Subgroups — #26225589

Order 12032.

1366 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20244312	C₂	1	4	yes	maximal, complemented
4	#20244191	C₂	1	4	yes	maximal, complemented
5	#20243323	C₂	1	4	yes	maximal
6	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	—	2	1	no	—
8	#15689176	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093948	C₂²	1	16	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15688993	C₂²	1	2	yes	—
14	#15688989	—	2	1	no	—
15	#15688983	—	2	1	no	—
16	#15688983	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	—	2	1	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
21	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	#13093983	—	4	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
37	#13093948	—	4	16	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₂²×Q₈×C₄₇	—	2	4	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	4	8	no	—
45	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
46	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	8	no	—
52	C₄⋊Q₈×C₄₇	—	2	8	no	—
53	C₄⋊Q₈×C₄₇	—	2	4	no	—
54	C₄⋊Q₈×C₄₇	—	2	4	no	—
55	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
64	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₇	D₄	1	4	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₄₇	—	2	8	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	16	no	—
73	C₂×Q₈×C₄₇	—	4	16	no	—
74	C₂×Q₈×C₄₇	—	4	8	no	—
75	C₂×Q₈×C₄₇	—	4	16	no	—
76	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	8	no	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	—	4	8	no	—
82	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
85	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
86	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	—	2	8	no	—
96	C₄⋊C₄×C₄₇	—	2	8	no	—
97	C₄⋊C₄×C₄₇	—	2	8	no	—
98	C₄⋊C₄×C₄₇	—	2	8	no	—
99	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₂²⋊Dic₄₇	—	4	4	no	—
106	C₄⋊Dic₄₇	—	2	8	no	—
107	C₄⋊Dic₄₇	—	4	4	no	—
108	C₄⋊Dic₄₇	—	4	4	no	—
109	C₄×Dic₄₇	—	4	4	no	—
110	C₂×C₄₇⋊C₈	—	2	4	no	—
111	C₂×C₄₇⋊C₈	—	2	8	no	—
112	C₂³×C₄₇	—	2	2	no	—
113	C₂³×C₄₇	—	2	2	no	—
114	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
116	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
117	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₄₇	—	2	16	no	—
119	Q₈×C₄₇	—	4	16	no	—
120	Q₈×C₄₇	—	4	8	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	2	8	no	—
127	C₂×C₄×C₄₇	—	4	16	no	—
128	C₂×C₄×C₄₇	—	4	8	no	—
129	C₂×C₄×C₄₇	#103	1	16	yes	complemented
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₄₇	#137	1	4	yes	—
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
143	C₂×Dic₄₇	—	4	4	no	—
144	C₂×Dic₄₇	—	4	4	no	—
145	C₂×Dic₄₇	—	4	4	no	—
146	C₄₇⋊C₈	—	4	4	no	—
147	#21804	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	4	no	—
151	C₂²×C₄₇	—	2	4	no	—
152	C₂²×C₄₇	C₂×#103	1	4	yes	—
153	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄₇	#535	1	2	yes	—
155	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#137	1	2	yes	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	4	8	no	—
161	C₄×C₄₇	C₂×#103	1	4	yes	—
162	C₄×C₄₇	C₂×#103	1	4	yes	—
163	C₄×C₄₇	#419	1	4	yes	—
164	C₄×C₄₇	#518	1	2	yes	—
165	C₄×C₄₇	#535	1	2	yes	—
166	Dic₄₇	—	8	4	no	—
167	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
168	C₂×#431	—	47	1	no	—
169	#2285	—	47	1	no	—
170	#1981	—	47	4	no	—
171	#1956	—	47	4	no	—
172	#1787	—	47	4	no	—
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	#2309	1	1	yes	characteristic
175	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#419	1	1	yes	characteristic
177	C₂×C₄₇	#3206	1	2	yes	—
178	C₂×C₄₇	#3209	1	2	yes	—
179	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
180	C₂×C₄⋊Q₈	—	2	1	no	—
181	C₂×C₄⋊Q₈	—	2	1	no	—
182	C₂×C₄⋊Q₈	—	2	4	no	—
183	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
184	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
185	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
186	C₂×#117	—	94	1	no	—
187	C₂²×C₄⋊C₄	—	2	1	no	—
188	C₂²×C₄⋊C₄	—	2	1	no	—
189	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
190	#431	—	94	4	no	—
191	C₂×C₄⋊C₈	—	47	2	no	—
192	C₂×#103	—	47	16	no	—
193	C₂×C₂²⋊C₈	—	47	2	no	—
194	C₂×C₄×C₈	—	47	2	no	—
