Subgroups — #26316242

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	C₂×#15689246	C₂	1	1	yes	maximal, characteristic
2	#20244557	C₂	1	4	yes	maximal, complemented
3	#20244284	C₂	1	2	yes	maximal
4	#20243511	C₂	1	2	yes	maximal, complemented
5	#20243439	C₂	1	2	yes	maximal, complemented
6	#20243431	C₂	1	2	yes	maximal, complemented
7	#20243272	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#3162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂×#13094030	C₂²	1	2	yes	—
11	#15689246	—	2	4	no	—
12	C₂×#13093960	C₂²	1	1	yes	characteristic
13	C₂×#13093960	C₂²	1	1	yes	characteristic
14	#15689161	C₂²	1	1	yes	characteristic
15	#15689141	C₂²	1	4	yes	—
16	C₂×#13093952	C₂²	1	2	yes	—
17	#15689096	C₂²	1	4	yes	—
18	#15689078	C₂²	1	2	yes	—
19	#15688893	C₂²	1	4	yes	—
20	#15688883	C₂²	1	2	yes	—
21	#15688879	C₂²	1	4	yes	—
22	C₄₇×#533	C₂²	1	2	yes	—
23	C₄₇×#501	—	2	4	no	—
24	C₄₇×#427	C₂²	1	2	yes	—
25	C₄₇×#427	C₂²	1	2	yes	—
26	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	C₂³	1	2	yes	—
28	#13094046	—	2	2	no	—
29	#13094046	—	2	2	no	—
30	C₂×Dic₁₈₈	Q₈	1	1	yes	characteristic, complemented
31	C₂×#10625325	Q₈	1	2	yes	complemented
32	C₂×D₃₇₆	Q₈	1	1	yes	characteristic, complemented
33	#13094030	—	2	2	no	—
34	#13094030	—	2	2	no	—
35	#13094030	—	2	2	no	—
36	#13094030	—	2	2	no	—
37	C₂×#10625329	C₂³	1	2	yes	—
38	C₂×#10625329	C₂³	1	2	yes	—
39	C₂×#10625324	—	2	2	no	—
40	C₂×C₈×D₄₇	—	2	2	no	—
41	#13093983	C₂³	1	2	yes	—
42	#13093960	—	2	2	no	—
43	#13093960	—	2	2	no	—
44	#13093958	—	2	1	no	—
45	#13093956	—	2	1	no	—
46	#13093952	—	2	4	no	—
47	#13093952	—	4	4	no	—
48	#13093938	—	4	4	no	—
49	#13093934	—	2	2	no	—
50	C₈×Dic₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₇	—	2	2	no	—
54	#125×C₄₇	—	2	2	no	—
55	C₂²⋊Q₈×C₄₇	—	4	2	no	—
56	C₄×Q₈×C₄₇	—	4	2	no	—
57	#117×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#107×C₄₇	—	2	2	no	—
60	#107×C₄₇	—	2	2	no	—
61	#106×C₄₇	—	2	2	no	—
62	#106×C₄₇	—	2	2	no	—
63	#97×C₄₇	D₄	1	4	yes	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₂²⋊D₄₇	—	2	2	no	—
68	#10625350	—	2	2	no	—
69	#10625350	—	2	2	no	—
70	#10625350	—	2	2	no	—
71	#10625350	—	2	2	no	—
72	C₂×D₁₈₈	C₂×Q₈	1	2	yes	—
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×Dic₉₄	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₄₇	#41	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
82	C₂×C₈×C₄₇	#41	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	4	4	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	1	no	—
88	C₄²×C₄₇	—	2	4	no	—
89	C₄²×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	#10625329	—	2	2	no	—
94	#10625329	—	2	2	no	—
95	#10625329	—	2	2	no	—
96	#10625329	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	Dic₁₈₈	—	2	2	no	—
100	D₃₇₆	—	2	2	no	—
101	#10625325	—	2	4	no	—
102	#10625324	—	4	2	no	—
103	#10625324	—	4	2	no	—
104	C₈×D₄₇	—	4	2	no	—
105	C₈×D₄₇	—	4	2	no	—
106	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	2	no	—
108	Q₈×C₄₇	—	8	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	1	no	—
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²⋊D₄₇	—	4	2	no	—
119	C₂²⋊D₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	D₁₈₈	—	2	2	no	—
123	D₁₈₈	—	2	2	no	—
124	C₄×D₄₇	—	4	2	no	—
125	C₄×D₄₇	—	4	2	no	—
126	Dic₉₄	—	2	2	no	—
127	Dic₉₄	—	2	2	no	—
128	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₄₇	#124	1	2	yes	—
131	C₈×C₄₇	#124	1	2	yes	—
132	C₄₇⋊C₈	—	4	2	no	—
133	C₄₇⋊C₈	—	4	2	no	—
134	#29155	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	#535	1	1	yes	characteristic
137	C₂²×C₄₇	#554	1	1	yes	characteristic
138	C₂²×C₄₇	#555	1	1	yes	characteristic
139	C₂×D₄₇	—	4	2	no	—
140	C₂×D₄₇	—	8	2	no	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	8	4	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	—	4	2	no	—
148	Dic₄₇	—	8	2	no	—
149	#3162	D₄₇	1	1	yes	characteristic, complemented
150	C₂×#444	—	47	1	no	—
