Subgroups — #26316241

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253912	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244560	C₂	1	4	yes	maximal, complemented
3	#20244287	C₂	1	2	yes	maximal
4	#20243513	C₂	1	2	yes	maximal, complemented
5	#20243442	C₂	1	2	yes	maximal, complemented
6	#20243434	C₂	1	2	yes	maximal, complemented
7	#20243272	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#444	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689532	—	2	4	no	—
11	C₂×#13093960	C₂²	1	1	yes	characteristic
12	#15689161	C₂²	1	2	yes	—
13	#15689161	C₂²	1	1	yes	characteristic
14	#15689161	C₂²	1	2	yes	—
15	#15689140	C₂²	1	4	yes	—
16	#15689097	C₂²	1	4	yes	—
17	#15689078	C₂²	1	2	yes	—
18	#15688895	C₂²	1	4	yes	—
19	#15688883	C₂²	1	2	yes	—
20	#15688880	C₂²	1	4	yes	—
21	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#131×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#444	—	2	4	no	—
24	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#104×C₄₇	C₂²	1	2	yes	—
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625329	C₂³	1	2	yes	—
28	#13094024	—	2	2	no	—
29	#13094016	—	2	2	no	—
30	#13094003	—	4	2	no	—
31	C₄×Dic₉₄	—	4	2	no	—
32	#13093983	C₂³	1	2	yes	—
33	#13093983	C₂³	1	4	yes	—
34	#13093960	—	2	2	no	—
35	#13093958	—	2	2	no	—
36	#13093958	—	2	2	no	—
37	#13093958	—	2	1	no	—
38	#13093956	—	2	2	no	—
39	#13093956	—	2	2	no	—
40	#13093956	—	2	1	no	—
41	#13093951	—	4	4	no	—
42	#13093939	—	4	4	no	—
43	#13093934	—	2	2	no	—
44	C₈×Dic₄₇	—	2	2	no	—
45	C₂×#41×C₄₇	C₂³	1	2	yes	—
46	#135×C₄₇	—	2	2	no	—
47	#135×C₄₇	—	2	2	no	—
48	C₂×Q₁₆×C₄₇	Q₈	1	1	yes	characteristic, complemented
49	C₂×SD₁₆×C₄₇	Q₈	1	2	yes	complemented
50	C₂×D₈×C₄₇	Q₈	1	1	yes	characteristic, complemented
51	#131×C₄₇	—	2	2	no	—
52	#131×C₄₇	—	2	2	no	—
53	#131×C₄₇	—	2	2	no	—
54	#131×C₄₇	—	2	2	no	—
55	C₂×M₁₆×C₄₇	—	2	2	no	—
56	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#108×C₄₇	—	2	2	no	—
62	#104×C₄₇	—	2	4	no	—
63	#97×C₄₇	D₄	1	4	yes	—
64	#41×C₄₇	—	2	2	no	—
65	#41×C₄₇	—	2	2	no	—
66	#41×C₄₇	—	2	2	no	—
67	#41×C₄₇	—	2	2	no	—
68	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	—	2	1	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₉₄	—	4	2	no	—
75	Q₁₆×C₄₇	—	2	2	no	—
76	SD₁₆×C₄₇	—	2	4	no	—
77	D₈×C₄₇	—	2	2	no	—
78	M₁₆×C₄₇	—	2	2	no	—
79	M₁₆×C₄₇	—	2	2	no	—
80	M₁₆×C₄₇	—	4	2	no	—
81	M₁₆×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	4	2	no	—
84	C₂×C₈×C₄₇	—	4	2	no	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₄₇	#41	1	2	yes	—
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
93	C₄²×C₄₇	—	2	1	no	—
94	C₄²×C₄₇	C₂×D₄	1	2	yes	—
95	C₂²⋊Dic₄₇	—	2	2	no	—
96	C₂²⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	4	2	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	2	4	no	—
103	#10625329	—	2	2	no	—
104	#10625329	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈×C₄₇	—	2	2	no	—
109	Q₈×C₄₇	—	2	2	no	—
110	D₄×C₄₇	—	2	2	no	—
111	D₄×C₄₇	—	2	2	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	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	8	1	no	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	Dic₉₄	—	8	4	no	—
128	C₈×C₄₇	—	4	2	no	—
129	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₄₇	#124	1	2	yes	—
132	C₈×C₄₇	#124	1	2	yes	—
133	C₄₇⋊C₈	—	4	2	no	—
134	#29155	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	—	8	2	no	—
138	C₂²×C₄₇	#535	1	1	yes	characteristic
139	C₂²×C₄₇	#554	1	1	yes	characteristic
140	C₂²×C₄₇	#555	1	1	yes	characteristic
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	#554	1	1	yes	characteristic
146	C₄×C₄₇	#555	1	1	yes	characteristic
147	Dic₄₇	—	8	4	no	—
148	Dic₄₇	—	8	2	no	—
149	#3162	—	47	1	no	—
150	C₂×#444	D₄₇	1	1	yes	characteristic
151	#2031	—	47	4	no	—
152	#1976	—	47	2	no	—
153	#1833	—	47	2	no	—
154	#1814	—	47	2	no	—
155	#1812	—	47	2	no	—
