Subgroups — #26379196

Order 12032.

1898 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	#20246569	C₂	1	4	yes	maximal, complemented
2	#20246494	C₂	1	4	yes	maximal, complemented
3	#20246345	C₂	1	8	yes	maximal, complemented
4	#20246146	C₂	1	8	yes	maximal
5	#20245997	C₂	1	4	yes	maximal
6	#20245786	C₂	1	2	yes	maximal
7	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
8	#15689030	C₂²	1	16	yes	—
9	#15689016	C₂²	1	16	yes	—
10	#15689009	C₂²	1	16	yes	—
11	#15689009	C₂²	1	16	yes	—
12	#15689009	C₂²	1	16	yes	—
13	#15688993	C₂²	1	4	yes	—
14	#15688991	C₂²	1	4	yes	—
15	#15688989	C₂²	1	4	yes	—
16	#15688989	C₂²	1	4	yes	—
17	#15688986	C₂²	1	4	yes	—
18	#15688983	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
22	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
23	#15688969	C₂²	1	4	yes	—
24	C₄₇×#533	C₄	1	16	yes	complemented
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂×#125×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
29	C₂×#117×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	8	yes	—
37	C₄₇⋊#95	C₂³	1	8	yes	—
38	C₄₇⋊#95	C₂³	1	8	yes	—
39	C₄₇⋊#95	C₂³	1	8	yes	—
40	C₄₇⋊#95	C₂³	1	16	yes	—
41	C₄₇⋊#95	C₂³	1	16	yes	—
42	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
46	#125×C₄₇	C₂×C₄	1	16	yes	—
47	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
48	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	—
49	#117×C₄₇	C₂×C₄	1	16	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₄₇	—	4	4	no	—
61	C₂×Q₈×C₄₇	—	4	4	no	—
62	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
73	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
75	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
76	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
81	C₂²⋊Dic₄₇	—	4	8	no	—
82	C₄⋊Dic₄₇	—	2	16	no	—
83	C₄⋊Dic₄₇	—	4	16	no	—
84	C₄×Dic₄₇	—	4	8	no	—
85	C₄×Dic₄₇	—	4	16	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
89	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
90	Q₈×C₄₇	—	4	16	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	4	8	no	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
103	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
104	C₂×Dic₄₇	—	2	16	no	—
105	C₂×Dic₄₇	—	2	16	no	—
106	C₂×Dic₄₇	—	2	16	no	—
107	#34375	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	4	no	—
110	C₂²×C₄₇	—	4	4	no	—
111	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
113	C₂²×C₄₇	#585	1	4	yes	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	2	16	no	—
118	C₄×C₄₇	#519	1	4	yes	—
119	Dic₄₇	—	4	16	no	—
120	C₂×#533	D₄₇	1	1	yes	characteristic
121	#2381	—	47	4	no	—
122	#2371	—	47	4	no	—
123	#2355	—	47	8	no	—
124	#2332	—	47	8	no	—
125	#2314	—	47	4	no	—
126	#2283	—	47	2	no	—
127	C₂×C₄₇	—	4	2	no	—
128	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
129	C₂×C₄₇	#3428	1	2	yes	—
130	C₂×C₄₇	#3557	1	4	yes	—
131	#533	Dic₄₇	1	16	yes	complemented
132	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
133	C₂×#125	C₂×D₄₇	1	2	yes	—
134	C₂×C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
135	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
136	C₂×#117	C₂×D₄₇	1	2	yes	—
137	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
138	#391	—	47	16	no	—
139	#388	—	47	16	no	—
140	#387	—	47	16	no	—
141	#387	—	47	16	no	—
142	#387	—	47	16	no	—
143	#384	—	47	4	no	—
144	#383	—	47	4	no	—
145	#382	—	47	4	no	—
146	#382	—	47	4	no	—
147	#381	—	47	4	no	—
148	#380	—	47	4	no	—
149	C₄×C₄⋊C₄	—	47	16	no	—
150	C₄×C₄⋊C₄	—	47	8	no	—
151	C₄×C₄⋊C₄	—	47	4	no	—
152	C₄×C₂²⋊C₄	—	47	4	no	—
153	#376	—	47	4	no	—
154	C₄₇	#34375	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
156	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
158	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
159	#125	C₂×Dic₄₇	1	16	yes	—
160	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
161	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
162	#117	C₂×Dic₄₇	1	16	yes	—
163	C₂×C₄⋊C₄	—	47	8	no	—
164	C₂×C₄⋊C₄	—	47	16	no	—
165	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
166	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
167	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
168	C₂×C₂²⋊C₄	—	47	8	no	—
169	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
170	C₂×C₄²	—	47	8	no	—
171	C₂×C₄²	—	47	16	no	—
172	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
173	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
177	#95	—	47	8	no	—
178	#95	—	47	8	no	—
179	#95	—	47	8	no	—
180	#95	—	47	8	no	—
181	#95	—	47	16	no	—
182	#95	—	47	16	no	—
183	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	—	47	16	no	—
190	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
197	C₄⋊C₄	—	94	16	no	—
198	C₄⋊C₄	—	188	16	no	—
199	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
200	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
201	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
202	C₂²⋊C₄	—	188	8	no	—
203	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
204	C₄²	—	188	8	no	—
205	C₄²	—	188	16	no	—
206	C₄²	C₂²×Dic₄₇	1	8	yes	—
207	C₄²	C₂²×Dic₄₇	1	4	yes	—
208	C₄²	C₂²×Dic₄₇	1	4	yes	—
209	C₄²	C₂²×Dic₄₇	1	16	yes	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	C₂³×Dic₄₇	1	2	yes	—
213	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
214	Q₈	—	4	16	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	94	16	no	—
223	C₂×C₄	—	94	16	no	—
224	C₂×C₄	—	94	16	no	—
225	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
226	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
227	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
228	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
229	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
230	C₂×C₄	#13094072	1	4	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	4	4	no	—
234	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
235	C₂²	C₂×#13094072	1	2	yes	—
236	C₂²	#15689959	1	4	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	2	16	no	—
241	C₄	—	188	16	no	—
242	C₄	#15689601	1	4	yes	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#15689601	1	1	yes	characteristic
245	C₂	#20256591	1	2	yes	—
246	C₂	#20257894	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂×C₄₇×#533	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#34375
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20246569	C₂	semidirect
2. #20246494	C₂	semidirect
3. #20246345	C₂	semidirect
24. C₄₇×#533	C₄	semidirect
131. #533	Dic₄₇	semidirect
154. C₄₇	#34375	semidirect

Maximal subgroups

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

subgroup	index
#20246569	2
#20246494	2
#20246345	2
#20246146	2
#20245997	2
#20245786	2
C₂×C₄₇×#533	2
#34375	47