Subgroups — #26216470

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20243518	C₂	1	4	yes	maximal, complemented
4	#20243515	C₂	1	4	yes	maximal, complemented
5	#20243104	C₂	1	4	yes	maximal
6	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	C₂²	1	2	yes	—
14	#15688983	—	2	2	no	—
15	C₄×C₄⋊Dic₄₇	—	2	2	no	—
16	#15688885	C₂²	1	16	yes	—
17	C₂×#117×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₂²	1	2	yes	—
20	C₄₇×#382	C₂²	1	2	yes	—
21	C₄₇×#380	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
33	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#95×C₄₇	—	2	4	no	—
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	4	yes	—
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₄⋊C₄×C₄₇	—	2	8	no	—
62	C₄⋊C₄×C₄₇	—	4	8	no	—
63	C₂²⋊C₄×C₄₇	—	4	4	no	—
64	C₄²×C₄₇	—	4	4	no	—
65	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
69	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²⋊Dic₄₇	—	4	4	no	—
71	C₄⋊Dic₄₇	—	2	8	no	—
72	C₄⋊Dic₄₇	—	4	8	no	—
73	C₄×Dic₄₇	—	4	4	no	—
74	C₂×C₄₇⋊C₈	—	2	16	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₄₇	#124	1	2	yes	—
81	C₂³×C₄₇	#124	1	2	yes	—
82	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	16	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	—	4	8	no	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₄₇	#117	1	8	yes	—
91	C₂×C₄×C₄₇	#124	1	2	yes	—
92	C₂×C₄×C₄₇	#124	1	2	yes	—
93	C₂×C₄×C₄₇	#125	1	8	yes	—
94	C₂×C₄×C₄₇	#135	1	8	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₄₇⋊C₈	—	2	8	no	—
102	#20760	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	#382	1	4	yes	—
107	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×#135	1	4	yes	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	8	4	no	—
113	C₄×C₄₇	#382	1	4	yes	—
114	C₄×C₄₇	#427	1	8	yes	—
115	C₄×C₄₇	C₂×#117	1	2	yes	—
116	C₄×C₄₇	C₂×#125	1	2	yes	—
117	Dic₄₇	—	8	4	no	—
118	C₂²×C₄×C₈	—	47	1	no	—
119	#2285	—	47	1	no	—
120	#2285	D₄₇	1	1	yes	characteristic
121	#1835	—	47	4	no	—
122	#1834	—	47	4	no	—
123	#1735	—	47	4	no	—
124	C₂×C₄₇	—	2	4	no	—
125	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#427	1	2	yes	—
127	C₂×C₄₇	#3131	1	2	yes	—
128	C₂×C₄₇	#3131	1	2	yes	—
129	C₂³×C₈	—	47	2	no	—
130	C₂×#117	—	47	2	no	—
131	C₂×#117	C₂×D₄₇	1	2	yes	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	C₂×C₄×C₈	—	47	8	no	—
134	C₂×C₄×C₈	—	47	2	no	—
135	C₂×C₄×C₈	—	47	2	no	—
136	#384	—	47	2	no	—
137	#384	C₂×D₄₇	1	2	yes	—
138	#382	—	47	2	no	—
139	#382	C₂×D₄₇	1	2	yes	—
140	#380	—	2	2	no	—
141	#380	—	94	2	no	—
142	C₄×C₄⋊C₄	—	2	2	no	—
143	C₄×C₄⋊C₄	—	94	2	no	—
144	#340	—	47	16	no	—
145	C₄₇	#20760	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
148	C₂²×C₈	—	47	8	no	—
149	C₂²×C₈	—	47	4	no	—
150	C₂²×C₈	—	94	4	no	—
151	C₂²×C₈	—	94	4	no	—
152	#117	—	2	8	no	—
153	#117	—	94	8	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	47	4	no	—
156	C₂×C₄⋊C₄	—	94	4	no	—
157	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
158	C₂×C₂²⋊C₄	—	2	2	no	—
159	C₂×C₂²⋊C₄	—	94	2	no	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	—	94	2	no	—
162	C₂×C₄²	Dic₉₄	1	4	yes	—
163	C₂×C₄²	C₄×D₄₇	1	4	yes	—
164	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	C₄×C₈	—	47	8	no	—
168	C₄×C₈	—	47	8	no	—
169	#95	—	2	4	no	—
170	#95	—	94	4	no	—
171	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	8	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	94	4	no	—
175	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
176	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
178	C₂²×C₄	#10625352	1	4	yes	—
179	C₂²×C₄	#10625354	1	4	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
184	C₂×C₈	—	47	8	no	—
185	C₂×C₈	—	94	16	no	—
186	C₂×C₈	—	94	4	no	—
187	C₂×C₈	—	94	4	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₄⋊C₄	—	94	8	no	—
191	C₄⋊C₄	—	188	8	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	188	4	no	—
194	C₄²	—	4	4	no	—
195	C₄²	—	188	4	no	—
196	C₄²	Dic₄₇⋊C₄	1	8	yes	—
197	C₄²	C₂×Dic₉₄	1	2	yes	—
198	C₄²	C₂×Dic₉₄	1	2	yes	—
199	C₄²	C₂×C₄×D₄₇	1	2	yes	—
200	C₄²	C₂×C₄×D₄₇	1	2	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
203	C₂³	#13094011	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	16	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	188	4	no	—
211	C₂×C₄	—	188	8	no	—
212	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
213	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
214	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
215	C₂×C₄	#13093982	1	8	yes	—
216	C₂×C₄	#13094011	1	2	yes	—
217	C₂×C₄	#13094013	1	2	yes	—
218	C₂×C₄	#13094015	1	8	yes	—
219	C₂×C₄	#13094044	1	8	yes	—
220	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625352	1	2	yes	—
223	C₂×C₄	C₂×#10625354	1	2	yes	—
224	C₈	—	94	8	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	#15688988	1	4	yes	—
229	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
230	C₂²	C₂×#13094011	1	1	yes	characteristic
231	C₂²	C₂×#13094013	1	1	yes	characteristic
232	C₂²	C₂×#13094044	1	4	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	8	4	no	—
235	C₄	—	376	4	no	—
236	C₄	#15688988	1	4	yes	—
237	C₄	#15689159	1	8	yes	—
238	C₄	C₂×#13093982	1	2	yes	—
239	C₄	C₂×#13094015	1	2	yes	—
240	C₂	—	2	4	no	—
241	C₂	C₂×#15688988	1	1	yes	characteristic
242	C₂	C₂×#15689159	1	2	yes	—
243	C₂	#20253710	1	2	yes	—
244	C₂	#20253713	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20760
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20243518	C₂	semidirect
4. #20243515	C₂	semidirect
9. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
145. C₄₇	#20760	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₇⋊C₈	2
#20245795	2
#20243518	2
#20243515	2
#20243104	2
C₄₇×#2285	2
#20760	47