156	#1773	—	47	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	8	2	no	—
161	C₂×C₄₇	#2563	1	1	yes	characteristic
162	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#131	C₂×D₄₇	1	2	yes	—
164	#533	—	47	2	no	—
165	#501	—	94	4	no	—
166	#444	—	2	4	no	—
167	C₂×#108	—	47	1	no	—
168	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
169	#427	—	47	1	no	—
170	#427	—	47	2	no	—
171	#427	—	47	2	no	—
172	#421	—	47	4	no	—
173	C₂×#104	C₂×D₄₇	1	2	yes	—
174	#410	—	47	4	no	—
175	#405	—	47	2	no	—
176	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
177	#344	—	47	4	no	—
178	#339	—	47	2	no	—
179	H(3,ℤ/4)	—	47	4	no	—
180	C₄₇	#29155	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	C₂²×D₄₇	1	2	yes	—
182	#135	—	2	2	no	—
183	#135	—	2	2	no	—
184	C₂×Q₁₆	Dic₉₄	1	1	yes	characteristic, complemented
185	C₂×SD₁₆	Dic₉₄	1	2	yes	complemented
186	C₂×D₈	Dic₉₄	1	1	yes	characteristic, complemented
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	47	2	no	—
193	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊Q₈	—	94	2	no	—
198	#125	—	94	2	no	—
199	C₂²⋊Q₈	—	188	2	no	—
200	C₄×Q₈	—	188	2	no	—
201	#117	—	47	4	no	—
202	#117	—	47	2	no	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	#108	—	2	2	no	—
205	#108	—	94	2	no	—
206	#107	—	94	1	no	—
207	#107	—	94	2	no	—
208	#107	—	94	2	no	—
209	#106	—	94	1	no	—
210	#106	—	94	2	no	—
211	#106	—	94	2	no	—
212	#104	—	2	4	no	—
213	#104	—	188	4	no	—
214	#99	—	188	4	no	—
215	#97	—	94	2	no	—
216	#97	C₂²⋊D₄₇	1	4	yes	—
217	C₄×C₈	—	94	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	188	2	no	—
223	C₂×Q₈	C₂×Dic₉₄	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Dic₉₄	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	94	2	no	—
237	M₁₆	—	94	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
243	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
244	C₂×C₈	#10625350	1	2	yes	—
245	C₂×C₈	#10625352	1	1	yes	characteristic
246	C₂×C₈	#10625352	1	1	yes	characteristic
247	C₂×C₈	#10625352	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	188	4	no	—
253	C₄⋊C₄	—	188	2	no	—
254	C₂²⋊C₄	—	94	2	no	—
255	C₂²⋊C₄	—	94	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	94	2	no	—
258	C₄²	—	94	4	no	—
259	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	#13094070	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	376	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
279	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
281	C₂×C₄	#13094070	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	188	2	no	—
284	C₈	#13094003	1	2	yes	—
285	C₈	#13094003	1	2	yes	—
286	C₈	#13094011	1	2	yes	—
287	C₈	#13094011	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#15689687	1	1	yes	characteristic
292	C₂²	#15689830	1	1	yes	characteristic
293	C₂²	#15689838	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	376	4	no	—
297	C₄	—	376	2	no	—
298	C₄	C₂×#13094003	1	1	yes	characteristic
299	C₄	C₂×#13094011	1	1	yes	characteristic
300	C₄	#15689830	1	1	yes	characteristic
301	C₄	#15689838	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	#20248561	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689838
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#444	C₂
socle	C₂×C₄₇	#2563
prime radical	C₂×#135×C₄₇	C₂²
hypercenter	C₂×#444	D₄₇
hypocenter	C₄₇	#29155
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
1. #20253912	C₂	semidirect
2. #20244560	C₂	semidirect
4. #20243513	C₂	semidirect
5. #20243442	C₂	semidirect
6. #20243434	C₂	semidirect
7. #20243272	C₂	semidirect
48. C₂×Q₁₆×C₄₇	Q₈	semidirect
49. C₂×SD₁₆×C₄₇	Q₈	semidirect
50. C₂×D₈×C₄₇	Q₈	semidirect
180. C₄₇	#29155	semidirect
184. C₂×Q₁₆	Dic₉₄	semidirect
185. C₂×SD₁₆	Dic₉₄	semidirect
186. C₂×D₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253912	2
#20244560	2
#20244287	2
#20243513	2
#20243442	2
#20243434	2
#20243272	2
C₂×C₄₇×#444	2
#29155	47