Subgroups — #26216941

Order 12032.

1646 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689150	C₂	1	2	yes	maximal, complemented
3	C₂×#15689087	C₂	1	1	yes	maximal, characteristic
4	#20243568	C₂	1	8	yes	maximal, complemented
5	#20243085	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2280	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×#10625329	C₂²	1	1	yes	characteristic
9	C₂³×C₄₇⋊C₈	—	2	8	no	—
10	C₂³×C₄₇⋊C₈	—	2	4	no	—
11	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
12	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15689150	—	2	8	no	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
15	#15689087	—	2	4	no	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
17	#15688878	C₂²	1	8	yes	complemented
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#381	C₄	1	2	yes	—
21	C₄₇×#380	C₄	1	4	yes	—
22	C₄₇×#379	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	32	no	—
29	C₂²×C₄₇⋊C₈	—	2	8	no	—
30	C₂²×C₄₇⋊C₈	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	—	2	4	no	—
32	C₂²×C₄₇⋊C₈	—	2	16	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
37	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
39	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	—	2	4	no	—
41	C₂³×C₄×C₄₇	—	2	4	no	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	D₄	1	4	yes	—
45	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	#117×C₄₇	—	2	4	no	—
49	#117×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
54	#95×C₄₇	C₂×C₄	1	8	yes	—
55	C₂⁴×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	—	2	8	no	—
57	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	16	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	—	2	8	no	—
80	C₂²⋊C₄×C₄₇	—	2	8	no	—
81	C₂²⋊C₄×C₄₇	—	2	16	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	#10625329	—	4	4	no	—
85	#10625329	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	2	32	no	—
87	C₂×C₄₇⋊C₈	—	2	16	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
95	C₂³×C₄₇	—	2	32	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	—	2	8	no	—
100	C₂³×C₄₇	—	2	16	no	—
101	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₄₇	C₄×D₄	1	4	yes	—
104	C₂³×C₄₇	#131	1	8	yes	—
105	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	16	no	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₄₇	#120	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
119	C₂×C₄×C₄₇	#123	1	4	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₄₇⋊C₈	—	2	8	no	—
125	C₄₇⋊C₈	—	4	2	no	—
126	C₄₇⋊C₈	—	4	2	no	—
127	#20834	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	32	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	8	no	—
132	C₂²×C₄₇	—	2	16	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	#385	1	4	yes	—
135	C₂²×C₄₇	#424	1	8	yes	complemented
136	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
138	C₂²×C₄₇	C₂×#131	1	4	yes	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	8	no	—
141	C₄×C₄₇	#385	1	4	yes	—
142	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
143	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
145	C₂⁴×C₈	—	47	1	no	—
146	C₂×#424	—	47	2	no	—
147	C₂×#407	—	47	1	no	—
148	#2280	D₄₇	1	1	yes	characteristic
149	#1850	—	47	8	no	—
150	#1729	—	47	2	no	—
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	—	2	8	no	—
153	C₂×C₄₇	C₂×#385	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#424	1	2	yes	—
155	C₂×C₄₇	#2929	1	4	yes	—
156	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×M₁₆	—	47	1	no	—
158	C₂²×M₁₆	—	47	1	no	—
159	C₂³×C₈	—	47	2	no	—
160	C₂³×C₈	—	47	4	no	—
161	C₂³×C₈	—	94	8	no	—
162	C₂³×C₈	—	94	4	no	—
163	C₂×#117	C₂×D₄₇	1	2	yes	—
164	#424	—	94	8	no	—
165	C₂×C₄⋊C₈	—	47	8	no	—
166	#407	—	94	4	no	—
167	C₂×C₂²⋊C₈	—	47	4	no	—
168	#381	Dic₄₇	1	2	yes	—
169	#380	Dic₄₇	1	4	yes	—
170	#379	Dic₄₇	1	2	yes	—
171	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
172	#336	—	47	8	no	—
173	C₄₇	#20834	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂³×C₄	—	2	4	no	—
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	Dic₉₄	1	4	yes	—
178	C₂³×C₄	D₁₈₈	1	4	yes	—
179	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	4	no	—
186	C₂×M₁₆	—	94	4	no	—
187	C₂²×C₈	—	47	8	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	—	94	16	no	—
192	C₂²×C₈	—	94	32	no	—
193	C₂²×C₈	—	94	8	no	—
194	C₂²×C₈	—	94	4	no	—
195	C₂²×C₈	—	94	4	no	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
202	C₄⋊C₈	—	94	16	no	—
203	C₂²⋊C₈	—	94	8	no	—
204	#95	C₂×Dic₄₇	1	8	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
208	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
209	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
218	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
219	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
220	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
221	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
223	C₂²×C₄	D₄×D₄₇	1	4	yes	—
224	C₂²×C₄	#10625352	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	M₁₆	—	188	4	no	—
230	M₁₆	—	188	4	no	—
231	C₂×C₈	—	47	8	no	—
232	C₂×C₈	—	94	32	no	—
233	C₂×C₈	—	94	16	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	4	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₂³	—	2	32	no	—
246	C₂³	—	2	8	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	2	16	no	—
251	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
252	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
253	C₂³	D₄×Dic₄₇	1	4	yes	—
254	C₂³	#13094031	1	8	yes	—
255	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
266	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
267	C₂×C₄	#13093994	1	4	yes	—
268	C₂×C₄	#13094001	1	8	yes	—
269	C₂×C₄	#13094008	1	4	yes	—
270	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
271	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
273	C₂×C₄	C₂×#10625352	1	2	yes	—
274	C₈	—	94	8	no	—
275	C₈	—	188	2	no	—
276	C₈	—	188	2	no	—
277	C₂²	—	2	32	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	16	no	—
283	C₂²	#15688998	1	4	yes	—
284	C₂²	#15689150	1	8	yes	complemented
285	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
286	C₂²	C₂×D₄×Dic₄₇	1	2	yes	—
287	C₂²	C₂×#13094031	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	#15688998	1	4	yes	—
291	C₄	C₂×#13093994	1	1	yes	characteristic
292	C₄	C₂×#13094001	1	2	yes	—
293	C₄	C₂×#13094008	1	1	yes	characteristic
294	C₂	—	2	4	no	—
295	C₂	—	2	8	no	—
296	C₂	C₂×#15688998	1	1	yes	characteristic
297	C₂	C₂×#15689150	1	2	yes	—
298	C₂	#20252083	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2280	C₂
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2280	D₄₇
hypocenter	C₄₇	#20834
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2280

Complements

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

N	H ≅ G/N	type
2. C₂×#15689150	C₂	semidirect
4. #20243568	C₂	semidirect
5. #20243085	C₂	semidirect
14. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
17. #15688878	C₂²	semidirect
135. C₂²×C₄₇	#424	semidirect
173. C₄₇	#20834	semidirect
284. C₂²	#15689150	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₄₇⋊C₈	2
C₂×#15689150	2
C₂×#15689087	2
#20243568	2
#20243085	2
C₄₇×#2280	2
#20834	47