Group #26118896 — subgroups

Subgroups — #26118896

Order 12032.

562 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15689410	C₂	1	2	yes	maximal
2	#20244182	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688872	C₂	1	2	yes	maximal, complemented
4	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
7	#15689410	—	2	4	no	—
8	#15689410	—	2	4	no	—
9	C₂×#13093955	—	2	2	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093944	—	2	2	no	—
13	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
14	#15688872	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#500	—	2	4	no	—
17	C₂×#107×C₄₇	—	2	2	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₄₇	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₄×D₁₈₈	—	2	2	no	—
24	C₄×D₁₈₈	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
26	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
27	#13093955	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
31	#13093944	—	2	8	no	—
32	(C₂×D₄₇)⋊C₈	—	4	4	no	—
33	C₂²×Q₈×C₄₇	—	2	2	no	—
34	C₂²×C₈×C₄₇	D₄	1	2	yes	—
35	C₄⋊Q₈×C₄₇	—	2	2	no	—
36	C₄⋊Q₈×C₄₇	—	2	2	no	—
37	C₄⋊Q₈×C₄₇	—	2	2	no	—
38	C₄⋊Q₈×C₄₇	D₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#107×C₄₇	—	2	8	no	—
43	C₄×C₈×C₄₇	—	2	2	no	—
44	C₂³×D₄₇	—	2	1	no	—
45	C₂×Q₈×C₄₇	—	2	4	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂×Q₈×C₄₇	—	8	4	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	1	no	—
53	C₂×D₁₈₈	—	2	2	no	—
54	C₂×D₁₈₈	—	2	2	no	—
55	C₂×D₁₈₈	—	2	1	no	—
56	C₂×D₁₈₈	Q₁₆	1	2	yes	—
57	C₂×C₄×D₄₇	—	4	1	no	—
58	C₂×C₄×D₄₇	—	4	1	no	—
59	C₂×C₄×D₄₇	—	4	2	no	—
60	C₂×C₄×D₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	C₂×D₄	1	2	yes	—
69	C₄²×C₄₇	C₂×D₄	1	2	yes	—
70	(C₂×D₄₇)⋊C₄	—	4	2	no	—
71	C₄⋊Dic₄₇	—	2	2	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	1	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂³×C₄₇	#121	1	1	yes	characteristic
78	C₂²×D₄₇	—	2	2	no	—
79	C₂²×D₄₇	—	4	1	no	—
80	C₂²×D₄₇	—	4	2	no	—
81	C₂²×D₄₇	—	4	1	no	—
82	C₂²×D₄₇	—	4	2	no	—
83	Q₈×C₄₇	—	8	4	no	—
84	Q₈×C₄₇	—	8	4	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	8	4	no	—
88	C₂×C₄×C₄₇	#120	1	2	yes	—
89	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#121	1	2	yes	—
91	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#137	1	2	yes	—
96	C₂×C₄×C₄₇	#137	1	2	yes	—
97	C₂×Dic₄₇	—	4	1	no	—
98	C₂×Dic₄₇	—	4	2	no	—
99	D₁₈₈	—	2	4	no	—
100	D₁₈₈	—	4	2	no	—
101	C₄×D₄₇	—	8	2	no	—
102	C₄×D₄₇	—	8	1	no	—
103	C₄×D₄₇	—	8	1	no	—
104	C₈×C₄₇	—	2	4	no	—
105	C₄₇⋊C₈	—	4	4	no	—
106	C₄₇⋊C₈	—	8	1	no	—
107	#10278	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	#392	1	1	yes	characteristic
109	C₂²×C₄₇	#461	1	2	yes	—
110	C₂²×C₄₇	#467	1	1	yes	characteristic
111	C₂²×C₄₇	#470	1	1	yes	characteristic
112	C₂²×C₄₇	#472	1	2	yes	—
113	C₂×D₄₇	—	4	4	no	—
114	C₂×D₄₇	—	4	2	no	—
115	C₂×D₄₇	—	8	2	no	—
116	C₂×D₄₇	—	8	1	no	—
117	C₂×D₄₇	—	8	1	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	8	4	no	—
120	C₄×C₄₇	#448	1	2	yes	—
121	C₄×C₄₇	#451	1	2	yes	—
122	C₄×C₄₇	#452	1	2	yes	—
123	C₄×C₄₇	#492	1	1	yes	characteristic
124	C₄×C₄₇	#497	1	1	yes	characteristic
125	Dic₄₇	—	8	1	no	—
126	C₂×#500	D₄₇	1	1	yes	characteristic, complemented
127	C₂×#476	—	47	2	no	—
128	#1954	—	47	1	no	—
129	C₂×#333	—	47	2	no	—
130	#1277	—	47	1	no	—
131	C₂×C₄₇	#1675	1	2	yes	—
132	C₂×C₄₇	#1695	1	2	yes	—
133	C₂×C₄₇	#2001	1	1	yes	characteristic
134	C₂×C₄₇	#2013	1	1	yes	characteristic
