Subgroups — #26232994

Order 12032.

1428 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	C₂×Q₈×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
2	#20244183	C₂	1	6	yes	maximal, complemented
3	#20243368	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₇⋊C₈	—	2	8	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	6	yes	—
11	#15688878	C₂²	1	3	yes	—
12	#15688878	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
14	C₄×#41×C₄₇	—	2	8	no	—
15	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₇	C₄	1	3	yes	—
18	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
19	C₂×#117×C₄₇	C₄	1	3	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
21	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
22	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	—	2	12	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	12	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	4	6	no	—
26	C₄×C₄₇⋊C₈	—	2	12	no	—
27	C₂×#41×C₄₇	—	2	4	no	—
28	C₂×#41×C₄₇	—	2	4	no	—
29	C₂×#41×C₄₇	C₈	1	8	yes	complemented
30	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₄₇	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
35	C₄×Q₈×C₄₇	—	2	4	no	—
36	C₄×Q₈×C₄₇	D₄	1	8	yes	—
37	C₄×D₄×C₄₇	—	2	12	no	—
38	C₄×D₄×C₄₇	—	2	12	no	—
39	#117×C₄₇	—	2	12	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₄₇	—	2	12	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
49	#41×C₄₇	—	2	32	no	—
50	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
52	C₂×Q₈×C₄₇	M₁₆	1	4	yes	—
53	C₂×D₄×C₄₇	—	2	6	no	—
54	C₂×D₄×C₄₇	—	2	6	no	—
55	C₂×D₄×C₄₇	C₂×C₈	1	12	yes	—
56	C₂²×C₄×C₄₇	—	2	6	no	—
57	C₂²×C₄×C₄₇	—	2	6	no	—
58	C₂²×C₄×C₄₇	—	2	6	no	—
59	C₂²×C₄×C₄₇	—	2	6	no	—
60	C₂²×C₄×C₄₇	—	2	6	no	—
61	C₂²×C₄×C₄₇	—	2	6	no	—
62	C₂²×C₄×C₄₇	—	2	3	no	—
63	C₂²×C₄×C₄₇	—	2	3	no	—
64	C₂²×C₄×C₄₇	—	4	3	no	—
65	C₂²×C₄×C₄₇	C₂×C₈	1	12	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	6	no	—
73	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	12	yes	—
74	C₂²⋊C₄×C₄₇	—	2	12	no	—
75	C₄²×C₄₇	—	2	6	no	—
76	C₄²×C₄₇	—	2	6	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	C₂×D₄	1	12	yes	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	6	no	—
82	C₂×C₄₇⋊C₈	—	2	6	no	—
83	C₂×C₄₇⋊C₈	—	2	6	no	—
84	C₂×C₄₇⋊C₈	#41	1	4	yes	complemented
85	C₂³×C₄₇	—	2	3	no	—
86	C₂³×C₄₇	—	2	3	no	—
87	C₂³×C₄₇	—	2	3	no	—
88	C₂³×C₄₇	—	2	3	no	—
89	C₂³×C₄₇	—	2	3	no	—
90	C₂³×C₄₇	—	2	3	no	—
91	C₂³×C₄₇	C₂²×C₈	1	6	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₄₇	C₂²⋊C₈	1	16	yes	complemented
94	D₄×C₄₇	—	2	24	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	3	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	3	no	—
101	C₂×C₄×C₄₇	—	2	24	no	—
102	C₂×C₄×C₄₇	—	2	3	no	—
103	C₂×C₄×C₄₇	—	2	3	no	—
104	C₂×C₄×C₄₇	—	2	3	no	—
105	C₂×C₄×C₄₇	—	4	12	no	—
106	C₂×C₄×C₄₇	—	4	6	no	—
107	C₂×C₄×C₄₇	—	4	6	no	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×C₈	1	6	yes	—
112	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
113	C₂×C₄×C₄₇	#131	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₄₇⋊C₈	—	2	4	no	—
119	C₄₇⋊C₈	—	4	6	no	—
120	#22530	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	6	no	—
122	C₂²×C₄₇	—	2	6	no	—
123	C₂²×C₄₇	—	2	6	no	—
124	C₂²×C₄₇	—	4	3	no	—
125	C₂²×C₄₇	—	4	3	no	—
126	C₂²×C₄₇	—	4	3	no	—
127	C₂²×C₄₇	—	4	3	no	—
128	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
130	C₂²×C₄₇	#520	1	1	yes	characteristic
131	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
134	C₂²×C₄₇	#568	1	1	yes	characteristic
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	6	no	—
138	C₄×C₄₇	—	4	3	no	—
139	C₄×C₄₇	C₂×C₂²⋊C₈	1	12	yes	—
140	C₄×C₄₇	#548	1	2	yes	—
141	C₄×C₄₇	#550	1	2	yes	—
142	C₂×C₄×#41	D₄₇	1	1	yes	characteristic
143	C₂×C₈×Q₈	—	47	2	no	—
144	#1954	—	47	6	no	—
145	#1797	—	47	3	no	—
146	C₄×C₂²⋊C₈	—	47	3	no	—
147	C₂×C₄₇	—	4	6	no	—
148	C₂×C₄₇	#2341	1	1	yes	characteristic
149	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×C₄₇	#2880	1	1	yes	characteristic
151	C₂×C₄₇	C₈×#41	1	1	yes	characteristic
152	C₂×C₄₇	#2971	1	1	yes	characteristic
153	C₂×C₄₇	#2988	1	1	yes	characteristic
154	C₂×C₄₇	#2991	1	1	yes	characteristic
155	C₂²×#41	Dic₄₇	1	1	yes	characteristic
156	C₄×#41	—	2	8	no	—
157	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×C₄×D₄	Dic₄₇	1	3	yes	—
160	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
