16863299	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×#14523970	1	1	yes	characteristic
246	C₄	#16863413	1	1	yes	characteristic
247	C₄	#16863513	1	2	yes	—
248	C₄	#16863668	1	2	yes	—
249	C₂	—	4	1	no	—
250	C₂	#22561098	1	1	yes	characteristic
251	C₂	#22561125	1	1	yes	characteristic
252	C₂	#22561129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523979

C₂

yes

maximal, characteristic

#22558804

C₂

yes

maximal

#22558630

C₂

yes

maximal

C₈×#11761612

C₂

yes

maximal, characteristic

C₆₇×#3170

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

characteristic

C₂×#14523979

—

C₄×#11761612

C₂²

yes

—

C₄×#11761612

C₂²

yes

characteristic

#16863161

—

#16863138

—

Q₈⋊(C₆₇⋊C₈)

—

#16863118

—

C₈×C₆₇⋊C₈

—

#533×C₆₇

C₂²

yes

—

#500×C₆₇

C₄

yes

characteristic

#494×C₆₇

C₄

yes

characteristic

#487×C₆₇

C₄

yes

—

#467×C₆₇

—

#467×C₆₇

—

C₄×Q₁₆×C₆₇

—

C₄×Q₁₆×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

D₄

yes

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

#107×C₆₇

—

#103×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

C₂²×C₄×C₆₇

yes

C₂²×C₄×C₆₇

yes

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

characteristic

100

C₂×C₄×C₆₇

#121

yes

characteristic

101

C₂×C₄×C₆₇

#127

yes

characteristic

102

C₈×C₆₇

—

103

C₈×C₆₇

—

104

C₈×C₆₇

—

105

C₈×C₆₇

#104

yes

—

106

C₆₇⋊C₈

—

107

C₆₇⋊C₈

—

108

C₆₇⋊C₈

—

109

C₂²×C₆₇

—

110

C₂²×C₆₇

—

111

C₂²×C₆₇

—

112

C₂²×C₆₇

#390

yes

characteristic

113

C₄×C₆₇

—

114

C₄×C₆₇

—

115

C₄×C₆₇

—

116

C₄×C₆₇

—

117

C₄×C₆₇

—

118

C₄×C₆₇

C₂×#104

yes

characteristic

119

C₄×C₆₇

#421

yes

characteristic

120

C₄×C₆₇

#443

yes

—

121

C₄×C₆₇

#473

yes

—

122

#9725

—

maximal, Sylow

123

C₂×C₆₇

—

124

C₂×C₆₇

#1974

yes

characteristic

125

C₂×C₆₇

#1979

yes

characteristic

126

C₂×C₆₇

#1980

yes

characteristic

127

#3170

D₆₇

yes

characteristic

128

C₄×#108

—

129

#1526

—

130

#1485

—

131

C₈×M₁₆

—

132

C₆₇

#9725

yes

characteristic, complemented, Sylow

133

#533

C₂×D₆₇

yes

—

134

#500

Dic₆₇

yes

characteristic

135

#494

Dic₆₇

yes

characteristic

136

#487

Dic₆₇

yes

—

137

#467

—

138

#467

—

139

C₄×Q₁₆

—

140

C₄×Q₁₆

—

141

#431

—

142

C₂×#108

—

134

—

143

C₄×M₁₆

—

144

C₄×M₁₆

—

145

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

146

#341

—

134

—

147

#330

—

134

—

148

Q₈⋊C₈

—

134

—

149

#322

—

134

—

150

C₈²

—

134

—

151

C₂×Q₁₆

—

152

C₂×Q₁₆

—

153

C₂×M₁₆

—

134

—

154

C₂×M₁₆

—

134

—

155

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

156

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

157

C₄⋊Q₈

C₂×Dic₆₇

yes

—

158

C₄⋊Q₈

C₂×Dic₆₇

yes

—

159

#125

—

160

C₂²⋊Q₈

—

161

C₂²⋊Q₈

—

162

C₄×Q₈

—

163

C₄×Q₈

—

164

#117

—

165

C₂×C₄²

C₂²×D₆₇

yes

characteristic

166

#108

—

134

—

167

#107

—

168

#107

—

169

C₄⋊C₈

—

170

#103

—

171

#103

—

172

C₂²⋊C₈

—

134

—

173

#97

—

174

#97

—

134

—

175

C₄×C₈

—

134

—

176

C₄×C₈

—

134

—

177

C₄×C₈

—

134

—

178

C₄×C₈

C₂×Dic₆₇

yes

characteristic

179

C₄×C₈

C₂×Dic₆₇

yes

characteristic

180

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

181

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

185

C₂²×C₄

#11761627

yes

characteristic

186

C₂²×C₄

#11761627

yes

characteristic

187

Q₁₆

—

188

Q₁₆

—

189

M₁₆

—

268

—

190

M₁₆

—

134

—

191

M₁₆

—

134

—

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

268

—

195

C₂×C₈

—

134

—

196

C₂×C₈

—

134

—

197

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

198

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

199

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

200

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

201

C₄⋊C₄

—

202

C₄⋊C₄

—

203

C₄⋊C₄

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

206

C₄⋊C₄

—

207

C₂²⋊C₄

—

208

C₄²

—

209

C₄²

C₂²×Dic₆₇

yes

characteristic

210

C₄²

D₄×D₆₇

yes

characteristic

211

C₄²

D₄×D₆₇

yes

characteristic

212

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

213

C₂³

#14524030

yes

characteristic

214

Q₈

—

215

Q₈

—

216

Q₈

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

224

C₂×C₄

D₄×Dic₆₇

yes

characteristic

225

C₂×C₄

D₄×Dic₆₇

yes

characteristic

226

C₂×C₄

#14524014

yes

characteristic

227

C₂×C₄

#14524014

yes

characteristic

228

C₂×C₄

#14524039

yes

characteristic

229

C₈

—

230

C₈

—

231

C₈

—

232

C₈

—

268

—

233

C₈

—

268

—

234

C₈

—

134

—

235

C₈

#14523970

yes

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

#16863299

yes

characteristic

240

C₄

—

241

C₄

—

242

C₄

—

243

C₄

—

244

C₄

—

245

C₄

C₂×#14523970

yes

characteristic

246

C₄

#16863413

yes

characteristic

247

C₄

#16863513

yes

—

248

C₄

#16863668

yes

—

249

C₂

—

250

C₂

#22561098

yes

characteristic

251

C₂

#22561125

yes

characteristic

252

C₂

#22561129

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#9725
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

subgroup

G/H

center

C₂×C₄

#14524014

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#3170

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

N	H ≅ G/N	type
132. C₆₇	#9725	semidirect

H ≅ G/N

type

132. C₆₇

#9725

semidirect

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

subgroup	index
C₄×#14523979	2
#22558804	2
#22558630	2
C₈×#11761612	2
C₆₇×#3170	2
#9725	67

subgroup

index

C₆₇×#3170

Subgroups — #30755027

Conjugacy classes of subgroups

Complements

Maximal subgroups