Group #30755641 — subgroups

Subgroups — #30755641

Order 17152.

694 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16863773	C₂	1	1	yes	maximal, characteristic
2	#22561014	C₂	1	2	yes	maximal, complemented
3	#22561014	C₂	1	2	yes	maximal, complemented
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₃₆	—	2	2	no	—
7	C₂×#14524040	C₂²	1	2	yes	—
8	#16863773	—	2	2	no	—
9	#16863773	C₄	1	4	yes	complemented
10	C₂×#14523963	—	2	4	no	—
11	C₂×#14523963	—	2	4	no	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863118	—	2	2	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₂²	1	2	yes	—
17	#340×C₆₇	—	2	2	no	—
18	C₂²×D₂₆₈	—	2	2	no	—
19	C₂²×D₂₆₈	—	2	2	no	—
20	C₂²×D₂₆₈	—	2	2	no	—
21	C₂×D₅₃₆	—	2	8	no	—
22	C₂×D₅₃₆	—	4	8	no	—
23	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
24	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
25	#14524040	—	2	2	no	—
26	#14524040	—	2	2	no	—
27	#14524040	—	2	2	no	—
28	#14524040	C₂×C₄	1	4	yes	—
29	#14523963	—	4	8	no	—
30	#14523963	—	4	8	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	1	no	—
33	C₄×C₆₇⋊C₈	—	2	1	no	—
34	C₄×C₆₇⋊C₈	—	2	1	no	—
35	C₂²×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₇	D₄	1	2	yes	—
42	C₂³×D₆₇	—	4	2	no	—
43	C₂×D₂₆₈	—	2	4	no	—
44	C₂×D₂₆₈	—	2	4	no	—
45	C₂×D₂₆₈	—	2	4	no	—
46	C₂×D₂₆₈	—	4	4	no	—
47	C₂×D₂₆₈	—	4	4	no	—
48	C₂×D₂₆₈	—	4	4	no	—
49	C₂×D₂₆₈	—	4	4	no	—
50	C₂×D₂₆₈	—	4	4	no	—
51	C₂×D₂₆₈	—	4	4	no	—
52	C₂×D₂₆₈	—	4	2	no	—
53	C₂×D₂₆₈	—	4	2	no	—
54	C₂×D₂₆₈	—	4	2	no	—
55	D₅₃₆	—	4	16	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	D₈	1	4	yes	complemented
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
67	C₄⋊C₄×C₆₇	—	4	4	no	—
68	C₄⋊C₄×C₆₇	—	4	4	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₆₇	#41	1	1	yes	characteristic
73	C₄²×C₆₇	#41	1	1	yes	characteristic
74	C₂³×C₆₇	#127	1	1	yes	characteristic
75	C₂²×D₆₇	—	4	4	no	—
76	C₂²×D₆₇	—	8	2	no	—
77	C₂²×D₆₇	—	8	2	no	—
78	C₂²×D₆₇	—	8	2	no	—
79	C₂²×D₆₇	—	8	2	no	—
80	C₂²×D₆₇	—	8	2	no	—
81	C₂²×D₆₇	—	8	2	no	—
82	C₂×C₄×C₆₇	—	8	2	no	—
83	C₂×C₄×C₆₇	—	8	2	no	—
84	C₂×C₄×C₆₇	—	8	2	no	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₆₇	#135	1	2	yes	—
93	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
94	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
95	D₂₆₈	—	4	8	no	—
96	D₂₆₈	—	4	8	no	—
97	D₂₆₈	—	4	8	no	—
98	D₂₆₈	—	8	4	no	—
99	D₂₆₈	—	8	4	no	—
100	D₂₆₈	—	8	4	no	—
101	C₈×C₆₇	—	2	4	no	—
102	C₆₇⋊C₈	—	2	4	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₂²×C₆₇	#390	1	1	yes	characteristic
105	C₂²×C₆₇	#493	1	1	yes	characteristic
106	C₂²×C₆₇	#495	1	1	yes	characteristic
107	C₂²×C₆₇	#496	1	2	yes	—
108	C₂²×C₆₇	#496	1	2	yes	—
109	C₂×D₆₇	—	8	4	no	—
110	C₂×D₆₇	—	8	4	no	—
111	C₂×D₆₇	—	8	4	no	—
112	C₂×D₆₇	—	16	2	no	—
113	C₂×D₆₇	—	16	2	no	—
114	C₂×D₆₇	—	16	2	no	—
115	C₂×D₆₇	—	16	2	no	—
116	C₄×C₆₇	—	16	2	no	—
117	C₄×C₆₇	#418	1	2	yes	—
118	C₄×C₆₇	C₄×D₈	1	2	yes	—
119	C₄×C₆₇	#442	1	2	yes	—
120	C₄×C₆₇	#466	1	2	yes	—
121	C₄×C₆₇	#469	1	2	yes	—
122	C₄×C₆₇	#490	1	2	yes	—
123	#9832	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	#1717	1	2	yes	—
125	C₂×C₆₇	#1721	1	2	yes	—
126	C₂×C₆₇	#1961	1	1	yes	characteristic
127	C₂×C₆₇	#1970	1	1	yes	characteristic
128	C₂×C₆₇	#1972	1	1	yes	characteristic
129	D₆₇	—	16	4	no	—
130	C₂×#493	—	67	1	no	—
131	#1957	—	67	2	no	—
