Group #26115099 — subgroups

Subgroups — #26115099

Order 12032.

726 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20243505	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×#13093935	—	2	2	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688878	—	2	2	no	—
9	D₄⋊(C₄₇⋊C₈)	—	2	8	no	—
10	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₇	—	2	2	no	—
12	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#390	—	2	2	no	—
14	C₄₇×#390	C₄	1	2	yes	—
15	C₄₇×#384	C₄	1	1	yes	characteristic
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	#13093935	—	2	8	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₄×C₄₇⋊C₈	D₄	1	4	yes	complemented
23	C₂²×D₄×C₄₇	—	2	2	no	—
24	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	—	2	2	no	—
26	#127×C₄₇	—	2	4	no	—
27	C₄×D₄×C₄₇	—	2	4	no	—
28	C₄×D₄×C₄₇	—	2	4	no	—
29	C₄×D₄×C₄₇	—	4	8	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄₇	—	2	2	no	—
40	C₂×D₄×C₄₇	—	2	8	no	—
41	C₂×D₄×C₄₇	—	2	4	no	—
42	C₂×D₄×C₄₇	—	2	2	no	—
43	C₂×D₄×C₄₇	—	2	4	no	—
44	C₂×D₄×C₄₇	—	4	4	no	—
45	C₂×D₄×C₄₇	—	4	4	no	—
46	C₂×D₄×C₄₇	M₁₆	1	4	yes	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	8	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₂²⋊C₄×C₄₇	—	4	8	no	—
59	C₂²⋊C₄×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	—	2	4	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	4	no	—
64	C₄²×C₄₇	C₂×D₄	1	2	yes	—
65	C₄²×C₄₇	#41	1	2	yes	—
66	C₂×C₄₇⋊C₈	—	2	8	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₂×C₄₇⋊C₈	—	4	4	no	—
70	C₂³×C₄₇	—	2	4	no	—
71	C₂³×C₄₇	—	4	2	no	—
72	C₂³×C₄₇	—	4	2	no	—
73	C₂³×C₄₇	—	4	4	no	—
74	C₂³×C₄₇	—	4	4	no	—
75	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	D₄×C₄₇	—	2	8	no	—
77	D₄×C₄₇	—	4	16	no	—
78	D₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	8	4	no	—
88	C₂×C₄×C₄₇	—	8	4	no	—
89	C₂×C₄×C₄₇	#104	1	4	yes	—
90	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	#121	1	2	yes	—
92	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₄₇	#131	1	2	yes	—
96	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₄₇⋊C₈	—	2	8	no	—
98	C₄₇⋊C₈	—	8	2	no	—
99	#9648	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	4	4	no	—
101	C₂²×C₄₇	—	4	8	no	—
102	C₂²×C₄₇	—	8	2	no	—
103	C₂²×C₄₇	—	8	2	no	—
104	C₂²×C₄₇	—	8	4	no	—
105	C₂²×C₄₇	#390	1	1	yes	characteristic
106	C₂²×C₄₇	#407	1	1	yes	characteristic
107	C₂²×C₄₇	#408	1	1	yes	characteristic
108	C₂²×C₄₇	#418	1	2	yes	—
109	C₂²×C₄₇	C₂×#104	1	2	yes	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	4	2	no	—
113	C₄×C₄₇	—	8	2	no	—
114	C₄×C₄₇	#435	1	4	yes	—
115	C₄×C₄₇	#490	1	2	yes	—
116	C₄×C₄₇	#496	1	2	yes	—
117	C₄×#127	D₄₇	1	1	yes	characteristic
118	#1831	—	47	1	no	—
119	C₄×#97	—	47	1	no	—
120	C₂×D₄⋊C₈	—	47	4	no	—
121	C₂×C₄₇	—	8	4	no	—
122	C₂×C₄₇	#1484	1	2	yes	—
123	C₂×C₄₇	#1490	1	2	yes	—
