Subgroups — #26220973

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244329	C₂	1	2	yes	maximal
2	#20244112	C₂	1	2	yes	maximal
3	#20243954	C₂	1	4	yes	maximal
4	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#20243134	C₂	1	2	yes	maximal
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	#15688878	C₂²	1	4	yes	—
17	#15688878	C₂²	1	2	yes	—
18	#15688878	C₂²	1	4	yes	—
19	#15688878	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
21	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₄³×C₄₇	C₂²	1	1	yes	characteristic
28	C₄³×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	#13093935	—	2	4	no	—
35	#13093935	—	2	4	no	—
36	C₄×C₄₇⋊C₈	—	2	2	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
39	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₄₇	—	2	8	no	—
41	C₄×Q₈×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#95×C₄₇	C₂×C₄	1	2	yes	—
59	#95×C₄₇	C₂×C₄	1	1	yes	characteristic
60	#95×C₄₇	C₂×C₄	1	2	yes	—
61	#95×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	M₁₆	1	8	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
88	C₄²×C₄₇	#41	1	4	yes	—
89	C₄²×C₄₇	#41	1	2	yes	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
98	C₂×C₄₇⋊C₈	#41	1	4	yes	—
99	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	Q₈×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	#117	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	#131	1	4	yes	—
123	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₄₇⋊C₈	—	2	4	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	4	4	no	—
130	#21340	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
135	C₂²×C₄₇	C₂×#131	1	2	yes	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	2	no	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	#446	1	8	yes	—
144	C₄×C₄₇	#529	1	2	yes	—
145	C₄×C₄₇	#533	1	2	yes	—
146	C₄²×Q₈	D₄₇	1	1	yes	characteristic
147	#1985	—	47	2	no	—
148	#1940	—	47	2	no	—
149	#1917	—	47	4	no	—
150	C₄×C₄⋊C₈	—	47	1	no	—
151	C₄×C₄⋊C₈	—	47	2	no	—
152	#1744	—	47	2	no	—
153	C₄×#97	—	47	1	no	—
154	C₂×C₄₇	#2295	1	1	yes	characteristic
155	C₂×C₄₇	#2920	1	1	yes	characteristic
156	C₂×C₄₇	#2922	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#446	1	2	yes	—
158	C₂×C₄₇	#2965	1	2	yes	—
159	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
160	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	47	4	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×#97	—	47	2	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	1	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
169	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
170	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
171	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
172	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
173	C₄³	C₂×D₄₇	1	1	yes	characteristic
174	C₄³	C₂×D₄₇	1	2	yes	—
175	#336	—	47	2	no	—
176	#336	—	47	4	no	—
177	#336	—	47	4	no	—
178	#336	—	47	4	no	—
179	C₄₇	#21340	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂²×C₈	—	47	2	no	—
182	C₂²×C₈	—	47	2	no	—
183	C₂²×C₈	—	47	4	no	—
184	C₄×Q₈	—	2	8	no	—
185	C₄×Q₈	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
191	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
193	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
194	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
197	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	94	8	no	—
203	C₄⋊C₈	—	94	4	no	—
204	#97	—	94	4	no	—
205	#97	—	94	4	no	—
206	C₄×C₈	—	47	4	no	—
207	C₄×C₈	—	94	2	no	—
208	C₄×C₈	—	94	4	no	—
209	#95	C₂×Dic₄₇	1	2	yes	—
210	#95	C₂×Dic₄₇	1	1	yes	characteristic
211	#95	C₂×Dic₄₇	1	2	yes	—
212	#95	C₂×Dic₄₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	47	4	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	#10625329	1	8	yes	—
247	C₄²	#10625350	1	4	yes	—
248	C₄²	Q₈×D₄₇	1	2	yes	—
249	C₄²	#10625354	1	2	yes	—
250	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#13093983	1	2	yes	—
268	C₂×C₄	#13093983	1	2	yes	—
269	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
270	C₂×C₄	C₂×#10625329	1	4	yes	—
271	C₂×C₄	C₂×#10625329	1	4	yes	—
272	C₂×C₄	C₂×#10625329	1	4	yes	—
273	C₂×C₄	#13094031	1	4	yes	—
274	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	#41×D₄₇	1	2	yes	—
276	C₂×C₄	C₂×#10625350	1	2	yes	—
277	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
278	C₈	—	94	4	no	—
279	C₈	—	94	4	no	—
280	C₈	—	188	4	no	—
281	C₂²	C₂×#13093983	1	1	yes	characteristic
282	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
283	C₂²	#41×Dic₄₇	1	1	yes	characteristic
284	C₂²	C₂²×#10625329	1	2	yes	—
285	C₂²	C₂×#13094031	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	#15689254	1	8	yes	—
294	C₄	#15689660	1	2	yes	—
295	C₄	#15689677	1	2	yes	—
296	C₂	#20245866	1	1	yes	characteristic
297	C₂	#20252021	1	1	yes	characteristic
298	C₂	#20252032	1	1	yes	characteristic
299	C₂	C₂×#15689254	1	2	yes	—
300	C₂	#20252355	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×Q₈×C₄₇	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×Q₈	D₄₇
hypocenter	C₄₇	#21340
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
38. C₄×C₄₇⋊C₈	Q₈	semidirect
179. C₄₇	#21340	semidirect

Maximal subgroups

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

subgroup	index
#20244329	2
#20244112	2
#20243954	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243134	2
C₄×#13093935	2
C₄²×Q₈×C₄₇	2
#21340	47