135	C₂×C₄₇	#2018	1	1	yes	characteristic
136	D₄₇	—	8	2	no	—
137	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
138	C₂×C₄×D₄	—	47	1	no	—
139	#500	—	2	4	no	—
140	#476	—	94	4	no	—
141	#476	—	94	4	no	—
142	C₂×#107	—	2	2	no	—
143	C₂×#106	—	94	2	no	—
144	C₂×C₄⋊C₈	—	47	2	no	—
145	C₂×C₄⋊C₈	—	47	1	no	—
146	C₂×#102	—	94	2	no	—
147	C₂×C₂²⋊C₈	—	94	2	no	—
148	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
149	#333	—	94	4	no	—
150	C₄₇	#10278	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	2	2	no	—
152	C₂²×D₄	—	47	1	no	—
153	C₂³×C₄	—	94	1	no	—
154	C₂²×C₈	—	94	2	no	—
155	C₂²×C₈	—	94	1	no	—
156	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	C₂²⋊D₄₇	1	4	yes	—
161	C₄×D₄	—	94	2	no	—
162	C₄×D₄	—	94	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	47	1	no	—
166	C₂×C₂²⋊C₄	—	94	1	no	—
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	#107	—	2	8	no	—
169	#106	—	94	8	no	—
170	C₄⋊C₈	—	47	4	no	—
171	C₄⋊C₈	—	94	2	no	—
172	C₄⋊C₈	—	94	2	no	—
173	#102	—	94	8	no	—
174	C₂²⋊C₈	—	188	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₂⁴	—	94	1	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	8	4	no	—
180	C₂×D₄	—	47	2	no	—
181	C₂×D₄	—	94	2	no	—
182	C₂×D₄	—	94	1	no	—
183	C₂×D₄	—	94	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	94	1	no	—
186	C₂²×C₄	—	188	1	no	—
187	C₂²×C₄	—	188	1	no	—
188	C₂²×C₄	—	188	2	no	—
189	C₂²×C₄	—	188	2	no	—
190	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	#10625352	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	—	188	1	no	—
197	C₂×C₈	—	188	2	no	—
198	C₂×C₈	—	188	2	no	—
199	C₂×C₈	—	188	2	no	—
200	C₂×C₈	#10625336	1	4	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	94	2	no	—
206	C₂²⋊C₄	—	188	2	no	—
207	C₄²	D₄×D₄₇	1	2	yes	—
208	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
209	C₂³	—	94	2	no	—
210	C₂³	—	188	1	no	—
211	C₂³	—	188	2	no	—
212	C₂³	—	188	1	no	—
213	C₂³	—	188	2	no	—
214	C₂³	#13093995	1	1	yes	characteristic
215	Q₈	—	8	4	no	—
216	Q₈	—	8	4	no	—
217	D₄	—	94	4	no	—
218	D₄	—	188	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	376	1	no	—
223	C₂×C₄	—	376	1	no	—
224	C₂×C₄	—	376	2	no	—
225	C₂×C₄	—	188	1	no	—
226	C₂×C₄	—	188	2	no	—
227	C₂×C₄	#13093992	1	2	yes	—
228	C₂×C₄	#13093993	1	1	yes	characteristic
229	C₂×C₄	#13093996	1	2	yes	—
230	C₂×C₄	#13094020	1	1	yes	characteristic
231	C₂×C₄	C₂×#10625336	1	2	yes	—
232	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
233	C₂×C₄	#13094047	1	1	yes	characteristic
234	C₂×C₄	#13094055	1	2	yes	—
235	C₂×C₄	#13094056	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	376	1	no	—
238	C₈	—	188	4	no	—
239	C₂²	—	376	1	no	—
240	C₂²	—	376	2	no	—
241	C₂²	—	376	1	no	—
242	C₂²	—	188	4	no	—
243	C₂²	—	188	2	no	—
244	C₂²	#15689031	1	1	yes	characteristic
245	C₂²	#15689325	1	2	yes	—
246	C₂²	#15689361	1	1	yes	characteristic
247	C₂²	#15689378	1	1	yes	characteristic
248	C₂²	#15689390	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	—	376	1	no	—
252	C₄	#15689262	1	2	yes	—
253	C₄	#15689277	1	2	yes	—
254	C₄	#15689284	1	2	yes	—
255	C₄	#15689497	1	1	yes	characteristic
256	C₄	#15689515	1	1	yes	characteristic
257	C₂	—	376	2	no	—
258	C₂	#20242784	1	2	yes	—
259	C₂	#20242908	1	2	yes	—
260	C₂	#20244395	1	1	yes	characteristic
261	C₂	#20244471	1	1	yes	characteristic
262	C₂	#20244495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689410