124	C₂×C₄₇	#1955	1	1	yes	characteristic
125	C₂×C₄₇	#1961	1	1	yes	characteristic
126	C₂×C₄₇	#1962	1	1	yes	characteristic
127	C₂×#127	Dic₄₇	1	1	yes	characteristic
128	C₂×C₄×D₄	—	2	2	no	—
129	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
130	C₂×C₄⋊C₈	—	47	2	no	—
131	C₂×#97	—	94	2	no	—
132	C₂×C₄×C₈	—	47	2	no	—
133	#390	—	2	2	no	—
134	#390	Dic₄₇	1	2	yes	—
135	#384	Dic₄₇	1	1	yes	characteristic
136	C₄³	C₂×D₄₇	1	1	yes	characteristic
137	#336	—	94	2	no	—
138	D₄⋊C₈	—	94	8	no	—
139	C₄₇	#9648	1	1	yes	characteristic, complemented, Sylow
140	C₂²×D₄	—	2	2	no	—
141	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
142	C₂³×C₄	—	2	2	no	—
143	C₂²×C₈	—	94	2	no	—
144	C₂²×C₈	—	94	2	no	—
145	#127	—	2	4	no	—
146	C₄×D₄	—	2	4	no	—
147	C₄×D₄	—	2	4	no	—
148	C₄×D₄	—	4	8	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₂²⋊C₄	—	2	2	no	—
153	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
156	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
157	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
158	C₄⋊C₈	—	94	4	no	—
159	#97	—	94	8	no	—
160	C₄×C₈	—	47	4	no	—
161	C₄×C₈	—	94	2	no	—
162	C₂⁴	—	2	2	no	—
163	C₂×D₄	—	2	8	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	#10625329	1	4	yes	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	8	no	—
174	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×C₈	—	94	8	no	—
180	C₂×C₈	—	94	4	no	—
181	C₂×C₈	—	188	2	no	—
182	C₂×C₈	—	188	4	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₂²⋊C₄	—	4	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	4	no	—
191	C₄²	D₄×D₄₇	1	2	yes	—
192	C₄²	#10625352	1	2	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	4	no	—
198	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
199	D₄	—	2	8	no	—
200	D₄	—	4	16	no	—
201	D₄	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	#13093951	1	4	yes	—
213	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
214	C₂×C₄	#13093995	1	2	yes	—
215	C₂×C₄	#13094011	1	1	yes	characteristic
216	C₂×C₄	#13094020	1	1	yes	characteristic
217	C₂×C₄	C₂×#10625329	1	2	yes	—
218	C₂×C₄	#13094031	1	2	yes	—
219	C₂×C₄	#13094050	1	4	yes	—
220	C₈	—	94	8	no	—
221	C₈	—	376	2	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	4	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	8	2	no	—
226	C₂²	—	8	2	no	—
227	C₂²	#15689026	1	1	yes	characteristic
228	C₂²	#15689087	1	1	yes	characteristic
229	C₂²	#15689090	1	1	yes	characteristic
230	C₂²	#15689128	1	2	yes	—
231	C₂²	C₂×#13093951	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	#15689195	1	4	yes	—
237	C₄	#15689487	1	2	yes	—
238	C₄	#15689510	1	2	yes	—
239	C₂	—	8	4	no	—
240	C₂	#20241805	1	2	yes	—
241	C₂	#20241831	1	2	yes	—
242	C₂	#20244187	1	1	yes	characteristic
243	C₂	#20244213	1	1	yes	characteristic
244	C₂	#20244217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243505

