Subgroups — #26246207

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
2	#20244956	C₂	1	8	yes	maximal, complemented
3	#20243109	C₂	1	2	yes	maximal
4	#20243109	C₂	1	2	yes	maximal
5	#20243082	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	#15688989	—	2	2	no	—
10	#15688989	C₂²	1	4	yes	—
11	C₂×C₄₇⋊#95	—	2	2	no	—
12	#15688885	C₂²	1	8	yes	—
13	#15688885	C₂²	1	8	yes	—
14	#15688878	C₂²	1	4	yes	—
15	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#389	C₂²	1	4	yes	—
19	C₄₇×#385	—	2	2	no	—
20	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	8	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	C₄₇⋊#95	—	2	8	no	—
30	C₂³×C₄×C₄₇	D₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	#117×C₄₇	—	2	2	no	—
33	#117×C₄₇	—	2	4	no	—
34	#117×C₄₇	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	—	2	1	no	—
41	C₂×C₄²×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	—	2	1	no	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	#95×C₄₇	—	2	4	no	—
46	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₄₇	#41	1	2	yes	—
55	C₂²×C₄×C₄₇	#41	1	4	yes	—
56	C₂²×C₄×C₄₇	#41	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	—	4	8	no	—
60	C₂²×Dic₄₇	—	4	4	no	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	4	8	no	—
64	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
65	C₂²⋊C₄×C₄₇	—	2	4	no	—
66	C₂²⋊C₄×C₄₇	—	4	2	no	—
67	C₄²×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	—	4	2	no	—
71	C₄⋊Dic₄₇	—	2	8	no	—
72	C₄⋊Dic₄₇	—	4	4	no	—
73	#10625329	—	4	8	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₄₇	#123	1	2	yes	—
80	C₂³×C₄₇	#124	1	2	yes	—
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	4	8	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₄₇	#123	1	2	yes	—
92	C₂×C₄×C₄₇	#124	1	2	yes	—
93	C₂×C₄×C₄₇	#125	1	8	yes	—
94	C₂×C₄×C₄₇	#126	1	8	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	—	8	8	no	—
102	C₄₇⋊C₈	—	4	4	no	—
103	#23746	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	#400	1	4	yes	—
108	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
111	C₂²×C₄₇	#575	1	4	yes	—
112	C₄×C₄₇	—	4	4	no	—
113	C₄×C₄₇	—	4	4	no	—
114	C₄×C₄₇	—	8	2	no	—
115	C₄×C₄₇	#400	1	4	yes	—
116	C₄×C₄₇	C₂×#125	1	2	yes	—
117	C₄×C₄₇	C₂×#126	1	2	yes	—
118	Dic₄₇	—	8	4	no	—
119	#2296	D₄₇	1	1	yes	characteristic
120	C₂×#382	—	47	1	no	—
121	#2101	—	47	8	no	—
122	#1736	—	47	2	no	—
123	#1736	—	47	2	no	—
124	#1728	—	47	1	no	—
125	C₂×C₄₇	—	2	4	no	—
126	C₂×C₄₇	C₂×#400	1	1	yes	characteristic
127	C₂×C₄₇	#3080	1	2	yes	—
128	C₂×C₄₇	#3093	1	2	yes	—
129	C₂×C₄₇	#3140	1	2	yes	—
130	C₂²×M₁₆	—	47	2	no	—
131	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
132	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	—	47	2	no	—
134	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
135	#389	C₂×D₄₇	1	4	yes	—
136	#385	—	2	2	no	—
137	#382	—	47	4	no	—
138	#382	—	94	2	no	—
139	C₄×C₄⋊C₄	—	2	2	no	—
140	C₂×#95	—	94	2	no	—
141	#340	—	47	8	no	—
142	#340	—	47	8	no	—
143	#336	—	47	4	no	—
144	C₄₇	#23746	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	—	94	2	no	—
146	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
148	C₂×M₁₆	—	94	4	no	—
149	C₂×M₁₆	—	94	4	no	—
150	C₂×M₁₆	—	94	4	no	—
151	C₂²×C₈	—	47	4	no	—
152	#117	—	2	2	no	—
153	#117	—	2	4	no	—
154	#117	Dic₉₄	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	94	8	no	—
158	C₂×C₄⋊C₄	—	94	2	no	—
159	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
161	C₂×C₂²⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
168	#95	—	2	4	no	—
169	#95	—	94	8	no	—
170	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	—	94	4	no	—
176	C₂²×C₄	—	188	8	no	—
177	C₂²×C₄	—	188	4	no	—
178	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
179	C₂²×C₄	#10625350	1	2	yes	—
180	C₂²×C₄	#10625352	1	4	yes	—
181	C₂²×C₄	#10625354	1	4	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	M₁₆	—	188	8	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₂×C₈	—	94	4	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	8	no	—
192	C₄⋊C₄	—	94	8	no	—
193	C₄⋊C₄	—	188	4	no	—
194	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	#13094003	1	2	yes	—
203	C₂³	#13094005	1	2	yes	—
204	C₂³	#13094013	1	2	yes	—
205	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	376	8	no	—
215	C₂×C₄	—	188	4	no	—
216	C₂×C₄	—	188	8	no	—
217	C₂×C₄	#13094003	1	2	yes	—
218	C₂×C₄	#13094005	1	2	yes	—
219	C₂×C₄	#13094013	1	2	yes	—
220	C₂×C₄	#13094015	1	8	yes	—
221	C₂×C₄	#13094019	1	8	yes	—
222	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
223	C₂×C₄	C₂×#10625352	1	2	yes	—
224	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
225	C₂×C₄	C₂×#10625354	1	2	yes	—
226	C₈	—	188	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	#15689062	1	4	yes	—
231	C₂²	C₂×#13094003	1	1	yes	characteristic
232	C₂²	C₂×#13094005	1	1	yes	characteristic
233	C₂²	C₂×#13094013	1	1	yes	characteristic
234	C₂²	#15689921	1	4	yes	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	376	4	no	—
239	C₄	#15689062	1	4	yes	—
240	C₄	C₂×#13094015	1	2	yes	—
241	C₄	C₂×#13094019	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×#15689062	1	1	yes	characteristic
244	C₂	#20253210	1	2	yes	—
245	C₂	#20253317	1	2	yes	—
246	C₂	#20253777	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2296	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#23746
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20244956	C₂	semidirect
144. C₄₇	#23746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688989	2
#20244956	2
#20243109	2
#20243082	2
C₄₇×#2296	2
#23746	47