132	#1957	—	67	2	no	—
133	#1834	D₆₇	1	1	yes	characteristic, complemented
134	C₂×#322	—	67	1	no	—
135	C₆₇	#9832	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	134	2	no	—
137	C₂×#127	—	67	2	no	—
138	#493	—	67	4	no	—
139	#493	—	134	2	no	—
140	C₂×#102	—	134	4	no	—
141	C₂×#102	—	134	4	no	—
142	C₂×C₄×C₈	—	67	1	no	—
143	C₂×C₄×C₈	—	67	1	no	—
144	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
145	#384	C₂×D₆₇	1	2	yes	—
146	#340	—	2	2	no	—
147	#322	—	134	2	no	—
148	C₂²×D₄	—	134	2	no	—
149	C₂²×D₄	—	134	2	no	—
150	C₂²×D₄	—	134	2	no	—
151	C₂×D₈	—	268	8	no	—
152	C₂×D₈	—	134	8	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	—	67	2	no	—
155	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
156	#127	—	67	4	no	—
157	#127	—	134	2	no	—
158	#127	—	134	2	no	—
159	#127	—	134	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
164	#102	—	268	8	no	—
165	#102	—	268	8	no	—
166	C₄×C₈	—	134	1	no	—
167	C₄×C₈	—	134	1	no	—
168	C₄×C₈	—	134	1	no	—
169	C₄×C₈	—	134	1	no	—
170	C₄×C₈	C₄×D₆₇	1	2	yes	—
171	C₄×C₈	D₂₆₈	1	2	yes	—
172	C₂⁴	—	268	2	no	—
173	C₂×D₄	—	268	2	no	—
174	C₂×D₄	—	268	2	no	—
175	C₂×D₄	—	268	2	no	—
176	C₂×D₄	—	268	4	no	—
177	C₂×D₄	—	268	4	no	—
178	C₂×D₄	—	268	4	no	—
179	C₂×D₄	—	268	4	no	—
180	C₂×D₄	—	268	4	no	—
181	C₂×D₄	—	268	4	no	—
182	C₂×D₄	—	134	4	no	—
183	C₂×D₄	—	134	4	no	—
184	C₂×D₄	—	134	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	D₈	—	268	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	67	4	no	—
192	C₂×C₈	—	134	2	no	—
193	C₂×C₈	—	134	4	no	—
194	C₂×C₈	—	134	2	no	—
195	C₂×C₈	—	134	2	no	—
196	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
201	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
202	C₄²	#11761629	1	1	yes	characteristic
203	C₄²	#11761629	1	1	yes	characteristic
204	C₂³	—	268	4	no	—
205	C₂³	—	536	2	no	—
206	C₂³	—	536	2	no	—
207	C₂³	—	536	2	no	—
208	C₂³	—	536	2	no	—
209	C₂³	—	536	2	no	—
210	C₂³	—	536	2	no	—
211	C₂³	#14524039	1	1	yes	characteristic
212	D₄	—	268	8	no	—
213	D₄	—	268	8	no	—
214	D₄	—	268	8	no	—
215	D₄	—	536	4	no	—
216	D₄	—	536	4	no	—
217	D₄	—	536	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#14524005	1	1	yes	characteristic
223	C₂×C₄	#14524005	1	1	yes	characteristic
224	C₂×C₄	#14524018	1	1	yes	characteristic
225	C₂×C₄	#14524018	1	1	yes	characteristic
226	C₂×C₄	#14524032	1	1	yes	characteristic
227	C₂×C₄	D₈×D₆₇	1	2	yes	—
228	C₂×C₄	#14524066	1	2	yes	—
229	C₂×C₄	#14524068	1	4	yes	—
230	C₂×C₄	#14524070	1	4	yes	—
231	C₈	—	2	4	no	—
232	C₈	—	268	2	no	—
233	C₈	—	134	4	no	—
234	C₂²	—	536	4	no	—
235	C₂²	—	536	4	no	—
236	C₂²	—	536	4	no	—
237	C₂²	—	1072	2	no	—
238	C₂²	—	1072	2	no	—
239	C₂²	—	1072	2	no	—
240	C₂²	—	1072	2	no	—
241	C₂²	#16863297	1	1	yes	characteristic
242	C₂²	#16863771	1	1	yes	characteristic
243	C₂²	#16863781	1	1	yes	characteristic
244	C₂²	#16863782	1	2	yes	—
245	C₂²	#16863786	1	2	yes	—
246	C₄	—	16	2	no	—
247	C₄	#16863400	1	2	yes	—
248	C₄	#16863478	1	2	yes	—
249	C₄	#16863507	1	2	yes	—
250	C₄	#16863633	1	2	yes	—
251	C₄	#16863650	1	2	yes	—
252	C₄	#16863762	1	2	yes	—
253	C₂	—	1072	4	no	—
254	C₂	#22559851	1	2	yes	—
255	C₂	#22559877	1	2	yes	—
256	C₂	#22561032	1	1	yes	characteristic
257	C₂	#22561073	1	1	yes	characteristic
258	C₂	#22561083	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863773

