Subgroups — #26224710

Order 12032.

1086 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	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689217	C₂	1	4	yes	maximal
3	#20245955	C₂	1	2	yes	maximal
4	#20244247	C₂	1	4	yes	maximal
5	#20244189	C₂	1	2	yes	maximal
6	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₈₈	C₂²	1	4	yes	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	C₂×#13093989	C₂²	1	4	yes	—
11	#15689503	C₄	1	8	yes	complemented
12	#15689217	—	2	16	no	—
13	C₂×#13093947	C₂²	1	8	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15689028	C₂²	1	4	yes	—
18	#15689019	—	2	4	no	—
19	C₄⋊C₄×Dic₄₇	—	2	4	no	—
20	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×#107×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	2	yes	—
24	C₂²×Dic₉₄	C₂³	1	4	yes	—
25	C₂×Dic₁₈₈	—	2	8	no	—
26	C₂×Dic₁₈₈	—	2	8	no	—
27	C₂×Dic₁₈₈	C₂×C₄	1	16	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	#13094024	—	2	4	no	—
30	#13094024	C₂×C₄	1	8	yes	—
31	#13093989	—	2	16	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×Dic₄₇⋊C₄	—	2	4	no	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	#13093947	—	2	16	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₈×Dic₄₇	—	2	4	no	—
42	C₈×Dic₄₇	D₄	1	8	yes	—
43	C₄₇⋊#95	—	2	8	no	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#107×C₄₇	—	2	8	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
56	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₉₄	—	2	8	no	—
58	C₂×Dic₉₄	—	2	8	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	C₂²×C₄	1	8	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₄₇	#41	1	4	yes	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₄⋊C₄×C₄₇	—	4	8	no	—
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	#41	1	2	yes	—
68	C₄⋊Dic₄₇	—	2	16	no	—
69	Dic₄₇⋊C₄	—	4	8	no	—
70	C₄×Dic₄₇	—	2	4	no	—
71	C₄×Dic₄₇	—	2	2	no	—
72	C₄×Dic₄₇	—	4	8	no	—
73	C₄×Dic₄₇	Q₁₆	1	8	yes	complemented
74	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
75	C₂×C₄₇⋊C₈	—	2	2	no	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
78	Dic₁₈₈	—	2	32	no	—
79	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	8	no	—
82	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
84	C₂×C₄×C₄₇	#135	1	4	yes	—
85	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
89	C₂×Dic₄₇	—	2	2	no	—
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	4	4	no	—
94	C₂×Dic₄₇	—	4	4	no	—
95	C₂×Dic₄₇	#127	1	2	yes	—
96	C₂×Dic₄₇	#127	1	2	yes	—
97	C₂×Dic₄₇	C₂×Q₁₆	1	4	yes	—
98	C₂×Dic₄₇	C₂×Q₁₆	1	4	yes	—
99	Dic₉₄	—	2	16	no	—
100	Dic₉₄	—	4	8	no	—
101	C₈×C₄₇	C₄×D₄	1	8	yes	—
102	C₄₇⋊C₈	—	2	4	no	—
103	C₄×#494	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
106	C₂²×C₄₇	#532	1	1	yes	characteristic
107	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
108	C₂²×C₄₇	C₂×#135	1	2	yes	—
109	C₄×C₄₇	—	8	4	no	—
110	C₄×C₄₇	C₄×Q₁₆	1	8	yes	complemented
111	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
112	C₄×C₄₇	#532	1	2	yes	—
113	Dic₄₇	—	2	2	no	—
114	Dic₄₇	—	2	4	no	—
115	Dic₄₇	—	4	8	no	—
116	Dic₄₇	#494	1	4	yes	complemented
117	C₂×#494	—	47	1	no	—
118	C₂×C₄×Q₁₆	—	47	4	no	—
119	C₄×C₄⋊Q₈	—	47	2	no	—
120	#1969	—	47	4	no	—
121	#1956	—	47	2	no	—
122	#1845	D₄₇	1	1	yes	characteristic
123	C₄²×C₈	—	47	1	no	—
124	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
125	C₂×C₄₇	C₂×C₄×Q₁₆	1	2	yes	—
126	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
127	C₂×C₄₇	#3147	1	1	yes	characteristic
128	C₂×C₄₇	#3170	1	2	yes	—
129	C₂²×Q₁₆	—	47	4	no	—
130	C₂×C₄⋊Q₈	—	47	2	no	—
131	C₂×C₄×Q₈	—	47	4	no	—
132	#494	—	47	8	no	—
