Subgroups — #26235122

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20246343	C₂	1	2	yes	maximal
3	#20244159	C₂	1	4	yes	maximal
4	#20244151	C₂	1	2	yes	maximal
5	#20243551	C₂	1	2	yes	maximal
6	#20243548	C₂	1	1	yes	maximal, characteristic
7	#107×Dic₄₇	C₂	1	2	yes	maximal
8	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₈×Dic₉₄	C₄	1	8	yes	complemented
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093955	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15689029	—	2	2	no	—
18	#15689010	—	2	4	no	—
19	#15688993	C₂²	1	2	yes	—
20	#15688988	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₄₇	—	2	2	no	—
23	#15688885	C₂²	1	4	yes	—
24	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
32	C₄×Dic₉₄	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	—	2	4	no	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093958	—	2	4	no	—
41	#13093958	—	4	2	no	—
42	#13093955	—	4	4	no	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
45	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	—	2	4	no	—
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#107×C₄₇	—	2	2	no	—
56	#107×C₄₇	Q₈	1	4	yes	complemented
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	#95×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
66	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
67	C₂×Dic₉₄	—	2	2	no	—
68	C₂×Dic₉₄	—	2	2	no	—
69	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₇	#41	1	2	yes	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
80	C₄²×C₄₇	—	4	4	no	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄⋊Dic₄₇	—	8	2	no	—
87	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
88	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
89	C₄×Dic₄₇	—	8	4	no	—
90	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
91	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
95	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	#135	1	2	yes	—
107	C₂×C₄×C₄₇	#135	1	2	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
123	Dic₉₄	—	2	8	no	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	C₄×Q₈	1	4	yes	—
126	C₄₇⋊C₈	—	2	4	no	—
127	#22727	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
130	C₂²×C₄₇	#550	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
133	C₂²×C₄₇	#576	1	1	yes	characteristic
134	C₂²×C₄₇	#578	1	1	yes	characteristic
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	—	8	2	no	—
138	C₄×C₄₇	#427	1	4	yes	—
139	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₄₇	#520	1	1	yes	characteristic
141	C₄×C₄₇	#557	1	1	yes	characteristic
142	C₄×C₄₇	Q₈²	1	1	yes	characteristic
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	8	4	no	—
146	C₂×C₈×Q₈	—	47	1	no	—
147	#2355	—	47	2	no	—
148	#1949	—	47	4	no	—
149	#1948	—	47	2	no	—
150	#1846	—	47	2	no	—
151	#1845	—	47	1	no	—
152	C₄×#107	—	47	2	no	—
153	C₄×#107	D₄₇	1	1	yes	characteristic
154	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
156	C₂×C₄₇	#2913	1	1	yes	characteristic
157	C₂×C₄₇	#3380	1	1	yes	characteristic
158	C₂×C₄₇	#3383	1	1	yes	characteristic
159	C₂×C₄₇	#3393	1	1	yes	characteristic
160	C₂×C₄₇	#3396	1	1	yes	characteristic
161	C₂×C₄×Q₈	—	47	1	no	—
162	C₈×Q₈	—	47	8	no	—
163	C₂×#107	—	47	2	no	—
164	C₂×#107	—	47	2	no	—
165	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
166	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
167	C₂×#106	—	47	4	no	—
168	C₂×C₄⋊C₈	—	47	2	no	—
169	C₂×C₄⋊C₈	—	47	1	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
172	#391	—	94	2	no	—
173	#387	—	94	4	no	—
174	#384	—	47	2	no	—
175	#382	—	47	4	no	—
176	C₄×C₄⋊C₄	—	47	4	no	—
177	C₄×C₄⋊C₄	—	94	2	no	—
178	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
179	#340	—	47	4	no	—
180	#340	C₂×D₄₇	1	2	yes	—
181	C₄₇	#22727	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	—	47	1	no	—
183	C₂²×C₈	—	47	2	no	—
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₄×Q₈	—	47	4	no	—
187	C₄×Q₈	—	47	4	no	—
188	C₂×C₄⋊C₄	—	47	2	no	—
189	C₂×C₄⋊C₄	—	47	1	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	47	2	no	—
197	C₂×C₄²	—	94	4	no	—
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	—	94	4	no	—
201	#107	—	188	2	no	—
202	#107	Dic₉₄	1	4	yes	complemented
203	#106	—	188	4	no	—
204	C₄⋊C₈	—	47	8	no	—
205	C₄⋊C₈	—	47	4	no	—
206	C₄×C₈	—	47	8	no	—
207	C₄×C₈	C₄×D₄₇	1	4	yes	—
208	#95	—	2	2	no	—
209	#95	—	94	4	no	—
210	#95	—	94	4	no	—
211	#95	—	94	4	no	—
212	C₂×Q₈	—	47	4	no	—
213	C₂×Q₈	—	94	2	no	—
214	C₂×Q₈	—	94	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	47	2	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	94	4	no	—
220	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	47	4	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
231	C₂×C₈	#10625350	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	47	8	no	—
236	C₄⋊C₄	—	47	4	no	—
237	C₄⋊C₄	—	94	4	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	376	2	no	—
240	C₄⋊C₄	—	188	4	no	—
241	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
242	C₄²	—	4	4	no	—
243	C₄²	—	47	4	no	—
244	C₄²	—	47	4	no	—
245	C₄²	—	376	4	no	—
246	C₄²	C₂×C₄×D₄₇	1	2	yes	—
247	C₄²	Q₈×D₄₇	1	2	yes	—
248	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
249	Q₈	—	94	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	47	4	no	—
259	C₂×C₄	—	47	4	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	—	94	2	no	—
262	C₂×C₄	—	94	2	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
269	C₂×C₄	#13094043	1	2	yes	—
270	C₂×C₄	#13094047	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
274	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
275	C₂×C₄	#13094070	1	1	yes	characteristic
276	C₂×C₄	#13094071	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	94	4	no	—
279	C₈	C₄×Dic₉₄	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
281	C₂²	D₄×Dic₉₄	1	1	yes	characteristic
282	C₂²	#15689794	1	1	yes	characteristic
283	C₂²	C₂×#13094043	1	1	yes	characteristic
284	C₂²	C₂×#13094047	1	1	yes	characteristic
285	C₂²	#15689925	1	1	yes	characteristic
286	C₂²	#15689937	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	94	4	no	—
291	C₄	—	94	4	no	—
292	C₄	—	376	4	no	—
293	C₄	#15689160	1	4	yes	—
294	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
295	C₄	#15689603	1	1	yes	characteristic
296	C₄	#15689851	1	1	yes	characteristic
297	C₄	Q₈×Dic₉₄	1	1	yes	characteristic
298	C₂	C₄⋊C₄×Dic₉₄	1	1	yes	characteristic
299	C₂	C₂×#15689160	1	1	yes	characteristic
300	C₂	#20251985	1	1	yes	characteristic
301	C₂	#20256253	1	1	yes	characteristic
302	C₂	#20256277	1	1	yes	characteristic
303	C₂	#20256374	1	1	yes	characteristic
304	C₂	#20256398	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	#22727
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₉₄	C₄	semidirect
56. #107×C₄₇	Q₈	semidirect
181. C₄₇	#22727	semidirect
202. #107	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
#20246343	2
#20244159	2
#20244151	2
#20243551	2
#20243548	2
#107×Dic₄₇	2
C₄×#107×C₄₇	2
#22727	47