Subgroups — #26109328

Order 12032.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20243545	C₂	1	1	yes	maximal, characteristic
3	#20243394	C₂	1	2	yes	maximal
4	#20241272	C₂	1	2	yes	maximal
5	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₉₄	—	2	2	no	—
8	C₈×Dic₉₄	C₄	1	4	yes	complemented
9	C₂×#13093956	—	2	1	no	—
10	C₂×#13093956	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₈	—	2	1	no	—
13	C₂×#13093949	—	2	2	no	—
14	C₂×C₈×Dic₄₇	—	2	1	no	—
15	#15688993	C₂²	1	2	yes	—
16	#15688886	—	2	2	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#334	—	2	2	no	—
20	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₄×Dic₉₄	—	2	1	no	—
23	C₄×Dic₉₄	—	2	1	no	—
24	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₉₄	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
30	C₂×Dic₄₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	#13093956	—	4	2	no	—
33	#13093956	—	4	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₄₇⋊C₈	—	2	4	no	—
37	#13093949	—	4	4	no	—
38	C₈×Dic₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	—	2	2	no	—
44	C₄⋊C₈×C₄₇	—	2	2	no	—
45	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	1	no	—
51	C₂²×Dic₄₇	—	2	1	no	—
52	C₂²×Dic₄₇	—	4	2	no	—
53	C₂×Dic₉₄	—	2	1	no	—
54	C₂×Dic₉₄	—	2	2	no	—
55	C₂×Dic₉₄	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₉₄	SD₁₆	1	4	yes	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
65	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₄₇	#41	1	1	yes	characteristic
70	C₄⋊Dic₄₇	—	2	1	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	8	2	no	—
76	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
77	Dic₄₇⋊C₄	—	2	2	no	—
78	Dic₄₇⋊C₄	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	4	1	no	—
83	C₂×C₄₇⋊C₈	—	4	1	no	—
84	C₂³×C₄₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#135	1	2	yes	—
96	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	4	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	4	2	no	—
102	C₂×Dic₄₇	—	8	2	no	—
103	C₂×Dic₄₇	—	8	2	no	—
104	C₂×Dic₄₇	—	8	2	no	—
105	Dic₉₄	—	2	2	no	—
106	Dic₉₄	—	2	4	no	—
107	Dic₉₄	#106	1	4	yes	complemented
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	—	8	2	no	—
110	C₈×C₄₇	#103	1	4	yes	complemented
111	C₄₇⋊C₈	—	4	2	no	—
112	#8643	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	#380	1	1	yes	characteristic
114	C₂²×C₄₇	#460	1	1	yes	characteristic
115	C₂²×C₄₇	#462	1	1	yes	characteristic
116	C₂²×C₄₇	#465	1	2	yes	—
117	C₂²×C₄₇	#471	1	2	yes	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
121	C₄×C₄₇	#417	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
123	C₄×C₄₇	#427	1	1	yes	characteristic
124	C₄×C₄₇	#465	1	2	yes	—
125	C₄×C₄₇	#475	1	1	yes	characteristic
126	C₄×C₄₇	#477	1	1	yes	characteristic
127	Dic₄₇	—	2	4	no	—
128	Dic₄₇	—	4	2	no	—
129	Dic₄₇	—	16	2	no	—
130	C₂×C₈×Q₈	—	47	1	no	—
131	#1844	—	47	1	no	—
132	#1803	—	47	2	no	—
133	C₂×#334	D₄₇	1	1	yes	characteristic
134	#1296	—	47	2	no	—
135	C₂×C₄₇	#1666	1	2	yes	—
136	C₂×C₄₇	#1672	1	2	yes	—
137	C₂×C₄₇	#1803	1	1	yes	characteristic
138	C₂×C₄₇	#1807	1	1	yes	characteristic
139	C₂×C₄₇	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	47	1	no	—
141	C₈×Q₈	—	47	4	no	—
142	C₈×Q₈	—	94	2	no	—
143	C₂×#106	—	94	1	no	—
144	C₂×#106	—	94	1	no	—
145	C₂×C₄⋊C₈	—	47	1	no	—
146	C₂×C₄⋊C₈	—	94	1	no	—
147	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
148	C₂×#103	—	94	2	no	—
149	C₂×C₄×C₈	—	94	1	no	—
150	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
151	#384	—	47	2	no	—
152	#340	—	94	2	no	—
153	#334	—	2	2	no	—
