Subgroups — #26218341

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
2	#20245870	C₂	1	1	yes	maximal, characteristic
3	#20243949	C₂	1	8	yes	maximal, complemented
4	#20243086	C₂	1	1	yes	maximal, characteristic
5	#20243085	C₂	1	1	yes	maximal, characteristic
6	#20243084	C₂	1	1	yes	maximal, characteristic
7	#20243081	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×#13093934	—	2	4	no	—
14	C₂×#13093934	C₂²	1	4	yes	—
15	#15689019	C₄	1	4	yes	—
16	#15688995	C₄	1	4	yes	—
17	C₄×Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	#15688878	C₂²	1	4	yes	—
19	#15688877	C₂²	1	4	yes	—
20	#15688877	C₂²	1	4	yes	—
21	#15688877	C₂²	1	4	yes	—
22	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#424	—	2	4	no	—
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
27	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	#13093983	—	2	2	no	—
37	#13093983	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
40	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	—	2	2	no	—
42	C₂×C₄×Dic₄₇	—	2	2	no	—
43	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	#13093934	—	2	8	no	—
47	C₄₇⋊#95	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₄₇	—	2	2	no	—
50	C₂×M₁₆×C₄₇	—	2	2	no	—
51	C₂×M₁₆×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	—	2	2	no	—
54	C₂²×C₈×C₄₇	D₄	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	#117×C₄₇	—	2	2	no	—
58	#117×C₄₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
62	C₄⋊C₈×C₄₇	—	2	8	no	—
63	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
77	M₁₆×C₄₇	—	4	4	no	—
78	C₂×C₈×C₄₇	—	2	8	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₂²⋊C₄×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₂²⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	Dic₄₇⋊C₄	—	4	8	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	2	2	no	—
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	#10625329	—	4	4	no	—
98	C₂×C₄₇⋊C₈	—	2	8	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	#117	1	2	yes	—
107	C₂³×C₄₇	C₄×D₄	1	2	yes	—
108	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	#117	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₄₇	#123	1	4	yes	—
117	C₂×C₄×C₄₇	#125	1	4	yes	—
118	C₂×C₄×C₄₇	#126	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	4	8	no	—
131	C₈×C₄₇	—	2	4	no	—
132	C₈×C₄₇	—	4	2	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	C₄₇⋊C₈	—	4	2	no	—
135	#21021	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	#387	1	4	yes	—
140	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₂²×C₄₇	#568	1	4	yes	—
144	C₄×C₄₇	—	4	4	no	—
145	C₄×C₄₇	#387	1	4	yes	—
146	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
150	Dic₄₇	—	4	4	no	—
151	Dic₄₇	—	4	4	no	—
152	C₂×#424	D₄₇	1	1	yes	characteristic
153	C₂²×#97	—	47	1	no	—
154	#2296	—	47	1	no	—
155	#1917	—	47	8	no	—
156	#1729	—	47	1	no	—
157	#1729	—	47	1	no	—
158	#1729	—	47	1	no	—
159	#1728	—	47	1	no	—
160	C₂×C₄₇	—	2	4	no	—
161	C₂×C₄₇	C₂×#387	1	1	yes	characteristic
162	C₂×C₄₇	#2923	1	2	yes	—
163	C₂×C₄₇	#2930	1	2	yes	—
164	C₂×C₄₇	#2961	1	2	yes	—
165	C₂²×M₁₆	—	47	1	no	—
166	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
167	C₂³×C₈	—	47	1	no	—
168	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#117	—	47	1	no	—
170	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₄²	—	47	1	no	—
172	#424	—	2	4	no	—
173	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
174	C₂×#97	—	47	4	no	—
175	C₂×#97	—	94	4	no	—
176	#389	—	47	4	no	—
177	#385	—	47	4	no	—
178	C₄×C₄⋊C₄	—	47	4	no	—
179	#336	—	47	4	no	—
180	#336	—	47	4	no	—
181	#336	—	47	4	no	—
182	#336	—	47	4	no	—
183	C₄₇	#21021	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	47	2	no	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	47	4	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
200	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
201	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	94	2	no	—
205	#117	—	94	2	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	47	8	no	—
208	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	47	2	no	—
210	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	47	2	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	—	94	2	no	—
215	C₂×C₄²	—	94	2	no	—
216	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
217	C₄⋊C₈	—	2	8	no	—
218	#97	—	94	8	no	—
219	#95	—	47	8	no	—
220	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	—	47	4	no	—
222	C₂²×C₄	—	47	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	188	4	no	—
225	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	#10625350	1	2	yes	—
228	C₂²×C₄	#10625352	1	2	yes	—
229	C₂²×C₄	#10625352	1	2	yes	—
230	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
231	C₂²×C₄	#10625354	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	M₁₆	—	4	4	no	—
235	M₁₆	—	188	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	94	8	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	188	8	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	94	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	94	4	no	—
258	C₄²	—	94	2	no	—
259	C₄²	—	94	2	no	—
260	C₄²	—	94	2	no	—
261	C₄²	—	94	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	#13093982	1	2	yes	—
264	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
265	C₂³	#13093989	1	2	yes	—
266	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	8	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	188	8	no	—
275	C₂×C₄	#13093982	1	2	yes	—
276	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
277	C₂×C₄	#13093989	1	2	yes	—
278	C₂×C₄	#13094000	1	4	yes	—
279	C₂×C₄	#13094005	1	4	yes	—
280	C₂×C₄	#13094014	1	4	yes	—
281	C₂×C₄	#13094019	1	4	yes	—
282	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
287	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	94	4	no	—
291	C₈	—	188	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	#15689004	1	4	yes	—
296	C₂²	C₂×#13093982	1	1	yes	characteristic
297	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₂²	C₂×#13093989	1	1	yes	characteristic
299	C₂²	#15689888	1	4	yes	—
300	C₄	—	4	4	no	—
301	C₄	—	188	4	no	—
302	C₄	—	188	4	no	—
303	C₄	#15689004	1	4	yes	—
304	C₄	C₂×#13094000	1	1	yes	characteristic
305	C₄	C₂×#13094005	1	1	yes	characteristic
306	C₄	C₂×#13094014	1	1	yes	characteristic
307	C₄	C₂×#13094019	1	1	yes	characteristic
308	C₂	—	2	4	no	—
309	C₂	C₂×#15689004	1	1	yes	characteristic
310	C₂	#20252038	1	2	yes	—
311	C₂	#20252089	1	2	yes	—
312	C₂	#20252327	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#424	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#21021
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
3. #20243949	C₂	semidirect
183. C₄₇	#21021	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093934	2
#20245870	2
#20243949	2
#20243086	2
#20243085	2
#20243084	2
#20243081	2
C₂×C₄₇×#424	2
#21021	47