Subgroups — #26124554

Order 12032.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688892	C₂	1	2	yes	maximal
2	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
3	#20243107	C₂	1	2	yes	maximal
4	#20243084	C₂	1	1	yes	maximal, characteristic
5	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	#15688892	—	2	4	no	—
10	#15688888	C₄	1	4	yes	complemented
11	#15688885	—	2	4	no	—
12	#15688877	—	2	2	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×#107×C₄₇	—	2	2	no	—
16	C₂×#107×C₄₇	—	2	4	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂×#10625329	—	2	1	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	C₂×C₄	1	4	yes	—
26	C₂×#10625329	Q₈	1	2	yes	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂³×C₄×C₄₇	—	2	2	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
47	#107×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	8	4	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₂²⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	2	4	no	—
75	C₄×Dic₄₇	—	4	1	no	—
76	C₄×Dic₄₇	—	4	1	no	—
77	#10625329	—	2	4	no	—
78	#10625329	—	2	4	no	—
79	#10625329	—	2	4	no	—
80	#10625329	—	4	2	no	—
81	#10625329	—	4	2	no	—
82	#10625329	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	4	1	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	1	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₄₇	#121	1	1	yes	characteristic
93	C₂³×C₄₇	#123	1	2	yes	—
94	C₂³×C₄₇	#125	1	1	yes	characteristic
95	C₂³×C₄₇	#126	1	2	yes	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	8	4	no	—
98	C₂×C₄×C₄₇	—	8	4	no	—
99	C₂×C₄×C₄₇	—	8	4	no	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#121	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#123	1	2	yes	—
111	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#124	1	2	yes	—
114	C₂×C₄×C₄₇	#125	1	2	yes	—
115	C₂×C₄×C₄₇	#125	1	2	yes	—
116	C₂×C₄×C₄₇	#126	1	2	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	C₄₇⋊C₈	—	4	2	no	—
124	#11270	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	#387	1	2	yes	—
131	C₂²×C₄₇	#389	1	1	yes	characteristic
132	C₂²×C₄₇	#394	1	1	yes	characteristic
133	C₂²×C₄₇	#397	1	1	yes	characteristic
134	C₂²×C₄₇	#400	1	2	yes	—
135	C₂²×C₄₇	#472	1	2	yes	—
136	C₂²×C₄₇	#473	1	2	yes	—
137	C₄×C₄₇	—	8	4	no	—
138	C₄×C₄₇	#382	1	1	yes	characteristic
139	C₄×C₄₇	#385	1	1	yes	characteristic
140	C₄×C₄₇	#388	1	2	yes	—
141	C₄×C₄₇	#393	1	1	yes	characteristic
142	C₄×C₄₇	#398	1	2	yes	—
143	C₄×C₄₇	#400	1	1	yes	characteristic
144	Dic₄₇	—	8	2	no	—
145	C₂²×#107	D₄₇	1	1	yes	characteristic
146	C₂×#343	—	47	2	no	—
147	C₂×#341	—	47	1	no	—
148	#1736	—	47	2	no	—
149	#1729	—	47	1	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#1442	1	1	yes	characteristic
153	C₂×C₄₇	#1952	1	2	yes	—
154	C₂×C₄₇	#2045	1	1	yes	characteristic
155	C₂×C₄₇	#2047	1	1	yes	characteristic
156	C₂×C₄₇	#2078	1	2	yes	—
157	C₂²×M₁₆	—	47	1	no	—
158	C₂²×M₁₆	—	47	2	no	—
159	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×#117	—	47	1	no	—
161	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
162	C₂×#107	—	2	2	no	—
163	C₂×#107	—	2	4	no	—
164	#343	—	94	4	no	—
165	#341	—	47	4	no	—
166	#340	—	94	4	no	—
167	#336	—	94	2	no	—
168	C₄₇	#11270	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	47	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	1	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	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	—	94	1	no	—
185	C₂²×C₈	Dic₉₄	1	2	yes	—
186	C₂²×C₈	C₄×D₄₇	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
188	#117	—	94	2	no	—
189	#117	—	94	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	47	2	no	—
196	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	94	1	no	—
198	C₂×C₄²	—	94	1	no	—
199	#107	—	2	8	no	—
200	C₂⁴	#10625352	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	—	94	2	no	—
206	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	#10625350	1	2	yes	—
209	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	#10625352	1	2	yes	—
211	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	#10625354	1	2	yes	—
213	C₂²×C₄	#10625354	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
216	M₁₆	—	94	4	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	—	94	4	no	—
219	M₁₆	—	188	2	no	—
220	M₁₆	—	188	2	no	—
221	M₁₆	—	188	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	—	188	1	no	—
229	C₂×C₈	—	188	2	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	1	no	—
232	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	94	4	no	—
238	C₂²⋊C₄	—	188	2	no	—
239	C₄²	—	188	1	no	—
240	C₄²	—	188	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	#13093989	1	1	yes	characteristic
244	C₂³	#13093995	1	1	yes	characteristic
245	C₂³	#13094005	1	2	yes	—
246	C₂³	#13094015	1	1	yes	characteristic
247	C₂³	#13094019	1	2	yes	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	188	2	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	—	188	2	no	—
255	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#13093982	1	2	yes	—
257	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
258	C₂×C₄	#13093986	1	1	yes	characteristic
259	C₂×C₄	#13093993	1	1	yes	characteristic
260	C₂×C₄	#13093996	1	2	yes	—
261	C₂×C₄	#13094000	1	2	yes	—
262	C₂×C₄	#13094000	1	1	yes	characteristic
263	C₂×C₄	#13094003	1	2	yes	—
264	C₂×C₄	#13094007	1	1	yes	characteristic
265	C₂×C₄	#13094009	1	2	yes	—
266	C₂×C₄	#13094011	1	1	yes	characteristic
267	C₂×C₄	#13094013	1	1	yes	characteristic
268	C₂×C₄	#13094013	1	2	yes	—
269	C₂×C₄	#13094014	1	2	yes	—
270	C₂×C₄	#13094015	1	2	yes	—
271	C₂×C₄	#13094019	1	2	yes	—
272	C₂×C₄	#13094023	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	188	4	no	—
275	C₈	—	188	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	#15689004	1	2	yes	—
282	C₂²	#15689019	1	1	yes	characteristic
283	C₂²	#15689037	1	1	yes	characteristic
284	C₂²	#15689049	1	1	yes	characteristic
285	C₂²	#15689062	1	2	yes	—
286	C₂²	#15689390	1	2	yes	—
287	C₂²	#15689395	1	2	yes	—
288	C₄	—	8	4	no	—
289	C₄	—	376	2	no	—
290	C₄	#15688988	1	1	yes	characteristic
291	C₄	#15688996	1	1	yes	characteristic
292	C₄	#15689013	1	2	yes	—
293	C₄	#15689034	1	1	yes	characteristic
294	C₄	#15689056	1	2	yes	—
295	C₄	#15689061	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#20241647	1	1	yes	characteristic
299	C₂	#20244171	1	2	yes	—
300	C₂	#20244641	1	1	yes	characteristic
301	C₂	#20244653	1	1	yes	characteristic
302	C₂	#20244829	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093995
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	#11270
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
10. #15688888	C₄	semidirect
168. C₄₇	#11270	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688892	2
C₂×#15688888	2
#20243107	2
#20243084	2
C₂²×#107×C₄₇	2
#11270	47