Subgroups — #26203975

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253963	C₂	1	4	yes	maximal, complemented
2	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243004	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243002	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242995	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242993	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242978	C₂	1	4	yes	maximal, complemented
8	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094046	—	2	2	no	—
11	#15689673	C₂²	1	2	yes	—
12	#15689526	C₂²	1	4	yes	—
13	#15689507	—	2	2	no	—
14	#15689503	C₂²	1	1	yes	characteristic
15	#15689500	C₂²	1	1	yes	characteristic
16	#15689499	C₂²	1	4	yes	—
17	#15689496	C₂²	1	2	yes	—
18	#15689471	C₂²	1	4	yes	—
19	#15689469	C₂²	1	4	yes	—
20	#15689357	—	2	8	no	—
21	C₄×#10625334	—	2	8	no	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
23	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
24	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×#135×C₄₇	—	2	2	no	—
27	C₄₇×#532	C₂²	1	2	yes	—
28	C₄₇×#495	—	2	2	no	—
29	C₄₇×#494	C₂²	1	1	yes	characteristic
30	C₄₇×#493	C₂²	1	1	yes	characteristic
31	C₄₇×#492	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625350	—	2	2	no	—
34	#13094046	—	4	8	no	—
35	C₂×Dic₁₈₈	—	2	2	no	—
36	C₂×#10625325	—	2	4	no	—
37	C₂×D₃₇₆	—	2	2	no	—
38	C₂×#10625334	—	2	8	no	—
39	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
40	#13094024	C₂³	1	2	yes	—
41	#13094021	C₂³	1	2	yes	—
42	#13094012	—	2	2	no	—
43	#13093998	—	2	4	no	—
44	C₄×D₁₈₈	—	2	4	no	—
45	#13093957	—	2	8	no	—
46	#13093945	—	2	8	no	—
47	#13093944	—	2	8	no	—
48	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
49	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
50	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
51	C₂×#41×C₄₇	—	2	2	no	—
52	#135×C₄₇	—	4	8	no	—
53	C₂×Q₁₆×C₄₇	—	2	2	no	—
54	C₂×SD₁₆×C₄₇	—	2	4	no	—
55	C₂×D₈×C₄₇	—	2	2	no	—
56	C₂²×C₈×C₄₇	D₄	1	2	yes	—
57	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
58	#127×C₄₇	C₂³	1	2	yes	—
59	#124×C₄₇	—	2	2	no	—
60	#121×C₄₇	—	2	4	no	—
61	C₄×D₄×C₄₇	—	2	4	no	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	D₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	#41×C₄₇	—	4	8	no	—
67	C₂×Q₈×C₄₇	—	2	2	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	2	2	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₇	—	4	2	no	—
75	#10625350	—	4	8	no	—
76	C₂×D₁₈₈	—	2	4	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×C₄×D₄₇	—	4	2	no	—
79	C₂×Dic₉₄	—	2	2	no	—
80	Q₁₆×C₄₇	—	4	4	no	—
81	SD₁₆×C₄₇	—	4	8	no	—
82	D₈×C₄₇	—	4	4	no	—
83	C₂×C₈×C₄₇	D₈	1	8	yes	complemented
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₂²⋊C₄×C₄₇	—	4	4	no	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
91	#10625334	—	4	16	no	—
92	(C₂×D₄₇)⋊C₄	—	4	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	8	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
97	Dic₁₈₈	—	4	4	no	—
98	D₃₇₆	—	4	4	no	—
99	#10625325	—	4	8	no	—
100	C₂³×C₄₇	—	4	2	no	—
101	C₂³×C₄₇	#127	1	1	yes	characteristic
102	C₂²×D₄₇	—	4	2	no	—
103	Q₈×C₄₇	—	4	4	no	—
104	D₄×C₄₇	—	4	4	no	—
105	D₄×C₄₇	—	4	8	no	—
106	D₄×C₄₇	—	8	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	8	4	no	—
110	C₂×C₄×C₄₇	#127	1	2	yes	—
111	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₄₇	—	8	4	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	D₁₈₈	—	4	4	no	—
118	D₁₈₈	—	4	8	no	—
119	C₄×D₄₇	—	8	4	no	—
120	Dic₉₄	—	4	4	no	—
121	C₈×C₄₇	C₂×D₈	1	4	yes	—
122	C₈×C₄₇	C₂×D₈	1	4	yes	—
123	C₄₇⋊C₈	—	2	4	no	—
124	C₄₇⋊C₈	#135	1	8	yes	complemented
125	#19603	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	8	2	no	—
128	C₂²×C₄₇	—	8	4	no	—
129	C₂²×C₄₇	#493	1	2	yes	—
130	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
131	C₂×D₄₇	—	8	2	no	—
132	C₂×D₄₇	—	8	4	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	8	2	no	—
135	C₄×C₄₇	#493	1	2	yes	—
136	C₄×C₄₇	C₂²×D₈	1	2	yes	—
137	C₄×C₄₇	C₂×#135	1	4	yes	—
138	Dic₄₇	—	8	2	no	—
139	#3169	—	47	4	no	—
140	#3147	—	47	1	no	—
141	#3147	D₄₇	1	1	yes	characteristic, complemented
142	#1713	—	47	1	no	—
143	#1712	—	47	1	no	—
144	#1709	—	47	1	no	—
145	#1709	—	47	1	no	—
146	#1706	—	47	4	no	—
147	C₂×C₈²	—	47	1	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	16	2	no	—
