Subgroups — #26122199

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	2	yes	maximal
2	#20243127	C₂	1	2	yes	maximal
3	C₂×C₄₇⋊#338	C₂	1	2	yes	maximal
4	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	C₂×#13093960	—	2	4	no	—
7	C₂×#13093960	—	2	4	no	—
8	C₂×#13093960	C₄	1	8	yes	complemented
9	#15688892	—	2	4	no	—
10	C₄₇⋊#338	—	2	4	no	—
11	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#405	—	2	4	no	—
15	C₂×#97×C₄₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
17	C₂×#10625329	—	2	4	no	—
18	C₂×#10625329	—	2	4	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	C₂×C₄	1	8	yes	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	#13093960	—	2	16	no	—
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂×M₁₆×C₄₇	—	2	2	no	—
28	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
29	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	D₄	1	2	yes	—
35	C₂²×C₈×C₄₇	D₄	1	4	yes	—
36	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	#97×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	#41	1	2	yes	—
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	M₁₆×C₄₇	—	2	4	no	—
57	M₁₆×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	C₄²	1	2	yes	—
61	C₂×C₈×C₄₇	C₄²	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₂²⋊C₄×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	#10625329	—	2	8	no	—
74	#10625329	—	2	8	no	—
75	#10625329	—	4	4	no	—
76	#10625329	—	4	4	no	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	4	4	no	—
79	C₂×C₄₇⋊C₈	—	4	4	no	—
80	C₂³×C₄₇	—	4	2	no	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
83	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
84	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₄₇	#121	1	2	yes	—
100	C₂×C₄×C₄₇	#121	1	2	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	#124	1	2	yes	—
103	C₂×C₄×C₄₇	#125	1	2	yes	—
104	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₈×C₄₇	—	2	4	no	—
107	C₈×C₄₇	#95	1	4	yes	—
108	C₈×C₄₇	#95	1	4	yes	—
109	C₄₇⋊C₈	—	4	8	no	—
110	#10867	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	—	4	2	no	—
114	C₂²×C₄₇	—	4	1	no	—
115	C₂²×C₄₇	—	4	1	no	—
116	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₇	#380	1	1	yes	characteristic
118	C₂²×C₄₇	#384	1	1	yes	characteristic
119	C₂²×C₄₇	#389	1	2	yes	—
120	C₂²×C₄₇	#391	1	2	yes	—
121	C₄×C₄₇	—	4	4	no	—
122	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
123	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₄₇	#380	1	1	yes	characteristic
125	C₄×C₄₇	#382	1	1	yes	characteristic
126	C₄×C₄₇	#389	1	2	yes	—
127	C₄×C₄₇	#390	1	2	yes	—
128	C₂²×#108	—	47	2	no	—
129	C₂×#405	D₄₇	1	1	yes	characteristic
130	#1742	—	47	2	no	—
131	C₂×#338	—	47	2	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	—	2	2	no	—
136	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
137	C₂×C₄₇	#1813	1	2	yes	—
138	C₂²×M₁₆	—	47	4	no	—
139	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
141	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
142	C₂×#108	—	47	8	no	—
143	C₂×#108	—	94	4	no	—
144	C₂×#108	—	94	4	no	—
145	#405	—	2	4	no	—
146	C₂×#97	Dic₄₇	1	2	yes	—
147	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
148	#343	—	94	4	no	—
149	#338	—	94	4	no	—
150	C₄₇	#10867	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	47	8	no	—
155	C₂×M₁₆	—	94	4	no	—
156	C₂×M₁₆	—	94	4	no	—
157	C₂×M₁₆	—	94	4	no	—
158	C₂×M₁₆	—	94	4	no	—
159	C₂×M₁₆	—	94	4	no	—
160	C₂×M₁₆	Dic₉₄	1	2	yes	—
161	C₂×M₁₆	D₁₈₈	1	2	yes	—
162	C₂²×C₈	—	94	4	no	—
163	C₂²×C₈	Dic₉₄	1	2	yes	—
164	C₂²×C₈	C₄×D₄₇	1	4	yes	—
165	C₂²×C₈	D₁₈₈	1	2	yes	—
166	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
167	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
170	#117	—	2	2	no	—
171	#117	—	2	2	no	—
172	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
173	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
174	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
175	#108	—	94	16	no	—
176	#97	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
180	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
183	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
186	C₂²×C₄	#10625350	1	2	yes	—
187	C₂²×C₄	#10625350	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	94	8	no	—
193	M₁₆	—	94	8	no	—
194	M₁₆	—	188	4	no	—
195	M₁₆	—	188	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	94	8	no	—
199	C₂×C₈	—	188	4	no	—
200	C₂×C₈	—	188	4	no	—
201	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
202	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
203	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
204	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
205	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
206	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
207	C₂×C₈	(C₂×D₄₇)⋊C₄	1	8	yes	—
208	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
209	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₂³	C₄×Dic₉₄	1	2	yes	—
217	C₂³	#13094003	1	2	yes	—
218	C₂³	#13094024	1	1	yes	characteristic
219	C₂³	#13094024	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
228	C₂×C₄	#13093983	1	1	yes	characteristic
229	C₂×C₄	#13093983	1	1	yes	characteristic
230	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
231	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
232	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
233	C₂×C₄	#13093998	1	2	yes	—
234	C₂×C₄	#13093998	1	2	yes	—
235	C₂×C₄	#13094003	1	2	yes	—
236	C₂×C₄	#13094012	1	2	yes	—
237	C₂×C₄	#13094016	1	2	yes	—
238	C₂×C₄	#13094021	1	1	yes	characteristic
239	C₂×C₄	#13094024	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	188	8	no	—
242	C₈	C₄₇⋊#95	1	4	yes	—
243	C₈	C₄₇⋊#95	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
250	C₂²	#15688983	1	1	yes	characteristic
251	C₂²	#15688993	1	1	yes	characteristic
252	C₂²	#15689021	1	2	yes	—
253	C₂²	#15689029	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
256	C₄	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
257	C₄	#15688983	1	1	yes	characteristic
258	C₄	#15688989	1	1	yes	characteristic
259	C₄	#15689021	1	2	yes	—
260	C₄	#15689025	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	#20241618	1	1	yes	characteristic
266	C₂	#20243438	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#10867
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. C₂×#13093960	C₄	semidirect
150. C₄₇	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
#20243127	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×#405	2
#10867	47