161	C₂×#117	Dic₄₇	1	3	yes	—
162	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
163	C₈×Q₈	—	94	8	no	—
164	C₂×C₄⋊C₈	—	47	6	no	—
165	C₂×C₂²⋊C₈	—	47	6	no	—
166	C₂×C₂²⋊C₈	—	47	6	no	—
167	C₂×C₄×C₈	—	47	6	no	—
168	#336	—	47	3	no	—
169	#336	—	47	1	no	—
170	C₄₇	#22530	1	1	yes	characteristic, complemented, Sylow
171	C₂×#41	—	2	4	no	—
172	C₂×#41	—	2	4	no	—
173	C₂×#41	C₄₇⋊C₈	1	8	yes	complemented
174	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂²×D₄	C₂×Dic₄₇	1	3	yes	—
176	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
177	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
178	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	—	47	6	no	—
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	C₂²⋊D₄₇	1	8	yes	—
183	C₄×D₄	—	2	12	no	—
184	C₄×D₄	—	2	12	no	—
185	#117	—	2	12	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
188	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
189	C₂×C₄²	—	2	12	no	—
190	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
191	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
192	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
194	C₄⋊C₈	—	94	12	no	—
195	C₂²⋊C₈	—	94	12	no	—
196	C₂²⋊C₈	—	188	6	no	—
197	C₄×C₈	—	94	12	no	—
198	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
199	#41	—	2	32	no	—
200	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
201	C₂×Q₈	#10625329	1	4	yes	—
202	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
203	C₂×D₄	—	2	6	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	C₂×C₄₇⋊C₈	1	12	yes	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	2	3	no	—
213	C₂²×C₄	—	2	3	no	—
214	C₂²×C₄	—	4	3	no	—
215	C₂²×C₄	C₂×C₄₇⋊C₈	1	12	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	3	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	3	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	6	no	—
226	C₂×C₈	—	94	6	no	—
227	C₂×C₈	—	94	6	no	—
228	C₄⋊C₄	—	2	6	no	—
229	C₄⋊C₄	C₂²⋊Dic₄₇	1	12	yes	—
230	C₂²⋊C₄	—	2	12	no	—
231	C₄²	—	2	6	no	—
232	C₄²	—	2	6	no	—
233	C₄²	—	2	2	no	—
234	C₄²	C₂×C₂²⋊D₄₇	1	12	yes	—
235	C₂³	—	2	3	no	—
236	C₂³	—	2	3	no	—
237	C₂³	—	2	3	no	—
238	C₂³	—	2	3	no	—
239	C₂³	—	2	3	no	—
240	C₂³	—	2	3	no	—
241	C₂³	C₂²×C₄₇⋊C₈	1	6	yes	—
242	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
243	Q₈	C₂²⋊(C₄₇⋊C₈)	1	16	yes	complemented
244	D₄	—	2	24	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	4	12	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	—	4	6	no	—
258	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
260	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
261	C₂×C₄	C₂²×C₄₇⋊C₈	1	6	yes	—
262	C₂×C₄	C₂×#10625329	1	6	yes	—
263	C₂×C₄	#13094031	1	2	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
265	C₂×C₄	#41×D₄₇	1	2	yes	—
266	C₂×C₄	#13094070	1	1	yes	characteristic
267	C₂×C₄	#13094071	1	1	yes	characteristic
268	C₈	—	94	4	no	—
269	C₈	—	188	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	4	3	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₂²	#41×Dic₄₇	1	1	yes	characteristic
279	C₂²	#15689606	1	1	yes	characteristic
280	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
281	C₂²	C₂²×#10625329	1	1	yes	characteristic
282	C₂²	C₂×#13094031	1	1	yes	characteristic
283	C₂²	#15689890	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
289	C₄	#15689780	1	2	yes	—
290	C₄	#15689792	1	2	yes	—
291	C₂	—	4	6	no	—
292	C₂	#20246229	1	1	yes	characteristic
293	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
294	C₂	#20251837	1	1	yes	characteristic
295	C₂	#41×C₄₇⋊C₈	1	1	yes	characteristic
296	C₂	#20252408	1	1	yes	characteristic
297	C₂	#20252499	1	1	yes	characteristic
298	C₂	#20252520	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄×#41×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×#41	D₄₇
hypocenter	C₄₇	#22530
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
2. #20244183	C₂	semidirect
3. #20243368	C₂	semidirect
4. C₄×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
29. C₂×#41×C₄₇	C₈	semidirect
84. C₂×C₄₇⋊C₈	#41	semidirect
93. Q₈×C₄₇	C₂²⋊C₈	semidirect
170. C₄₇	#22530	semidirect
173. C₂×#41	C₄₇⋊C₈	semidirect
243. Q₈	C₂²⋊(C₄₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₄₇⋊C₈	2
#20244183	2
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₂×C₄×#41×C₄₇	2
#22530	47