150	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
151	C₂×C₄₇	#3147	1	2	yes	—
152	D₄₇	—	16	2	no	—
153	C₂×#135	—	2	2	no	—
154	C₂×#135	—	94	2	no	—
155	#532	—	47	2	no	—
156	#532	C₂×D₄₇	1	2	yes	—
157	#500	—	47	4	no	—
158	#495	—	2	2	no	—
159	#495	—	94	2	no	—
160	#494	—	47	1	no	—
161	#494	C₂×D₄₇	1	1	yes	characteristic
162	#493	—	47	4	no	—
163	#493	—	47	1	no	—
164	#493	C₂×D₄₇	1	1	yes	characteristic
165	#492	—	47	2	no	—
166	#492	C₂×D₄₇	1	2	yes	—
167	#486	—	47	4	no	—
168	#486	—	47	4	no	—
169	#466	—	94	8	no	—
170	C₄×D₈	—	94	8	no	—
171	C₂×C₄×C₈	—	47	2	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₈²	—	47	2	no	—
174	C₈²	—	47	1	no	—
175	C₈²	—	47	1	no	—
176	C₄₇	#19603	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	2	2	no	—
178	C₂×#41	—	94	2	no	—
179	#135	—	4	8	no	—
180	#135	—	188	8	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	94	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	94	4	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	94	8	no	—
187	C₂×D₈	—	94	2	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
190	C₄⋊Q₈	—	47	2	no	—
191	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
192	#127	—	47	2	no	—
193	#127	C₂²×D₄₇	1	2	yes	—
194	#124	—	2	2	no	—
195	#124	—	94	2	no	—
196	#121	—	2	4	no	—
197	#121	—	94	4	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	94	4	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	—	94	8	no	—
202	#102	—	94	8	no	—
203	#102	—	94	8	no	—
204	C₄×C₈	—	47	4	no	—
205	C₄×C₈	—	47	2	no	—
206	C₄×C₈	—	47	2	no	—
207	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
208	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	#41	—	4	8	no	—
211	#41	—	188	8	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	94	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	94	4	no	—
218	C₂×D₄	—	94	2	no	—
219	C₂×D₄	—	188	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	188	2	no	—
222	C₂²×C₄	D₄×D₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	188	4	no	—
226	SD₁₆	—	4	8	no	—
227	SD₁₆	—	188	8	no	—
228	D₈	—	4	4	no	—
229	D₈	—	188	16	no	—
230	D₈	—	188	4	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	47	4	no	—
233	C₂×C₈	—	94	8	no	—
234	C₂×C₈	#10625334	1	8	yes	complemented
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	188	4	no	—
241	C₄²	D₄×D₄₇	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄²	C₂³×D₄₇	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	188	2	no	—
246	C₂³	#13094020	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	Q₈	—	188	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	8	4	no	—
252	D₄	—	376	4	no	—
253	D₄	—	188	4	no	—
254	D₄	—	188	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	376	4	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	#13094020	1	2	yes	—
261	C₂×C₄	#13094020	1	1	yes	characteristic
262	C₂×C₄	C₂×#10625334	1	4	yes	—
263	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
265	C₈	—	47	8	no	—
266	C₈	—	94	4	no	—
267	C₈	C₂×#10625334	1	4	yes	—
268	C₈	C₂×#10625334	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	376	2	no	—
273	C₂²	—	376	4	no	—
274	C₂²	#15689499	1	2	yes	—
275	C₂²	C₂×#13094020	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	376	2	no	—
279	C₄	#15689499	1	2	yes	—
280	C₄	C₂²×#10625334	1	2	yes	—
281	C₄	#135×D₄₇	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	—	752	2	no	—
285	C₂	C₂×#15689499	1	1	yes	characteristic
286	C₂	#20253812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3147	C₂
socle	C₂²×C₄₇	C₂×#127
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#19603
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20253963	C₂	semidirect
2. #20253816	C₂	semidirect
3. #20243004	C₂	semidirect
4. #20243002	C₂	semidirect
5. #20242995	C₂	semidirect
6. #20242993	C₂	semidirect
7. #20242978	C₂	semidirect
8. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
9. C₄₇×#3147	C₂	semidirect
83. C₂×C₈×C₄₇	D₈	semidirect
124. C₄₇⋊C₈	#135	semidirect
141. #3147	D₄₇	semidirect
176. C₄₇	#19603	semidirect
234. C₂×C₈	#10625334	semidirect

Maximal subgroups

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

subgroup	index
#20253963	2
#20253816	2
#20243004	2
#20243002	2
#20242995	2
#20242993	2
#20242978	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#3147	2
#19603	47