Group #26104253 — subgroups

Subgroups — #26104253

Order 12032.

684 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 207 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×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	#20241209	C₂	1	4	yes	maximal, complemented
3	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
6	#15688878	C₂²	1	4	yes	—
7	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₄₇	—	2	2	no	—
9	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
10	C₄₇×#390	—	2	2	no	—
11	C₄₇×#390	C₂²	1	2	yes	—
12	C₄₇×#384	C₄	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
16	C₄×C₄₇⋊C₈	—	4	4	no	—
17	C₂²×D₄×C₄₇	—	2	2	no	—
18	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₄₇	—	2	2	no	—
20	#127×C₄₇	—	2	2	no	—
21	#127×C₄₇	C₈	1	4	yes	complemented
22	C₄×D₄×C₄₇	—	4	4	no	—
23	C₄×D₄×C₄₇	—	4	4	no	—
24	C₄×D₄×C₄₇	—	4	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₂⁴×C₄₇	—	2	2	no	—
35	C₂×D₄×C₄₇	—	2	8	no	—
36	C₂×D₄×C₄₇	—	2	2	no	—
37	C₂×D₄×C₄₇	—	4	2	no	—
38	C₂×D₄×C₄₇	—	4	2	no	—
39	C₂×D₄×C₄₇	—	4	4	no	—
40	C₂×D₄×C₄₇	—	4	4	no	—
41	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₂²⋊C₄×C₄₇	—	4	4	no	—
53	C₂²⋊C₄×C₄₇	—	4	4	no	—
54	C₂²⋊C₄×C₄₇	—	4	4	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	—	4	1	no	—
58	C₄²×C₄₇	C₂×C₈	1	2	yes	—
59	C₄²×C₄₇	M₁₆	1	2	yes	—
60	C₄²×C₄₇	D₈	1	4	yes	—
61	C₄²×C₄₇	SD₁₆	1	4	yes	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	4	4	no	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	—	4	2	no	—
67	C₂³×C₄₇	—	4	2	no	—
68	C₂³×C₄₇	—	4	4	no	—
69	C₂³×C₄₇	—	4	4	no	—
70	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	D₄×C₄₇	—	4	4	no	—
72	D₄×C₄₇	—	4	4	no	—
73	D₄×C₄₇	—	4	16	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	8	8	no	—
83	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄×C₄₇	#100	1	2	yes	—
85	C₂×C₄×C₄₇	#102	1	4	yes	—
86	C₂×C₄×C₄₇	#102	1	4	yes	—
87	C₂×C₄×C₄₇	#104	1	4	yes	—
88	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
92	C₄₇⋊C₈	—	4	8	no	—
93	#7569	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	4	4	no	—
95	C₂²×C₄₇	—	4	8	no	—
96	C₂²×C₄₇	—	8	2	no	—
97	C₂²×C₄₇	—	8	4	no	—
98	C₂²×C₄₇	—	8	2	no	—
99	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₄₇	#410	1	1	yes	characteristic
101	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×#102	1	2	yes	—
103	C₂²×C₄₇	C₂×#104	1	2	yes	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	4	1	no	—
109	C₄×C₄₇	—	8	2	no	—
110	C₄×C₄₇	D₄⋊C₈	1	8	yes	—
111	C₄×C₄₇	#331	1	4	yes	—
112	C₄×#127	D₄₇	1	1	yes	characteristic
113	C₄×C₄⋊C₈	—	47	2	no	—
114	#1270	—	47	4	no	—
115	C₂×C₄₇	—	8	4	no	—
116	C₂×C₄₇	#1460	1	1	yes	characteristic
117	C₂×C₄₇	C₂×D₄⋊C₈	1	2	yes	—
118	C₂×C₄₇	#1502	1	2	yes	—
119	C₂×C₄₇	#1525	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