C₂

yes

maximal, characteristic, complemented

C₄×#13093935

C₂

yes

maximal, characteristic, complemented

C₂×D₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₄×#127×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×#13093935

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

—

D₄⋊(C₄₇⋊C₈)

—

C₂×#127×C₄₇

C₄

yes

characteristic

C₂×C₄×D₄×C₄₇

—

C₂×C₄×D₄×C₄₇

C₂²

yes

—

C₄₇×#390

—

C₄₇×#390

C₄

yes

—

C₄₇×#384

C₄

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

complemented

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

—

#127×C₄₇

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9648

—

maximal, Sylow

100

C₂²×C₄₇

—

101

C₂²×C₄₇

—

102

C₂²×C₄₇

—

103

C₂²×C₄₇

—

104

C₂²×C₄₇

—

105

C₂²×C₄₇

#390

yes

characteristic

106

C₂²×C₄₇

#407

yes

characteristic

107

C₂²×C₄₇

#408

yes

characteristic

108

C₂²×C₄₇

#418

yes

—

109

C₂²×C₄₇

C₂×#104

yes

—

110

C₄×C₄₇

—

111

C₄×C₄₇

—

112

C₄×C₄₇

—

113

C₄×C₄₇

—

114

C₄×C₄₇

#435

yes

—

115

C₄×C₄₇

#490

yes

—

116

C₄×C₄₇

#496

yes

—

117

C₄×#127

D₄₇

yes

characteristic

118

#1831

—

119

C₄×#97

—

120

C₂×D₄⋊C₈

—

121

C₂×C₄₇

—

122

C₂×C₄₇

#1484

yes

—

123

C₂×C₄₇

#1490

yes

—

124

C₂×C₄₇

#1955

yes

characteristic

125

C₂×C₄₇

#1961

yes

characteristic

126

C₂×C₄₇

#1962

yes

characteristic

127

C₂×#127

Dic₄₇

yes

characteristic

128

C₂×C₄×D₄

—

129

C₂×C₄×D₄

C₂×D₄₇

yes

—

130

C₂×C₄⋊C₈

—

131

C₂×#97

—

132

C₂×C₄×C₈

—

133

#390

—

134

#390

Dic₄₇

yes

—

135

#384

Dic₄₇

yes

characteristic

136

C₄³

C₂×D₄₇

yes

characteristic

137

#336

—

138

D₄⋊C₈

—

139

C₄₇

#9648

yes

characteristic, complemented, Sylow

140

C₂²×D₄

—

141

C₂²×D₄

C₂×Dic₄₇

yes

—

142

C₂³×C₄

—

143

C₂²×C₈

—

144

C₂²×C₈

—

145

#127

—

146

C₄×D₄

—

147

C₄×D₄

—

148

C₄×D₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

151

C₂×C₂²⋊C₄

—

152

C₂×C₂²⋊C₄

—

153

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

154

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

155

C₂×C₄²

C₂²⋊D₄₇

yes

—

156

C₂×C₄²

C₂²⋊D₄₇

yes

—

157

C₂×C₄²

C₂²×D₄₇

yes

characteristic

158

C₄⋊C₈

—

159

#97

—

160

C₄×C₈

—

161

C₄×C₈

—

162

C₂⁴

—

163

C₂×D₄

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

#10625329

yes

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

175

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

176

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

177

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

178

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

188

—

182

C₂×C₈

—

188

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₂²⋊C₄

—

186

C₂²⋊C₄

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

191

C₄²

D₄×D₄₇

yes

—

192

C₄²

#10625352

yes

—

193

C₂³

—

194

C₂³

—

195

C₂³

—

196

C₂³

—

197

C₂³

—

198

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

199

D₄

—

200

D₄

—

201

D₄

—

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

#13093951

yes

—

213

C₂×C₄

D₄×Dic₄₇

yes

—

214

C₂×C₄

#13093995

yes

—

215

C₂×C₄

#13094011

yes

characteristic

216

C₂×C₄

#13094020

yes

characteristic

217

C₂×C₄

C₂×#10625329

yes

—

218

C₂×C₄

#13094031

yes

—

219

C₂×C₄

#13094050

yes

—

220

C₈

—

221

C₈

—

376

—

222

C₂²

—

223

C₂²

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

#15689026

yes

characteristic

228

C₂²

#15689087

yes

characteristic

229

C₂²

#15689090

yes

characteristic

230

C₂²

#15689128

yes

—

231

C₂²

C₂×#13093951

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

#15689195

yes

—

237

C₄

#15689487

yes

—

238

C₄

#15689510

yes

—

239

C₂

—

240

C₂

#20241805

yes

—

241

C₂

#20241831

yes

—

242

C₂

#20244187

yes

characteristic

243

C₂

#20244213

yes

characteristic

244

C₂

#20244217

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#9648
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

prime radical

C₂×#127×C₄₇

C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

N	H ≅ G/N	type
1. #20243505	C₂	semidirect
2. C₄×#13093935	C₂	semidirect
3. C₂×D₄⋊(C₄₇⋊C₈)	C₂	semidirect
22. C₄×C₄₇⋊C₈	D₄	semidirect
139. C₄₇	#9648	semidirect

H ≅ G/N

type

1. #20243505

C₂

semidirect

2. C₄×#13093935

C₂

semidirect

3. C₂×D₄⋊(C₄₇⋊C₈)

C₂

semidirect

22. C₄×C₄₇⋊C₈

D₄

semidirect

139. C₄₇

#9648

semidirect

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

subgroup	index
#20243505	2
C₄×#13093935	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#127×C₄₇	2
#9648	47

subgroup

index

#20243505

C₄×#13093935

C₂×D₄⋊(C₄₇⋊C₈)

C₄×#127×C₄₇

#9648

Subgroups — #26115099

Conjugacy classes of subgroups

Complements

Maximal subgroups