C₂

yes

maximal, characteristic

#22561014

C₂

yes

maximal, complemented

#22561014

C₂

yes

maximal, complemented

C₂×#16863118

C₂

yes

maximal, characteristic, complemented

C₆₇×#1834

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₃₆

—

C₂×#14524040

C₂²

yes

—

#16863773

—

#16863773

C₄

yes

complemented

C₂×#14523963

—

C₂×#14523963

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863118

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

#340×C₆₇

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂²×C₆₇⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#14523963

—

#14523963

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₈

yes

complemented

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

C₄²×C₆₇

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

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

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

100

D₂₆₈

—

101

C₈×C₆₇

—

102

C₆₇⋊C₈

—

103

C₆₇⋊C₈

—

104

C₂²×C₆₇

#390

yes

characteristic

105

C₂²×C₆₇

#493

yes

characteristic

106

C₂²×C₆₇

#495

yes

characteristic

107

C₂²×C₆₇

#496

yes

—

108

C₂²×C₆₇

#496

yes

—

109

C₂×D₆₇

—

110

C₂×D₆₇

—

111

C₂×D₆₇

—

112

C₂×D₆₇

—

113

C₂×D₆₇

—

114

C₂×D₆₇

—

115

C₂×D₆₇

—

116

C₄×C₆₇

—

117

C₄×C₆₇

#418

yes

—

118

C₄×C₆₇

C₄×D₈

yes

—

119

C₄×C₆₇

#442

yes

—

120

C₄×C₆₇

#466

yes

—

121

C₄×C₆₇

#469

yes

—

122

C₄×C₆₇

#490

yes

—

123

#9832

—

maximal, Sylow

124

C₂×C₆₇

#1717

yes

—

125

C₂×C₆₇

#1721

yes

—

126

C₂×C₆₇

#1961

yes

characteristic

127

C₂×C₆₇

#1970

yes

characteristic

128

C₂×C₆₇

#1972

yes

characteristic

129

D₆₇

—

130

C₂×#493

—

131

#1957

—

132

#1957

—

133

#1834

D₆₇

yes

characteristic, complemented

134

C₂×#322

—

135

C₆₇

#9832

yes

characteristic, complemented, Sylow

136

C₂²×D₈

—

134

—

137

C₂×#127

—

138

#493

—

139

#493

—

134

—

140

C₂×#102

—

134

—

141

C₂×#102

—

134

—

142

C₂×C₄×C₈

—

143

C₂×C₄×C₈

—

144

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

145

#384

C₂×D₆₇

yes

—

146

#340

—

147

#322

—

134

—

148

C₂²×D₄

—

134

—

149

C₂²×D₄

—

134

—

150

C₂²×D₄

—

134

—

151

C₂×D₈

—

268

—

152

C₂×D₈

—

134

—

153

C₂²×C₈

—

154

C₂²×C₈

—

155

C₂²×C₈

C₂²⋊D₆₇

yes

—

156

#127

—

157

#127

—

134

—

158

#127

—

134

—

159

#127

—

134

—

160

C₂×C₄⋊C₄

—

161

C₂×C₄⋊C₄

—

162

C₂×C₄⋊C₄

—

163

C₂×C₄²

C₂²×D₆₇

yes

characteristic

164

#102

—

268

—

165

#102

—

268

—

166

C₄×C₈

—

134

—

167

C₄×C₈

—

134

—

168

C₄×C₈

—

134

—

169

C₄×C₈

—

134

—

170

C₄×C₈

C₄×D₆₇

yes

—

171

C₄×C₈

D₂₆₈

yes

—

172

C₂⁴

—

268

—

173

C₂×D₄

—

268

—

174

C₂×D₄

—

268

—

175

C₂×D₄

—

268

—

176

C₂×D₄

—

268

—

177

C₂×D₄

—

268

—

178

C₂×D₄

—

268

—

179

C₂×D₄

—

268

—

180

C₂×D₄

—

268

—

181

C₂×D₄

—

268

—

182

C₂×D₄

—

134

—

183

C₂×D₄

—

134

—

184

C₂×D₄

—

134

—

185

C₂²×C₄

—

186

