Subgroups — #26236078

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246337	C₂	1	2	yes	maximal, complemented
2	#20244265	C₂	1	2	yes	maximal
3	#20244017	C₂	1	4	yes	maximal, complemented
4	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243388	C₂	1	2	yes	maximal, complemented
6	#20243287	C₂	1	2	yes	maximal, complemented
7	#20243255	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	—	2	2	no	—
10	C₂×#13093956	C₂²	1	2	yes	—
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13093945	C₂²	1	4	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	#15689026	C₂²	1	4	yes	—
16	#15689016	—	2	4	no	—
17	#15688993	C₂²	1	2	yes	—
18	#15688983	—	2	2	no	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688885	C₂²	1	4	yes	—
22	#15688877	C₂²	1	2	yes	—
23	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
25	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₄₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	#13093983	—	4	8	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093956	—	2	4	no	—
40	#13093956	—	4	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
42	#13093945	—	4	4	no	—
43	#13093934	—	2	4	no	—
44	C₄₇⋊#95	—	2	4	no	—
45	C₄₇⋊#95	D₄	1	2	yes	—
46	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₇	—	2	2	no	—
48	C₂×D₈×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₄₇	—	2	4	no	—
54	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#107×C₄₇	C₂×C₄	1	4	yes	—
59	#102×C₄₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₂⁴×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	2	no	—
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	8	no	—
70	C₂²×C₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	4	4	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
77	D₈×C₄₇	—	2	8	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	#41	1	2	yes	—
83	C₂×C₈×C₄₇	#41	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂²⋊C₄×C₄₇	—	2	8	no	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	#41	1	2	yes	—
89	C₂²⋊Dic₄₇	—	4	8	no	—
90	C₂²⋊Dic₄₇	—	8	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	—	8	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	—	4	8	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂³×C₄₇	—	2	4	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	—	4	2	no	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	4	2	no	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₄₇	—	2	4	no	—
111	D₄×C₄₇	—	2	4	no	—
112	D₄×C₄₇	—	4	2	no	—
113	D₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	16	no	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₄₇	C₂³×C₄	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×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₈×C₄₇	—	2	2	no	—
135	C₈×C₄₇	#117	1	4	yes	complemented
136	C₄₇⋊C₈	—	4	2	no	—
137	#22813	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	4	4	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	8	2	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	8	2	no	—
145	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₄₇	#547	1	1	yes	characteristic
147	C₂²×C₄₇	#550	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₄₇	#577	1	2	yes	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	#442	1	4	yes	—
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	#520	1	1	yes	characteristic
155	C₄×C₄₇	#555	1	1	yes	characteristic
156	C₄×C₄₇	#556	1	1	yes	characteristic
157	Dic₄₇	—	4	4	no	—
158	Dic₄₇	—	8	4	no	—
159	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
160	#2354	—	47	2	no	—
161	#1972	—	47	2	no	—
162	#1926	—	47	4	no	—
163	#1844	—	47	1	no	—
164	#1802	—	47	2	no	—
165	#1776	—	47	2	no	—
166	#1770	—	47	1	no	—
167	C₂×C₄₇	—	8	4	no	—
168	C₂×C₄₇	#2354	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#442	1	1	yes	characteristic
170	C₂×C₄₇	#2952	1	1	yes	characteristic
171	C₂×C₄₇	#3354	1	2	yes	—
172	C₂×C₄₇	#3397	1	2	yes	—
173	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
175	C₂×#117	—	94	2	no	—
176	C₄×D₈	Dic₄₇	1	8	yes	complemented
177	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#106	—	47	2	no	—
179	C₂×#106	—	47	2	no	—
180	C₂×C₄⋊C₈	—	47	1	no	—
181	C₂×#102	—	47	4	no	—
182	C₂×#102	C₂×D₄₇	1	2	yes	—
183	C₂×#97	—	47	2	no	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	#390	—	47	4	no	—
186	#388	—	94	4	no	—
187	#384	—	47	2	no	—
188	#380	—	94	2	no	—
189	C₄×C₄⋊C₄	—	47	1	no	—
190	C₄×C₄⋊C₄	—	47	4	no	—
191	#340	—	47	4	no	—
192	#336	—	47	2	no	—
193	C₄₇	#22813	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	2	no	—
198	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
199	C₂²×C₈	—	47	2	no	—
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	C₂×Dic₄₇	1	8	yes	—
204	#117	—	188	8	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	94	4	no	—
212	C₂×C₂²⋊C₄	—	94	4	no	—
213	C₂×C₄²	—	47	2	no	—
214	C₂×C₄²	—	94	4	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	#107	C₂×Dic₄₇	1	4	yes	—
217	#106	—	94	4	no	—
218	#106	—	188	2	no	—
219	C₄⋊C₈	—	94	2	no	—
220	#102	—	188	4	no	—
221	#102	C₂×Dic₄₇	1	8	yes	—
222	#97	—	94	4	no	—
223	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
224	#95	—	47	2	no	—
225	#95	—	94	4	no	—
226	C₂⁴	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	47	4	no	—
238	C₂²×C₄	—	94	4	no	—
239	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	D₈	—	2	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	#10625350	1	2	yes	—
249	C₂×C₈	#10625350	1	2	yes	—
250	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
251	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	94	4	no	—
254	C₄⋊C₄	—	94	4	no	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	—	376	2	no	—
257	C₄⋊C₄	—	188	4	no	—
258	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	376	4	no	—
261	C₂²⋊C₄	—	188	8	no	—
262	C₄²	—	94	2	no	—
263	C₄²	—	94	2	no	—
264	C₄²	—	188	8	no	—
265	C₄²	#10625352	1	2	yes	—
266	C₄²	C₂²×Dic₄₇	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	188	4	no	—
288	C₂×C₄	—	188	4	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
291	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	#13094048	1	4	yes	—
293	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
297	C₂×C₄	#13094070	1	1	yes	characteristic
298	C₂×C₄	#13094071	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	188	2	no	—
301	C₈	#13093983	1	4	yes	complemented
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
310	C₂²	#15689778	1	1	yes	characteristic
311	C₂²	#15689794	1	1	yes	characteristic
312	C₂²	C₂×#13094048	1	2	yes	—
313	C₂²	#15689930	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	376	4	no	—
317	C₄	—	188	4	no	—
318	C₄	#15689235	1	4	yes	—
319	C₄	C₂×#13093983	1	1	yes	characteristic
320	C₄	#15689606	1	1	yes	characteristic
321	C₄	#15689838	1	1	yes	characteristic
322	C₄	#15689843	1	1	yes	characteristic
323	C₂	—	8	4	no	—
324	C₂	#20246337	1	1	yes	characteristic
325	C₂	C₂×#15689235	1	1	yes	characteristic
326	C₂	#20252272	1	1	yes	characteristic
327	C₂	#20255985	1	2	yes	—
328	C₂	#20256406	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#22813
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20246337	C₂	semidirect
3. #20244017	C₂	semidirect
4. #20243545	C₂	semidirect
5. #20243388	C₂	semidirect
6. #20243287	C₂	semidirect
7. #20243255	C₂	semidirect
25. C₄×D₈×C₄₇	C₄	semidirect
135. C₈×C₄₇	#117	semidirect
176. C₄×D₈	Dic₄₇	semidirect
193. C₄₇	#22813	semidirect
301. C₈	#13093983	semidirect

Maximal subgroups

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

subgroup	index
#20246337	2
#20244265	2
#20244017	2
#20243545	2
#20243388	2
#20243287	2
#20243255	2
C₂×C₄×D₈×C₄₇	2
#22813	47