Subgroups — #26378934

Order 12032.

1898 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246574	C₂	1	4	yes	maximal, complemented
2	#20246569	C₂	1	4	yes	maximal, complemented
3	#20246554	C₂	1	8	yes	maximal
4	#20245886	C₂	1	4	yes	maximal
5	#20245866	C₂	1	8	yes	maximal, complemented
6	#20245786	C₂	1	2	yes	maximal
7	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
8	#15689030	C₂²	1	16	yes	—
9	#15689022	C₂²	1	16	yes	—
10	#15689016	C₂²	1	16	yes	—
11	#15689009	C₂²	1	16	yes	—
12	#15689009	C₂²	1	16	yes	—
13	#15689009	C₂²	1	16	yes	—
14	#15688991	C₂²	1	4	yes	—
15	#15688989	C₂²	1	4	yes	—
16	#15688986	C₂²	1	4	yes	—
17	#15688983	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
21	#15688969	C₂²	1	4	yes	—
22	#15688969	C₂²	1	4	yes	—
23	C₄²×Dic₄₇	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	16	yes	—
39	C₄₇⋊#95	C₂³	1	8	yes	—
40	C₄₇⋊#95	C₂³	1	8	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	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	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	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₄₇	—	4	8	no	—
61	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	4	4	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
73	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
74	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
75	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
80	C₂²⋊Dic₄₇	—	4	8	no	—
81	C₄⋊Dic₄₇	—	4	8	no	—
82	C₄⋊Dic₄₇	—	4	16	no	—
83	C₄×Dic₄₇	—	2	16	no	—
84	C₄×Dic₄₇	—	4	16	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
88	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
89	Q₈×C₄₇	—	4	16	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	16	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
102	C₂×Dic₄₇	—	2	16	no	—
103	C₂×Dic₄₇	—	2	16	no	—
104	C₂×Dic₄₇	—	2	16	no	—
105	#34348	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	2	4	no	—
108	C₂²×C₄₇	—	4	4	no	—
109	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
111	C₂²×C₄₇	#585	1	4	yes	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	—	2	4	no	—
115	C₄×C₄₇	—	2	16	no	—
116	C₄×C₄₇	#519	1	4	yes	—
117	Dic₄₇	—	4	16	no	—
118	C₂×#533	D₄₇	1	1	yes	characteristic
119	#2382	—	47	4	no	—
120	#2381	—	47	4	no	—
121	#2378	—	47	8	no	—
122	#2298	—	47	4	no	—
123	#2295	—	47	8	no	—
124	#2283	—	47	2	no	—
125	C₂×C₄₇	—	4	2	no	—
126	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
127	C₂×C₄₇	#3428	1	2	yes	—
128	C₂×C₄₇	#3531	1	4	yes	—
129	#533	Dic₄₇	1	16	yes	complemented
130	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
131	C₂×#125	C₂×D₄₇	1	2	yes	—
132	C₂×C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
133	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
134	C₂×#117	C₂×D₄₇	1	2	yes	—
135	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
136	#391	—	47	16	no	—
137	#389	—	47	16	no	—
138	#388	—	47	16	no	—
139	#387	—	47	16	no	—
140	#387	—	47	16	no	—
141	#387	—	47	16	no	—
142	#383	—	47	4	no	—
143	#382	—	47	4	no	—
144	#381	—	47	4	no	—
145	#380	—	47	4	no	—
146	C₄×C₄⋊C₄	—	47	4	no	—
147	C₄×C₄⋊C₄	—	47	8	no	—
148	C₄×C₂²⋊C₄	—	47	4	no	—
149	#376	—	47	4	no	—
150	#376	—	47	4	no	—
151	C₄³	—	47	4	no	—
152	C₄₇	#34348	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
154	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
155	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
156	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
157	#125	C₂×Dic₄₇	1	16	yes	—
158	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
159	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
160	#117	C₂×Dic₄₇	1	16	yes	—
161	C₂×C₄⋊C₄	—	47	8	no	—
162	C₂×C₄⋊C₄	—	47	16	no	—
163	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
164	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
165	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
166	C₂×C₂²⋊C₄	—	47	8	no	—
167	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
168	C₂×C₄²	—	47	8	no	—
169	C₂×C₄²	—	47	16	no	—
170	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
171	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
175	#95	—	47	8	no	—
176	#95	—	47	8	no	—
177	#95	—	47	16	no	—
178	#95	—	47	8	no	—
179	#95	—	47	8	no	—
180	#95	—	47	16	no	—
181	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
182	C₂×Q₈	—	4	8	no	—
183	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	47	16	no	—
187	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
188	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
189	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
193	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	C₄⋊C₄	—	188	16	no	—
195	C₄⋊C₄	—	188	8	no	—
196	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
197	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
198	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
199	C₂²⋊C₄	—	188	8	no	—
200	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
201	C₄²	—	94	16	no	—
202	C₄²	—	188	16	no	—
203	C₄²	C₂²×Dic₄₇	1	8	yes	—
204	C₄²	C₂²×Dic₄₇	1	4	yes	—
205	C₄²	C₂²×Dic₄₇	1	4	yes	—
206	C₄²	C₂²×Dic₄₇	1	16	yes	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	4	no	—
209	C₂³	C₂³×Dic₄₇	1	2	yes	—
210	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
211	Q₈	—	4	16	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	—	94	16	no	—
219	C₂×C₄	—	94	16	no	—
220	C₂×C₄	—	94	16	no	—
221	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
222	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
223	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
224	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
225	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
226	C₂×C₄	#13094072	1	4	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	4	no	—
230	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
231	C₂²	C₂×#13094072	1	2	yes	—
232	C₂²	#15689959	1	4	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	16	no	—
237	C₄	—	188	16	no	—
238	C₄	#15689601	1	4	yes	—
239	C₂	—	4	2	no	—
240	C₂	C₂×#15689601	1	1	yes	characteristic
241	C₂	#20256591	1	2	yes	—
242	C₂	#20257535	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#34348
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20246574	C₂	semidirect
2. #20246569	C₂	semidirect
5. #20245866	C₂	semidirect
24. C₄₇×#533	C₄	semidirect
129. #533	Dic₄₇	semidirect
152. C₄₇	#34348	semidirect

Maximal subgroups

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

subgroup	index
#20246574	2
#20246569	2
#20246554	2
#20245886	2
#20245866	2
#20245786	2
C₂×C₄₇×#533	2
#34348	47