121	C₂×#127	Dic₄₇	1	1	yes	characteristic
122	C₂×C₄×D₄	—	2	2	no	—
123	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
124	C₂×C₄⋊C₈	—	94	2	no	—
125	C₂×C₄×C₈	—	94	2	no	—
126	#390	—	2	2	no	—
127	#390	C₂×D₄₇	1	2	yes	—
128	#384	Dic₄₇	1	1	yes	characteristic
129	C₄³	C₂×D₄₇	1	1	yes	characteristic
130	#336	—	47	4	no	—
131	C₄₇	#7569	1	1	yes	characteristic, complemented, Sylow
132	C₂²×D₄	—	2	2	no	—
133	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
134	C₂³×C₄	—	2	2	no	—
135	C₂²×C₈	—	94	4	no	—
136	#127	—	2	2	no	—
137	#127	C₄₇⋊C₈	1	4	yes	complemented
138	C₄×D₄	—	4	4	no	—
139	C₄×D₄	—	4	4	no	—
140	C₄×D₄	—	4	4	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
143	C₂×C₂²⋊C₄	—	2	2	no	—
144	C₂×C₂²⋊C₄	—	2	4	no	—
145	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
149	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
150	C₄⋊C₈	—	94	8	no	—
151	C₄×C₈	—	188	4	no	—
152	C₂⁴	—	2	2	no	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂×D₄	—	4	4	no	—
159	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	4	8	no	—
163	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
165	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₈	—	94	8	no	—
169	C₂×C₈	—	188	4	no	—
170	C₂×C₈	—	188	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₂²⋊C₄	—	4	4	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	4	1	no	—
179	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
180	C₄²	#10625329	1	2	yes	—
181	C₄²	#10625334	1	4	yes	—
182	C₄²	#10625335	1	4	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	—	4	2	no	—
185	C₂³	—	4	2	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	4	4	no	—
188	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
189	D₄	—	4	4	no	—
190	D₄	—	4	4	no	—
191	D₄	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
202	C₂×C₄	#13093941	1	2	yes	—
203	C₂×C₄	#13093945	1	4	yes	—
204	C₂×C₄	#13093945	1	4	yes	—
205	C₂×C₄	#13093951	1	4	yes	—
206	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
208	C₂×C₄	C₂×#10625334	1	2	yes	—
209	C₂×C₄	C₂×#10625335	1	2	yes	—
210	C₈	—	188	8	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	4	8	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	2	no	—
215	C₂²	—	8	4	no	—
216	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
217	C₂²	#15689097	1	1	yes	characteristic
218	C₂²	C₂×#13093941	1	1	yes	characteristic
219	C₂²	C₂×#13093945	1	2	yes	—
220	C₂²	C₂×#13093951	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	8	2	no	—
227	C₄	D₄⋊(C₄₇⋊C₈)	1	8	yes	—
228	C₄	#15688867	1	4	yes	—
229	C₂	—	8	4	no	—
230	C₂	#20241709	1	1	yes	characteristic
231	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	2	yes	—
232	C₂	#20241883	1	2	yes	—
233	C₂	#20241980	1	1	yes	characteristic
234	C₂	C₂×#15688867	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