133	C₄×Q₁₆	—	94	16	no	—
134	C₂×#107	C₂×D₄₇	1	4	yes	—
135	C₂×#103	—	47	8	no	—
136	C₂×C₄×C₈	—	47	2	no	—
137	C₂×C₄×C₈	—	47	2	no	—
138	C₂×C₄×C₈	—	47	1	no	—
139	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
140	#391	—	47	4	no	—
141	#389	—	94	4	no	—
142	#384	C₂×D₄₇	1	2	yes	—
143	C₄×C₄⋊C₄	—	94	4	no	—
144	C₄³	—	47	1	no	—
145	C₄₇	C₄×#494	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	47	4	no	—
147	C₂×Q₁₆	—	47	16	no	—
148	C₂×Q₁₆	—	94	8	no	—
149	C₂×Q₁₆	—	94	8	no	—
150	C₂²×C₈	—	47	2	no	—
151	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
152	C₄⋊Q₈	—	47	8	no	—
153	C₄⋊Q₈	—	94	4	no	—
154	C₄×Q₈	—	94	16	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	94	4	no	—
157	C₂×C₄⋊C₄	—	94	4	no	—
158	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
159	C₂×C₄²	—	47	2	no	—
160	C₂×C₄²	—	47	1	no	—
161	C₂×C₄²	—	47	2	no	—
162	C₂×C₄²	—	47	1	no	—
163	C₂×C₄²	—	94	4	no	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	#107	—	2	8	no	—
166	#103	—	94	16	no	—
167	C₄×C₈	—	47	8	no	—
168	C₄×C₈	—	94	4	no	—
169	C₄×C₈	—	94	2	no	—
170	C₄×C₈	C₄×D₄₇	1	4	yes	—
171	#95	—	94	8	no	—
172	C₂×Q₈	—	47	8	no	—
173	C₂×Q₈	—	94	4	no	—
174	C₂×Q₈	—	94	8	no	—
175	C₂×Q₈	—	94	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	47	1	no	—
178	C₂²×C₄	—	47	2	no	—
179	C₂²×C₄	—	47	1	no	—
180	C₂²×C₄	—	94	4	no	—
181	C₂²×C₄	D₄×D₄₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	Q₁₆	—	94	32	no	—
184	C₂×C₈	—	47	4	no	—
185	C₂×C₈	—	94	2	no	—
186	C₂×C₈	—	94	2	no	—
187	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
188	C₂×C₈	D₄×D₄₇	1	4	yes	—
189	C₂×C₈	#10625354	1	4	yes	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	4	8	no	—
192	C₄⋊C₄	—	94	16	no	—
193	C₄⋊C₄	—	188	8	no	—
194	C₄²	—	47	8	no	—
195	C₄²	—	47	4	no	—
196	C₄²	—	94	4	no	—
197	C₄²	—	94	2	no	—
198	C₄²	—	188	8	no	—
199	C₄²	C₂×C₄×D₄₇	1	2	yes	—
200	C₄²	#10625354	1	2	yes	—
201	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
202	Q₈	—	94	16	no	—
203	Q₈	—	188	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	47	4	no	—
207	C₂×C₄	—	47	2	no	—
208	C₂×C₄	—	47	4	no	—
209	C₂×C₄	—	47	2	no	—
210	C₂×C₄	—	94	2	no	—
211	C₂×C₄	—	94	2	no	—
212	C₂×C₄	—	94	2	no	—
213	C₂×C₄	—	94	8	no	—
214	C₂×C₄	—	188	4	no	—
215	C₂×C₄	—	188	4	no	—
216	C₂×C₄	#13093986	1	4	yes	—
217	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
218	C₂×C₄	#13094043	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
221	C₂×C₄	C₂×#10625354	1	2	yes	—
222	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
223	C₈	—	94	4	no	—
224	C₈	#13093986	1	8	yes	—
225	C₂²	C₂×#13093986	1	1	yes	characteristic
226	C₂²	#127×D₄₇	1	1	yes	characteristic
227	C₂²	#15689674	1	1	yes	characteristic
228	C₂²	C₂×Q₁₆×D₄₇	1	2	yes	—
229	C₂²	C₂×#13094043	1	2	yes	—
230	C₄	—	8	4	no	—
231	C₄	—	47	4	no	—
232	C₄	—	94	4	no	—
233	C₄	—	94	2	no	—
234	C₄	—	188	8	no	—
235	C₄	#15689217	1	8	yes	complemented
236	C₄	C₂×#13093986	1	2	yes	—
237	C₄	#15689674	1	2	yes	—
238	C₂	#20245950	1	1	yes	characteristic
239	C₂	C₂×#15689217	1	2	yes	—
240	C₂	#494×D₄₇	1	1	yes	characteristic
241	C₂	#20253815	1	1	yes	characteristic
242	C₂	#20253975	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1845	D₄₇
hypocenter	C₄₇	C₄×#494
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
11. #15689503	C₄	semidirect
73. C₄×Dic₄₇	Q₁₆	semidirect
110. C₄×C₄₇	C₄×Q₁₆	semidirect
116. Dic₄₇	#494	semidirect
145. C₄₇	C₄×#494	semidirect
235. C₄	#15689217	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689503	2
C₂×#15689217	2
#20245955	2
#20244247	2
#20244189	2
C₄×C₈×Dic₄₇	2
C₄₇×#1845	2
C₄×#494	47