154	C₄₇	#8643	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	47	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	94	1	no	—
158	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
159	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
160	C₄×Q₈	—	47	2	no	—
161	C₄×Q₈	—	47	2	no	—
162	C₄×Q₈	—	94	1	no	—
163	C₄×Q₈	—	94	1	no	—
164	C₂×C₄⋊C₄	—	47	1	no	—
165	C₂×C₄⋊C₄	—	94	1	no	—
166	C₂×C₄⋊C₄	—	94	2	no	—
167	C₂×C₄⋊C₄	—	94	2	no	—
168	C₂×C₄⋊C₄	—	94	2	no	—
169	C₂×C₄²	—	94	1	no	—
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	#106	—	188	2	no	—
172	#106	—	188	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	47	2	no	—
176	C₄⋊C₈	—	94	4	no	—
177	C₄⋊C₈	—	94	1	no	—
178	#103	—	188	4	no	—
179	C₄×C₈	—	94	4	no	—
180	C₄×C₈	C₄×D₄₇	1	2	yes	—
181	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
182	C₂×Q₈	—	47	2	no	—
183	C₂×Q₈	—	47	4	no	—
184	C₂×Q₈	—	94	1	no	—
185	C₂×Q₈	—	94	2	no	—
186	C₂²×C₄	—	94	1	no	—
187	C₂²×C₄	—	94	1	no	—
188	C₂²×C₄	—	188	2	no	—
189	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	#10625354	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	94	2	no	—
198	C₂×C₈	—	188	1	no	—
199	C₂×C₈	—	188	1	no	—
200	C₂×C₈	#10625325	1	2	yes	—
201	C₂×C₈	Dic₁₈₈	1	2	yes	—
202	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
203	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
204	C₄⋊C₄	—	47	2	no	—
205	C₄⋊C₄	—	94	2	no	—
206	C₄⋊C₄	—	94	1	no	—
207	C₄⋊C₄	—	94	2	no	—
208	C₄⋊C₄	—	376	2	no	—
209	C₄⋊C₄	—	188	2	no	—
210	C₄⋊C₄	—	188	2	no	—
211	C₄⋊C₄	—	188	2	no	—
212	C₄⋊C₄	—	188	2	no	—
213	C₄²	—	94	2	no	—
214	C₄²	—	94	2	no	—
215	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₄²	#10625352	1	1	yes	characteristic
217	C₄²	Q₈×D₄₇	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂³	#13093999	1	1	yes	characteristic
220	Q₈	—	47	4	no	—
221	Q₈	—	94	4	no	—
222	Q₈	—	94	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	94	2	no	—
225	C₂×C₄	—	94	4	no	—
226	C₂×C₄	—	94	2	no	—
227	C₂×C₄	—	376	2	no	—
228	C₂×C₄	—	376	2	no	—
229	C₂×C₄	—	376	2	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	#13093982	1	1	yes	characteristic
234	C₂×C₄	#13093995	1	1	yes	characteristic
235	C₂×C₄	#13094000	1	1	yes	characteristic
236	C₂×C₄	#13094004	1	1	yes	characteristic
237	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
238	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
240	C₂×C₄	#13094045	1	2	yes	—
241	C₂×C₄	#13094052	1	1	yes	characteristic
242	C₂×C₄	#13094057	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	8	2	no	—
245	C₈	—	188	2	no	—
246	C₈	#13093949	1	4	yes	complemented
247	C₂²	#15688980	1	1	yes	characteristic
248	C₂²	#15689324	1	1	yes	characteristic
249	C₂²	#15689334	1	1	yes	characteristic
250	C₂²	#15689353	1	2	yes	—
251	C₂²	#15689387	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	94	4	no	—
255	C₄	—	188	2	no	—
256	C₄	—	752	2	no	—
257	C₄	C₂×#13093949	1	1	yes	characteristic
258	C₄	#15689125	1	1	yes	characteristic
259	C₄	#106×D₄₇	1	1	yes	characteristic
260	C₄	#15689158	1	1	yes	characteristic
261	C₄	#15689349	1	2	yes	—
262	C₄	#15689409	1	1	yes	characteristic
263	C₄	#15689419	1	1	yes	characteristic
264	C₂	#20242737	1	2	yes	—
265	C₂	#20242771	1	2	yes	—
266	C₂	#20243394	1	1	yes	characteristic
267	C₂	#20243410	1	1	yes	characteristic
268	C₂	#20243444	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#8643
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₉₄	C₄	semidirect
107. Dic₉₄	#106	semidirect
110. C₈×C₄₇	#103	semidirect
154. C₄₇	#8643	semidirect
246. C₈	#13093949	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
#20243545	2
#20243394	2
#20241272	2
C₂×C₄₇×#334	2
#8643	47