#20241209

C₂

yes

maximal, complemented

C₄×#127×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄×C₄₇⋊C₈

—

#15688878

C₂²

yes

—

C₂×#127×C₄₇

C₄

yes

characteristic

C₂×C₄×D₄×C₄₇

—

C₂×C₄×D₄×C₄₇

C₄

yes

—

C₄₇×#390

—

C₄₇×#390

C₂²

yes

—

C₄₇×#384

C₄

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₈

yes

complemented

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

C₂×C₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

SD₁₆

yes

—

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

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

#100

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₄₇⋊C₈

—

#7569

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

100

C₂²×C₄₇

#410

yes

characteristic

101

C₂²×C₄₇

C₂×#100

yes

characteristic

102

C₂²×C₄₇

C₂×#102

yes

—

103

C₂²×C₄₇

C₂×#104

yes

—

104

C₄×C₄₇

—

105

C₄×C₄₇

—

106

C₄×C₄₇

—

107

C₄×C₄₇

—

108

C₄×C₄₇

—

109

C₄×C₄₇

—

110

C₄×C₄₇

D₄⋊C₈

yes

—

111

C₄×C₄₇

#331

yes

—

112

C₄×#127

D₄₇

yes

characteristic

113

C₄×C₄⋊C₈

—

114

#1270

—

115

C₂×C₄₇

—

116

C₂×C₄₇

#1460

yes

characteristic

117

C₂×C₄₇

C₂×D₄⋊C₈

yes

—

118

C₂×C₄₇

#1502

yes

—

119

C₂×C₄₇

#1525

yes

characteristic

120

C₂×C₄₇

C₂×#331

yes

characteristic

121

C₂×#127

Dic₄₇

yes

characteristic

122

C₂×C₄×D₄

—

123

C₂×C₄×D₄

Dic₄₇

yes

—

124

C₂×C₄⋊C₈

—

125

C₂×C₄×C₈

—

126

#390

—

127

#390

C₂×D₄₇

yes

—

128

#384

Dic₄₇

yes

characteristic

129

C₄³

C₂×D₄₇

yes

characteristic

130

#336

—

131

C₄₇

#7569

yes

characteristic, complemented, Sylow

132

C₂²×D₄

—

133

C₂²×D₄

C₂×Dic₄₇

yes

—

134

C₂³×C₄

—

135

C₂²×C₈

—

136

#127

—

137

#127

C₄₇⋊C₈

yes

complemented

138

C₄×D₄

—

139

C₄×D₄

—

140

C₄×D₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

143

C₂×C₂²⋊C₄

—

144

C₂×C₂²⋊C₄

—

145

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

146

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

147

C₂×C₄²

C₂²⋊D₄₇

yes

—

148

C₂×C₄²

C₂²⋊D₄₇

yes

—

149

C₂×C₄²

C₂²×D₄₇

yes

characteristic

150

C₄⋊C₈

—

151

C₄×C₈

—

188

—

152

C₂⁴

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

C₂×C₄₇⋊C₈

yes

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

164

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

165

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

166

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

167

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

168

C₂×C₈

—

169

C₂×C₈

—

188

—

170

C₂×C₈

—

188

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂×C₄₇⋊C₈

yes

—

180

C₄²

#10625329

yes

—

181

C₄²

#10625334

yes

—

182

C₄²

#10625335

yes

—

183

C₂³

—

184

C₂³

—

185

C₂³

—

186

C₂³

—

187

C₂³

—

188

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

189

D₄

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

202

C₂×C₄

#13093941

yes

—

203

C₂×C₄

#13093945

yes

—

204

C₂×C₄

#13093945

yes

—

205

C₂×C₄

#13093951

yes

—

206

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

207

C₂×C₄

C₂×#10625329

yes

characteristic

208

C₂×C₄

C₂×#10625334

yes

—

209

C₂×C₄

C₂×#10625335

yes

—

210

C₈

—

188

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

217

C₂²

#15689097

yes

characteristic

218

C₂²

C₂×#13093941

yes

characteristic

219

C₂²

C₂×#13093945

yes

—

220

C₂²

C₂×#13093951

yes

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

D₄⋊(C₄₇⋊C₈)

yes

—

228

C₄

#15688867

yes

—

229

C₂

—

230

C₂

#20241709

yes

characteristic

231

C₂

C₂×D₄⋊(C₄₇⋊C₈)

yes

—

232

C₂

#20241883

yes

—

233

C₂

#20241980

yes

characteristic

234

C₂

C₂×#15688867

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
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	#7569
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

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 ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. #20241209	C₂	semidirect
21. #127×C₄₇	C₈	semidirect
131. C₄₇	#7569	semidirect
137. #127	C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. C₄×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

2. #20241209

C₂

semidirect

21. #127×C₄₇

C₈

semidirect

131. C₄₇

#7569

semidirect

137. #127

C₄₇⋊C₈

semidirect

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
#20241209	2
C₄×#127×C₄₇	2
#7569	47

subgroup

index

C₄×C₄⋊(C₄₇⋊C₈)

#20241209

C₄×#127×C₄₇

#7569

Subgroups — #26104253

Conjugacy classes of subgroups

Complements

Maximal subgroups