C₂

yes

maximal

#20244182

C₂

yes

maximal, characteristic

C₂×#15688872

C₂

yes

maximal, complemented

#20241223

C₂

yes

maximal, characteristic, complemented

C₂×C₄₇×#500

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₁₈₈

C₂²

yes

—

—

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×#13093944

—

C₂×(C₂×D₄₇)⋊C₈

—

#15688872

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#500

—

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×D₁₈₈

D₄

yes

characteristic

C₂²×C₄×D₄₇

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄×D₁₈₈

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

#13093944

—

(C₂×D₄₇)⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₄×C₈×C₄₇

—

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₁₆

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#121

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

100

D₁₈₈

—

101

C₄×D₄₇

—

102

C₄×D₄₇

—

103

C₄×D₄₇

—

104

C₈×C₄₇

—

105

C₄₇⋊C₈

—

106

C₄₇⋊C₈

—

107

#10278

—

maximal, Sylow

108

C₂²×C₄₇

#392

yes

characteristic

109

C₂²×C₄₇

#461

yes

—

110

C₂²×C₄₇

#467

yes

characteristic

111

C₂²×C₄₇

#470

yes

characteristic

112

C₂²×C₄₇

#472

yes

—

113

C₂×D₄₇

—

114

C₂×D₄₇

—

115

C₂×D₄₇

—

116

C₂×D₄₇

—

117

C₂×D₄₇

—

118

C₄×C₄₇

—

119

C₄×C₄₇

—

120

C₄×C₄₇

#448

yes

—

121

C₄×C₄₇

#451

yes

—

122

C₄×C₄₇

#452

yes

—

123

C₄×C₄₇

#492

yes

characteristic

124

C₄×C₄₇

#497

yes

characteristic

125

Dic₄₇

—

126

C₂×#500

D₄₇

yes

characteristic, complemented

127

C₂×#476

—

128

#1954

—

129

C₂×#333

—

130

#1277

—

131

C₂×C₄₇

#1675

yes

—

132

C₂×C₄₇

#1695

yes

—

133

C₂×C₄₇

#2001

yes

characteristic

134

C₂×C₄₇

#2013

yes

characteristic

135

C₂×C₄₇

#2018

yes

characteristic

136

D₄₇

—

137

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

138

C₂×C₄×D₄

—

139

#500

—

140

#476

—

141

#476

—

142

C₂×#107

—

143

C₂×#106

—

144

C₂×C₄⋊C₈

—

145

C₂×C₄⋊C₈

—

146

C₂×#102

—

147

C₂×C₂²⋊C₈

—

148

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

149

#333

—

150

C₄₇

#10278

yes

characteristic, complemented, Sylow

151

C₂²×Q₈

—

152

C₂²×D₄

—

153

C₂³×C₄

—

154

C₂²×C₈

—

155

C₂²×C₈

—

156

C₂²×C₈

C₂²⋊D₄₇

yes

—

157

C₄⋊Q₈

—

158

C₄⋊Q₈

—

159

C₄⋊Q₈

—

160

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

161

C₄×D₄

—

162

C₄×D₄

—

163

C₂×C₄⋊C₄

—

164

C₂×C₄⋊C₄

—

165

C₂×C₄⋊C₄

—

166

C₂×C₂²⋊C₄

—

167

C₂×C₄²

C₂²×D₄₇

yes

characteristic

168

#107

—

169

#106

—

170

C₄⋊C₈

—

171

C₄⋊C₈

—

172

C₄⋊C₈

—

173

#102

—

174

C₂²⋊C₈

—

188

—

175

C₄×C₈

—

176

C₂⁴

—

177

C₂×Q₈

—

178

C₂×Q₈

—

179

C₂×Q₈

—

180

C₂×D₄

—

181

C₂×D₄

—

182

C₂×D₄

—

183

C₂×D₄

—

184

C₂²×C₄

—

185

C₂²×C₄

—

186

C₂²×C₄

—

188

—

187

C₂²×C₄

—

188

—

188