C₂²×C₄

D₄×D₆₇

yes

characteristic

187

C₂²×C₄

D₄×D₆₇

yes

characteristic

188

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

189

D₈

—

268

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

—

134

—

193

C₂×C₈

—

134

—

194

C₂×C₈

—

134

—

195

C₂×C₈

—

134

—

196

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

197

C₄⋊C₄

—

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₄²

C₂×C₄×D₆₇

yes

characteristic

201

C₄²

C₂×D₂₆₈

yes

characteristic

202

C₄²

#11761629

yes

characteristic

203

C₄²

#11761629

yes

characteristic

204

C₂³

—

268

—

205

C₂³

—

536

—

206

C₂³

—

536

—

207

C₂³

—

536

—

208

C₂³

—

536

—

209

C₂³

—

536

—

210

C₂³

—

536

—

211

C₂³

#14524039

yes

characteristic

212

D₄

—

268

—

213

D₄

—

268

—

214

D₄

—

268

—

215

D₄

—

536

—

216

D₄

—

536

—

217

D₄

—

536

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

222

C₂×C₄

#14524005

yes

characteristic

223

C₂×C₄

#14524005

yes

characteristic

224

C₂×C₄

#14524018

yes

characteristic

225

C₂×C₄

#14524018

yes

characteristic

226

C₂×C₄

#14524032

yes

characteristic

227

C₂×C₄

D₈×D₆₇

yes

—

228

C₂×C₄

#14524066

yes

—

229

C₂×C₄

#14524068

yes

—

230

C₂×C₄

#14524070

yes

—

231

C₈

—

232

C₈

—

268

—

233

C₈

—

134

—

234

C₂²

—

536

—

235

C₂²

—

536

—

236

C₂²

—

536

—

237

C₂²

—

1072

—

238

C₂²

—

1072

—

239

C₂²

—

1072

—

240

C₂²

—

1072

—

241

C₂²

#16863297

yes

characteristic

242

C₂²

#16863771

yes

characteristic

243

C₂²

#16863781

yes

characteristic

244

C₂²

#16863782

yes

—

245

C₂²

#16863786

yes

—

246

C₄

—

247

C₄

#16863400

yes

—

248

C₄

#16863478

yes

—

249

C₄

#16863507

yes

—

250

C₄

#16863633

yes

—

251

C₄

#16863650

yes

—

252

C₄

#16863762

yes

—

253

C₂

—

1072

—

254

C₂

#22559851

yes

—

255

C₂

#22559877

yes

—

256

C₂

#22561032

yes

characteristic

257

C₂

#22561073

yes

characteristic

258

C₂

#22561083

yes

characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#16863773	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#9832
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1834

C₂

C₂³×C₆₇

C₂

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
2. #22561014	C₂	semidirect
3. #22561014	C₂	semidirect
4. C₂×#16863118	C₂	semidirect
5. C₆₇×#1834	C₂	semidirect
9. #16863773	C₄	semidirect
60. C₂×C₆₇⋊C₈	D₈	semidirect
133. #1834	D₆₇	semidirect
135. C₆₇	#9832	semidirect

H ≅ G/N

type

2. #22561014

C₂

semidirect

3. #22561014

C₂

semidirect

4. C₂×#16863118

C₂

semidirect

5. C₆₇×#1834

C₂

semidirect

9. #16863773

C₄

semidirect

60. C₂×C₆₇⋊C₈

D₈

semidirect

133. #1834

D₆₇

semidirect

135. C₆₇

#9832

semidirect

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

subgroup	index
C₂×#16863773	2
#22561014	2
C₂×#16863118	2
C₆₇×#1834	2
#9832	67

subgroup

index

C₆₇×#1834

Subgroups — #30755641

Conjugacy classes of subgroups

Complements

Maximal subgroups