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	#2563	1	1	yes	characteristic
161	D₄₇	—	8	2	no	—
162	C₂×#135	—	47	1	no	—
163	C₂×#131	—	47	2	no	—
164	#533	C₂×D₄₇	1	2	yes	—
165	#501	—	2	4	no	—
166	#444	—	94	4	no	—
167	C₂×#108	—	47	1	no	—
168	C₂×#108	—	47	1	no	—
169	#427	—	47	1	no	—
170	#427	C₂×D₄₇	1	2	yes	—
171	#427	C₂×D₄₇	1	2	yes	—
172	#421	—	47	4	no	—
173	C₂×#104	—	47	2	no	—
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	—	47	2	no	—
182	#135	—	94	2	no	—
183	#135	—	94	2	no	—
184	C₂×Q₁₆	—	47	1	no	—
185	C₂×SD₁₆	—	47	2	no	—
186	C₂×D₈	—	47	1	no	—
187	#131	—	94	2	no	—
188	#131	—	94	2	no	—
189	#131	—	94	2	no	—
190	#131	—	94	2	no	—
191	C₂×M₁₆	—	47	2	no	—
192	C₂×M₁₆	—	47	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	#125	—	2	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	C₄×Q₈	—	4	2	no	—
201	#117	—	47	2	no	—
202	#117	C₂²×D₄₇	1	4	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	#108	—	94	2	no	—
205	#108	—	94	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	—	94	1	no	—
209	#106	—	2	2	no	—
210	#106	—	2	2	no	—
211	#106	—	94	1	no	—
212	#104	—	94	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	—	94	2	no	—
219	#41	—	94	2	no	—
220	#41	—	94	2	no	—
221	#41	—	94	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	47	2	no	—
224	C₂×D₄	—	47	2	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	Q₁₆	—	94	2	no	—
230	SD₁₆	—	94	4	no	—
231	D₈	—	94	2	no	—
232	M₁₆	—	94	2	no	—
233	M₁₆	—	94	2	no	—
234	M₁₆	—	94	2	no	—
235	M₁₆	—	94	2	no	—
236	M₁₆	—	188	2	no	—
237	M₁₆	—	188	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₂×C₈	—	188	2	no	—
242	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
243	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
244	C₂×C₈	#10625354	1	2	yes	—
245	C₂×C₈	#10625354	1	1	yes	characteristic
246	C₂×C₈	#10625354	1	1	yes	characteristic
247	C₂×C₈	#10625354	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	94	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	94	2	no	—
259	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₂³	—	94	2	no	—
261	C₂³	#13094069	1	1	yes	characteristic
262	Q₈	—	8	4	no	—
263	Q₈	—	94	2	no	—
264	Q₈	—	94	2	no	—
265	D₄	—	94	2	no	—
266	D₄	—	94	2	no	—
267	D₄	—	188	2	no	—
268	D₄	—	188	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
279	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
281	C₂×C₄	#13094069	1	1	yes	characteristic
282	C₈	—	188	2	no	—
283	C₈	—	188	2	no	—
284	C₈	#13094000	1	2	yes	—
285	C₈	#13094000	1	2	yes	—
286	C₈	#13094013	1	2	yes	—
287	C₈	#13094013	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	376	2	no	—
290	C₂²	—	188	2	no	—
291	C₂²	#15689690	1	1	yes	characteristic
292	C₂²	#15689834	1	1	yes	characteristic
293	C₂²	#15689839	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	8	4	no	—
296	C₄	—	376	2	no	—
297	C₄	—	188	2	no	—
298	C₄	C₂×#13094000	1	1	yes	characteristic
299	C₄	C₂×#13094013	1	1	yes	characteristic
300	C₄	#15689834	1	1	yes	characteristic
301	C₄	#15689839	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	376	2	no	—
306	C₂	#20248562	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689839
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3162	C₂
socle	C₂×C₄₇	#2563
prime radical	C₂×#13094046	C₂²
hypercenter	#3162	D₄₇
hypocenter	C₄₇	#29155
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3162

Complements

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

N	H ≅ G/N	type
2. #20244557	C₂	semidirect
4. #20243511	C₂	semidirect
5. #20243439	C₂	semidirect
6. #20243431	C₂	semidirect
7. #20243272	C₂	semidirect
8. C₄₇×#3162	C₂	semidirect
30. C₂×Dic₁₈₈	Q₈	semidirect
31. C₂×#10625325	Q₈	semidirect
32. C₂×D₃₇₆	Q₈	semidirect
149. #3162	D₄₇	semidirect
180. C₄₇	#29155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689246	2
#20244557	2
#20244284	2
#20243511	2
#20243439	2
#20243431	2
#20243272	2
C₄₇×#3162	2
#29155	47