Subgroups — #26122508

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251807	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688893	C₂	1	2	yes	maximal
3	#20243127	C₂	1	1	yes	maximal, characteristic
4	#20243082	C₂	1	2	yes	maximal
5	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689079	—	2	4	no	—
9	C₄×#10625329	—	2	2	no	—
10	#15688893	C₄	1	8	yes	complemented
11	#15688892	—	2	2	no	—
12	#15688878	—	2	4	no	—
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#389	—	2	2	no	—
17	C₄₇×#385	C₄	1	4	yes	—
18	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	C₂×C₄	1	4	yes	—
29	C₂×#10625329	C₂×C₄	1	4	yes	—
30	C₂×#10625329	Q₈	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	#13093935	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	—	2	1	no	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	#95×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₄⋊C₄×C₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	4	8	no	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₂²⋊C₄×C₄₇	—	2	4	no	—
72	C₂²⋊C₄×C₄₇	C₄²	1	4	yes	—
73	C₄²×C₄₇	—	2	4	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	1	no	—
79	#10625329	—	2	4	no	—
80	#10625329	—	2	4	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	#117	1	1	yes	characteristic
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	#117	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#125	1	2	yes	—
122	C₂×C₄×C₄₇	#126	1	2	yes	—
123	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
124	C₄₇⋊C₈	—	2	8	no	—
125	C₄₇⋊C₈	—	4	4	no	—
126	#10919	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	4	1	no	—
130	C₂²×C₄₇	—	4	1	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄₇	#386	1	1	yes	characteristic
135	C₂²×C₄₇	#387	1	2	yes	—
136	C₂²×C₄₇	#391	1	1	yes	characteristic
137	C₄×C₄₇	—	4	4	no	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	#385	1	2	yes	—
141	C₄×C₄₇	#387	1	2	yes	—
142	C₄×C₄₇	#388	1	2	yes	—
143	C₄×C₄₇	#389	1	2	yes	—
144	#2873	—	47	1	no	—
145	#2296	D₄₇	1	1	yes	characteristic
146	C₂×#344	—	47	2	no	—
147	#1742	—	47	1	no	—
148	#1728	—	47	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	#1436	1	1	yes	characteristic
154	C₂×C₄₇	#1891	1	2	yes	—
155	C₂²×M₁₆	—	47	2	no	—
156	C₂²×M₁₆	—	47	2	no	—
157	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
158	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
159	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
160	#405	—	94	4	no	—
161	C₄×M₁₆	—	94	2	no	—
162	#389	—	2	2	no	—
163	#385	Dic₄₇	1	4	yes	—
164	C₄×C₄⋊C₄	—	2	2	no	—
165	#344	—	47	8	no	—
166	#343	—	94	2	no	—
167	#336	—	94	4	no	—
168	C₄₇	#10919	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	2	no	—
178	C₂×M₁₆	—	94	2	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂²×C₈	—	47	4	no	—
184	C₂²×C₈	—	94	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₄×D₄₇	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
200	#97	—	94	4	no	—
201	C₄×C₈	—	94	4	no	—
202	#95	—	2	4	no	—
203	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
209	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
210	C₂²×C₄	#10625350	1	2	yes	—
211	C₂²×C₄	D₄×D₄₇	1	2	yes	—
212	C₂²×C₄	#10625352	1	2	yes	—
213	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
214	C₂²×C₄	#10625354	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	M₁₆	—	94	4	no	—
218	M₁₆	—	94	4	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	94	4	no	—
222	M₁₆	—	94	4	no	—
223	M₁₆	—	188	2	no	—
224	M₁₆	—	188	2	no	—
225	C₂×C₈	—	47	8	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	188	2	no	—
230	C₂×C₈	—	188	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	8	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	C₄×Dic₄₇	1	4	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
245	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
246	C₂³	#13093983	1	1	yes	characteristic
247	C₂³	D₄×Dic₄₇	1	2	yes	—
248	C₂³	Q₈×Dic₄₇	1	2	yes	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
259	C₂×C₄	#13093982	1	2	yes	—
260	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
261	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
262	C₂×C₄	#13093986	1	2	yes	—
263	C₂×C₄	#13093989	1	2	yes	—
264	C₂×C₄	#13093992	1	1	yes	characteristic
265	C₂×C₄	#13093996	1	2	yes	—
266	C₂×C₄	#13094000	1	2	yes	—
267	C₂×C₄	#13094002	1	2	yes	—
268	C₂×C₄	#13094005	1	1	yes	characteristic
269	C₂×C₄	#13094009	1	2	yes	—
270	C₂×C₄	#13094013	1	1	yes	characteristic
271	C₂×C₄	#13094014	1	2	yes	—
272	C₂×C₄	#13094019	1	2	yes	—
273	C₂×C₄	#13094022	1	1	yes	characteristic
274	C₈	—	94	8	no	—
275	C₈	—	188	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
282	C₂²	C₄⋊C₄×Dic₄₇	1	2	yes	—
283	C₂²	#15689003	1	1	yes	characteristic
284	C₂²	#15689009	1	2	yes	—
285	C₂²	#15689030	1	1	yes	characteristic
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	#15688994	1	2	yes	—
290	C₄	#15689004	1	2	yes	—
291	C₄	#15689013	1	2	yes	—
292	C₄	#15689018	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	#20241620	1	1	yes	characteristic
298	C₂	#20243807	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2296	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#10919
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
10. #15688893	C₄	semidirect
92. C₂×C₄₇⋊C₈	C₄⋊C₄	semidirect
168. C₄₇	#10919	semidirect

Maximal subgroups

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

subgroup	index
#20251807	2
C₂×#15688893	2
#20243127	2
#20243082	2
C₄₇×#2296	2
#10919	47