Subgroups — #26376472

Order 12032.

2250 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246353	C₂	1	6	yes	maximal, complemented
2	#20246309	C₂	1	2	yes	maximal, complemented
3	#20246265	C₂	1	6	yes	maximal
4	Q₈×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
5	#20246039	C₂	1	6	yes	maximal, complemented
6	#20245786	C₂	1	3	yes	maximal
7	#20245786	C₂	1	3	yes	maximal
8	#20245753	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
11	#15689030	C₂²	1	12	yes	—
12	#15689022	C₂²	1	12	yes	—
13	#15689009	C₂²	1	24	yes	—
14	#15688991	C₂²	1	6	yes	—
15	#15688991	C₂²	1	6	yes	—
16	#15688989	C₂²	1	6	yes	—
17	#15688989	C₂²	1	6	yes	—
18	#15688986	C₂²	1	1	yes	characteristic
19	#15688986	C₂²	1	3	yes	—
20	#15688983	C₂²	1	6	yes	—
21	#15688983	C₂²	1	6	yes	—
22	C₄⋊C₄×Dic₄₇	C₂²	1	12	yes	—
23	C₄×C₄⋊Dic₄₇	C₂²	1	6	yes	—
24	C₄×C₄⋊Dic₄₇	C₂²	1	6	yes	—
25	C₄×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
26	C₄×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
27	#15688969	C₂²	1	6	yes	—
28	#15688969	C₂²	1	6	yes	—
29	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₄	1	16	yes	complemented
31	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
34	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
36	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
37	Q₈×Dic₄₇	—	2	16	no	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
40	C₂×C₂²⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₄₇	C₂³	1	6	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
44	C₄₇⋊#95	C₂³	1	2	yes	—
45	C₄₇⋊#95	C₂³	1	6	yes	—
46	C₄₇⋊#95	C₂³	1	6	yes	—
47	C₄₇⋊#95	C₂³	1	12	yes	—
48	C₄₇⋊#95	C₂³	1	12	yes	—
49	C₄₇⋊#95	C₂³	1	12	yes	—
50	C₄₇⋊#95	C₂³	1	12	yes	—
51	C₄₇⋊#95	C₂³	1	2	yes	—
52	C₂²×Q₈×C₄₇	—	2	8	no	—
53	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₂²×Q₈×C₄₇	C₂³	1	3	yes	—
55	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
57	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
59	C₄×Q₈×C₄₇	C₂×C₄	1	24	yes	—
60	C₄×Q₈×C₄₇	C₂×C₄	1	24	yes	—
61	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
62	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	24	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
67	C₂×C₄²×C₄₇	C₂×C₄	1	24	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
72	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×Q₈×C₄₇	—	2	32	no	—
74	C₂×Q₈×C₄₇	—	2	12	no	—
75	C₂×Q₈×C₄₇	C₂²×C₄	1	12	yes	—
76	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×Q₈×C₄₇	#41	1	8	yes	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	12	no	—
80	C₂²×C₄×C₄₇	—	4	6	no	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
90	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
91	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
92	C₄⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
93	C₄⋊C₄×C₄₇	C₂²×C₄	1	12	yes	—
94	C₄⋊C₄×C₄₇	C₂²×C₄	1	12	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
99	C₂²⋊Dic₄₇	—	2	4	no	—
100	C₂²⋊Dic₄₇	—	2	12	no	—
101	C₄⋊Dic₄₇	—	2	24	no	—
102	C₄⋊Dic₄₇	—	4	12	no	—
103	C₄×Dic₄₇	—	2	24	no	—
104	C₄×Dic₄₇	—	4	12	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
108	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
109	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
110	Q₈×C₄₇	—	2	24	no	—
111	Q₈×C₄₇	#117	1	16	yes	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	6	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	6	no	—
117	C₂×C₄×C₄₇	—	2	6	no	—
118	C₂×C₄×C₄₇	—	2	24	no	—
119	C₂×C₄×C₄₇	—	4	12	no	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
125	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	12	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	12	no	—
131	C₂×Dic₄₇	—	2	12	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	#34192	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	#519	1	4	yes	—
138	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₂²×#41	1	2	yes	—
140	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
141	C₂²×C₄₇	#585	1	2	yes	—
142	C₄×C₄₇	—	2	6	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	6	no	—
146	C₄×C₄₇	—	2	6	no	—
147	C₄×C₄₇	C₂×#117	1	12	yes	—
148	Dic₄₇	—	4	4	no	—
149	Dic₄₇	—	4	12	no	—
150	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
151	#2356	—	47	6	no	—
152	#2351	—	47	2	no	—
153	#2345	—	47	6	no	—
154	Q₈×C₂²⋊C₄	—	47	2	no	—
155	#2320	—	47	6	no	—
156	#2283	—	47	3	no	—
157	#2283	—	47	3	no	—
158	#2276	—	47	2	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
162	C₂×C₄₇	#3428	1	1	yes	characteristic
163	C₂×C₄₇	#3516	1	4	yes	—
164	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×Q₈	—	47	4	no	—
166	C₂×C₄×Q₈	Dic₄₇	1	16	yes	complemented