C₂²×C₄

—

188

—

189

C₂²×C₄

—

188

—

190

C₂²×C₄

D₄×D₄₇

yes

characteristic

191

C₂²×C₄

#10625352

yes

characteristic

192

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

193

C₂×C₈

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

—

188

—

197

C₂×C₈

—

188

—

198

C₂×C₈

—

188

—

199

C₂×C₈

—

188

—

200

C₂×C₈

#10625336

yes

—

201

C₄⋊C₄

—

202

C₄⋊C₄

—

203

C₄⋊C₄

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

206

C₂²⋊C₄

—

188

—

207

C₄²

D₄×D₄₇

yes

—

208

C₄²

C₂×C₂²⋊D₄₇

yes

—

209

C₂³

—

210

C₂³

—

188

—

211

C₂³

—

188

—

212

C₂³

—

188

—

213

C₂³

—

188

—

214

C₂³

#13093995

yes

characteristic

215

Q₈

—

216

Q₈

—

217

D₄

—

218

D₄

—

188

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

376

—

223

C₂×C₄

—

376

—

224

C₂×C₄

—

376

—

225

C₂×C₄

—

188

—

226

C₂×C₄

—

188

—

227

C₂×C₄

#13093992

yes

—

228

C₂×C₄

#13093993

yes

characteristic

229

C₂×C₄

#13093996

yes

—

230

C₂×C₄

#13094020

yes

characteristic

231

C₂×C₄

C₂×#10625336

yes

—

232

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

233

C₂×C₄

#13094047

yes

characteristic

234

C₂×C₄

#13094055

yes

—

235

C₂×C₄

#13094056

yes

—

236

C₈

—

237

C₈

—

376

—

238

C₈

—

188

—

239

C₂²

—

376

—

240

C₂²

—

376

—

241

C₂²

—

376

—

242

C₂²

—

188

—

243

C₂²

—

188

—

244

C₂²

#15689031

yes

characteristic

245

C₂²

#15689325

yes

—

246

C₂²

#15689361

yes

characteristic

247

C₂²

#15689378

yes

characteristic

248

C₂²

#15689390

yes

—

249

C₄

—

250

C₄

—

251

C₄

—

376

—

252

C₄

#15689262

yes

—

253

C₄

#15689277

yes

—

254

C₄

#15689284

yes

—

255

C₄

#15689497

yes

characteristic

256

C₄

#15689515

yes

characteristic

257

C₂

—

376

—

258

C₂

#20242784

yes

—

259

C₂

#20242908

yes

—

260

C₂

#20244395

yes

characteristic

261

C₂

#20244471

yes

characteristic

262

C₂

#20244495

yes

characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#10278
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂³

#13093995

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#500

C₂

socle

C₂³×C₄₇

#121

prime radical

C₂²×D₁₈₈

D₄

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
3. C₂×#15688872	C₂	semidirect
4. #20241223	C₂	semidirect
5. C₂×C₄₇×#500	C₂	semidirect
126. C₂×#500	D₄₇	semidirect
150. C₄₇	#10278	semidirect

H ≅ G/N

type

3. C₂×#15688872

C₂

semidirect

4. #20241223

C₂

semidirect

5. C₂×C₄₇×#500

C₂

semidirect

126. C₂×#500

D₄₇

semidirect

150. C₄₇

#10278

semidirect

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

subgroup	index
C₂×#15689410	2
#20244182	2
C₂×#15688872	2
#20241223	2
C₂×C₄₇×#500	2
#10278	47

subgroup

index

C₂×C₄₇×#500

Subgroups — #26118896

Conjugacy classes of subgroups

Complements

Maximal subgroups