195	#384	—	47	2	no	—
196	#382	—	94	1	no	—
197	#380	—	47	2	no	—
198	#380	—	94	1	no	—
199	C₄×C₄⋊C₄	—	47	2	no	—
200	C₄×C₄⋊C₄	—	94	1	no	—
201	C₄₇	#21804	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	4	8	no	—
205	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
206	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₂²×C₈	—	47	4	no	—
212	C₄⋊Q₈	—	2	8	no	—
213	C₄⋊Q₈	—	2	8	no	—
214	C₄⋊Q₈	—	2	4	no	—
215	C₄⋊Q₈	—	2	4	no	—
216	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
217	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
218	#117	—	188	4	no	—
219	C₂×C₄⋊C₄	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	8	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	47	4	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₂²⋊C₄	—	94	2	no	—
229	C₂×C₄²	—	94	2	no	—
230	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
231	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
234	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊C₈	—	94	4	no	—
237	#103	—	188	16	no	—
238	C₂²⋊C₈	—	94	4	no	—
239	C₄×C₈	—	94	4	no	—
240	#95	—	94	2	no	—
241	#95	—	94	2	no	—
242	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	16	no	—
246	C₂×Q₈	—	4	16	no	—
247	C₂×Q₈	—	4	8	no	—
248	C₂×Q₈	—	4	16	no	—
249	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	4	8	no	—
255	C₂²×C₄	—	94	4	no	—
256	C₂²×C₄	#10625336	1	4	yes	—
257	C₂²×C₄	#10625337	1	4	yes	—
258	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
259	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
260	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
261	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
268	C₂×C₈	—	94	4	no	—
269	C₂×C₈	—	94	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	8	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	94	8	no	—
275	C₄⋊C₄	—	188	4	no	—
276	C₄⋊C₄	—	188	4	no	—
277	C₂²⋊C₄	—	188	4	no	—
278	C₄²	—	188	4	no	—
279	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
280	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
281	C₄²	C₂²×Dic₄₇	1	2	yes	—
282	C₄²	C₂²×Dic₄₇	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
284	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
288	C₂³	C₂×#10625336	1	2	yes	—
289	C₂³	C₂×#10625337	1	2	yes	—
290	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
291	Q₈	—	2	16	no	—
292	Q₈	—	4	16	no	—
293	Q₈	—	4	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	—	188	4	no	—
304	C₂×C₄	—	188	4	no	—
305	C₂×C₄	#13093948	1	16	yes	complemented
306	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
307	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
309	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
310	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
311	C₂×C₄	C₂×#10625336	1	2	yes	—
312	C₂×C₄	C₂×#10625336	1	2	yes	—
313	C₂×C₄	C₂×#10625337	1	2	yes	—
314	C₂×C₄	C₂×#10625337	1	2	yes	—
315	C₂×C₄	#13094055	1	4	yes	—
316	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₂×C₄	#13094069	1	4	yes	—
319	C₈	—	188	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	C₂×#13093948	1	4	yes	—
325	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
326	C₂²	#15689690	1	2	yes	—
327	C₂²	C₂²×#10625336	1	1	yes	characteristic
328	C₂²	C₂²×#10625337	1	1	yes	characteristic
329	C₂²	C₂×#13094055	1	2	yes	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	8	no	—
333	C₄	—	376	4	no	—
334	C₄	C₂×#13093948	1	4	yes	—
335	C₄	C₂×#13093948	1	4	yes	—
336	C₄	#15689132	1	4	yes	—
337	C₄	#15689598	1	2	yes	—
338	C₄	#15689690	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	#20245970	1	1	yes	characteristic
341	C₂	C₂²×#13093948	1	1	yes	characteristic
342	C₂	C₂×#15689132	1	1	yes	characteristic
343	C₂	#20254313	1	2	yes	—
344	C₂	#20254336	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#21804
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #20244312	C₂	semidirect
4. #20244191	C₂	semidirect
23. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
129. C₂×C₄×C₄₇	#103	semidirect
183. C₂×C₄⋊Q₈	Dic₄₇	semidirect
201. C₄₇	#21804	semidirect
305. C₂×C₄	#13093948	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20245795	2
#20244312	2
#20244191	2
#20243323	2
C₂²×C₄⋊Q₈×C₄₇	2
#21804	47