167	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
168	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
170	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
172	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
173	#391	—	47	12	no	—
174	#389	—	47	12	no	—
175	#387	—	47	24	no	—
176	#383	—	47	6	no	—
177	#383	—	47	6	no	—
178	#382	—	47	6	no	—
179	#382	—	47	6	no	—
180	#381	—	47	1	no	—
181	#381	—	47	3	no	—
182	#380	—	47	6	no	—
183	#380	—	47	6	no	—
184	C₄×C₄⋊C₄	—	47	12	no	—
185	C₄×C₄⋊C₄	—	47	6	no	—
186	C₄×C₄⋊C₄	—	47	6	no	—
187	C₄×C₂²⋊C₄	—	47	6	no	—
188	C₄×C₂²⋊C₄	—	47	6	no	—
189	#376	—	47	6	no	—
190	#376	—	47	6	no	—
191	C₄₇	#34192	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	2	8	no	—
193	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
194	C₂²×Q₈	C₂²×D₄₇	1	3	yes	—
195	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
199	C₄×Q₈	—	94	16	no	—
200	C₄×Q₈	C₂×Dic₄₇	1	24	yes	—
201	C₄×Q₈	C₂×Dic₄₇	1	24	yes	—
202	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
203	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
204	C₂×C₄⋊C₄	—	47	12	no	—
205	C₂×C₄⋊C₄	—	47	12	no	—
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	24	yes	—
207	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
209	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
210	C₂×C₂²⋊C₄	—	47	2	no	—
211	C₂×C₂²⋊C₄	—	47	6	no	—
212	C₂×C₄²	—	47	12	no	—
213	C₂×C₄²	—	47	12	no	—
214	C₂×C₄²	C₂×Dic₄₇	1	24	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
217	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
218	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
219	#95	—	47	2	no	—
220	#95	—	47	6	no	—
221	#95	—	47	12	no	—
222	#95	—	47	6	no	—
223	#95	—	47	12	no	—
224	#95	—	47	12	no	—
225	#95	—	47	12	no	—
226	#95	—	47	2	no	—
227	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
228	C₂×Q₈	—	2	32	no	—
229	C₂×Q₈	—	2	12	no	—
230	C₂×Q₈	#10625350	1	8	yes	—
231	C₂×Q₈	C₂²×Dic₄₇	1	12	yes	—
232	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	—	4	6	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	—	47	12	no	—
238	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
239	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
240	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
244	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
245	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
247	C₄⋊C₄	—	94	24	no	—
248	C₄⋊C₄	—	188	12	no	—
249	C₄⋊C₄	C₂²×Dic₄₇	1	24	yes	—
250	C₄⋊C₄	C₂²×Dic₄₇	1	12	yes	—
251	C₄⋊C₄	C₂²×Dic₄₇	1	12	yes	—
252	C₂²⋊C₄	—	94	4	no	—
253	C₂²⋊C₄	—	94	12	no	—
254	C₄²	—	94	24	no	—
255	C₄²	—	188	12	no	—
256	C₄²	C₂²×Dic₄₇	1	12	yes	—
257	C₄²	C₂²×Dic₄₇	1	12	yes	—
258	C₄²	C₂²×Dic₄₇	1	12	yes	—
259	C₄²	C₂²×Dic₄₇	1	12	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₂³×Dic₄₇	1	2	yes	—
263	C₂³	#13094072	1	4	yes	—
264	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
265	Q₈	—	2	24	no	—
266	Q₈	#13093983	1	16	yes	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	24	no	—
274	C₂×C₄	—	4	12	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	12	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	—	94	12	no	—
279	C₂×C₄	—	94	12	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
282	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
283	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
284	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
285	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
286	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
287	C₂×C₄	C₂×#10625350	1	12	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	8	no	—
291	C₂²	#15689601	1	4	yes	—
292	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
293	C₂²	C₂²×#10625350	1	2	yes	—
294	C₂²	C₂×#13094072	1	2	yes	—
295	C₂²	#15689959	1	2	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	6	no	—
300	C₄	—	2	6	no	—
301	C₄	—	188	4	no	—
302	C₄	—	188	12	no	—
303	C₄	C₂×#13093983	1	12	yes	—
304	C₂	—	2	4	no	—
305	C₂	C₂²×#13093983	1	1	yes	characteristic
306	C₂	C₂×#15689601	1	1	yes	characteristic
307	C₂	#20256591	1	1	yes	characteristic
308	C₂	#20257381	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	#34192
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20246353	C₂	semidirect
2. #20246309	C₂	semidirect
5. #20246039	C₂	semidirect
8. #20245753	C₂	semidirect
30. C₂×C₄×Q₈×C₄₇	C₄	semidirect
166. C₂×C₄×Q₈	Dic₄₇	semidirect
191. C₄₇	#34192	semidirect

Maximal subgroups

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

subgroup	index
#20246353	2
#20246309	2
#20246265	2
Q₈×C₂²⋊Dic₄₇	2
#20246039	2
#20245786	2
#20245753	2
C₂²×C₄×Q₈×C₄₇	2
#34192	47