Subgroups — #26124920

Order 12032.

722 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×#15689163	C₂	1	2	yes	maximal
2	C₂×#15688892	C₂	1	2	yes	maximal
3	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
4	#20243084	C₂	1	1	yes	maximal, characteristic
5	C₂²×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	1	yes	characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	1	yes	characteristic
9	#15689163	—	2	4	no	—
10	#15689163	—	2	4	no	—
11	C₂×#13093957	—	2	2	no	—
12	C₂×#13093955	—	2	2	no	—
13	#15688892	—	2	4	no	—
14	#15688888	C₄	1	4	yes	complemented
15	#15688877	—	2	2	no	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂×#106×C₄₇	—	2	2	no	—
19	C₂×#106×C₄₇	—	2	4	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	1	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	C₂×C₄	1	4	yes	—
27	C₂×#10625329	D₄	1	4	yes	—
28	C₂×#10625329	Q₈	1	2	yes	—
29	C₂×#10625329	Q₈	1	4	yes	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	#13093983	—	2	2	no	—
33	#13093983	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	#13093957	—	2	8	no	—
38	#13093955	—	2	8	no	—
39	C₂³×C₄×C₄₇	—	2	2	no	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
52	#106×C₄₇	—	2	8	no	—
53	C₂⁴×C₄₇	#41	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	8	4	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₂²⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	—	4	1	no	—
82	C₄×Dic₄₇	—	4	1	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	2	4	no	—
86	#10625329	—	2	4	no	—
87	#10625329	—	4	2	no	—
88	#10625329	C₄⋊C₄	1	8	yes	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	4	1	no	—
94	C₂×C₄₇⋊C₈	—	4	1	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
97	C₂³×C₄₇	#121	1	1	yes	characteristic
98	C₂³×C₄₇	#121	1	2	yes	—
99	C₂³×C₄₇	#125	1	2	yes	—
100	C₂³×C₄₇	#125	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	8	4	no	—
103	C₂×C₄×C₄₇	—	8	4	no	—
104	C₂×C₄×C₄₇	—	8	4	no	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#120	1	2	yes	—
110	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×Dic₄₇	—	4	2	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₈×C₄₇	—	2	4	no	—
126	C₄₇⋊C₈	—	2	8	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	#11330	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	#389	1	2	yes	—
135	C₂²×C₄₇	#389	1	1	yes	characteristic
136	C₂²×C₄₇	#392	1	1	yes	characteristic
137	C₂²×C₄₇	#398	1	1	yes	characteristic
138	C₂²×C₄₇	#398	1	2	yes	—
139	C₂²×C₄₇	#471	1	4	yes	—
140	C₄×C₄₇	—	8	4	no	—
141	C₄×C₄₇	#384	1	1	yes	characteristic
142	C₄×C₄₇	#385	1	2	yes	—
143	C₄×C₄₇	#385	1	1	yes	characteristic
144	C₄×C₄₇	#395	1	1	yes	characteristic
145	C₄×C₄₇	#395	1	2	yes	—
146	C₄×C₄₇	#399	1	1	yes	characteristic
147	Dic₄₇	—	8	2	no	—
148	C₂×#428	—	47	2	no	—
149	C₂²×#106	D₄₇	1	1	yes	characteristic
150	C₂×#343	—	47	2	no	—
151	C₂×#341	—	47	1	no	—
152	#1729	—	47	1	no	—
153	C₂×C₄₇	—	2	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1444	1	1	yes	characteristic
156	C₂×C₄₇	#1951	1	2	yes	—
157	C₂×C₄₇	#2019	1	2	yes	—
158	C₂×C₄₇	#2090	1	2	yes	—
159	C₂²×M₁₆	—	47	2	no	—
160	C₂²×M₁₆	—	47	1	no	—
161	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	1	no	—
163	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
164	#428	—	94	4	no	—
165	#428	—	94	4	no	—
166	C₂×#107	—	94	2	no	—
167	C₂×#106	—	2	2	no	—
168	C₂×#106	—	2	4	no	—
169	C₂×#106	—	94	2	no	—
170	#343	—	94	4	no	—
171	#341	—	47	4	no	—
172	#336	—	94	2	no	—
173	C₄₇	#11330	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	47	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	1	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	94	2	no	—
189	C₂²×C₈	—	94	1	no	—
190	C₂²×C₈	Dic₉₄	1	2	yes	—
191	C₂²×C₈	C₄×D₄₇	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
193	#117	—	94	2	no	—
194	#117	—	94	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	47	2	no	—
201	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
202	C₂×C₂²⋊C₄	—	94	1	no	—
203	C₂×C₄²	—	94	1	no	—
204	#107	—	94	8	no	—
205	#106	—	2	8	no	—
206	#106	—	94	8	no	—
207	C₂⁴	#10625352	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	8	4	no	—
212	C₂²×C₄	—	94	2	no	—
213	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	#10625350	1	2	yes	—
216	C₂²×C₄	D₄×D₄₇	1	2	yes	—
217	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
219	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	#10625354	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
223	M₁₆	—	47	8	no	—
224	M₁₆	—	94	4	no	—
225	M₁₆	—	94	4	no	—
226	M₁₆	—	94	4	no	—
227	M₁₆	—	94	4	no	—
228	M₁₆	—	188	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	188	1	no	—
238	C₂×C₈	—	188	1	no	—
239	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	94	2	no	—
246	C₂²⋊C₄	—	188	2	no	—
247	C₄²	—	188	1	no	—
248	C₄²	—	188	1	no	—
249	C₂³	—	2	4	no	—
250	C₂³	#13093989	1	1	yes	characteristic
251	C₂³	#13093995	1	1	yes	characteristic
252	C₂³	#13093996	1	2	yes	—
253	C₂³	#13094014	1	2	yes	—
254	C₂³	#13094015	1	1	yes	characteristic
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	4	no	—
261	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
264	C₂×C₄	#13093986	1	1	yes	characteristic
265	C₂×C₄	#13093992	1	2	yes	—
266	C₂×C₄	#13093993	1	1	yes	characteristic
267	C₂×C₄	#13094000	1	2	yes	—
268	C₂×C₄	#13094000	1	1	yes	characteristic
269	C₂×C₄	#13094002	1	2	yes	—
270	C₂×C₄	#13094002	1	2	yes	—
271	C₂×C₄	#13094003	1	2	yes	—
272	C₂×C₄	#13094007	1	1	yes	characteristic
273	C₂×C₄	#13094009	1	2	yes	—
274	C₂×C₄	#13094020	1	1	yes	characteristic
275	C₂×C₄	#13094022	1	1	yes	characteristic
276	C₂×C₄	#13094022	1	2	yes	—
277	C₂×C₄	#13094023	1	2	yes	—
278	C₂×C₄	#13094023	1	1	yes	characteristic
279	C₈	—	2	4	no	—
280	C₈	—	94	8	no	—
281	C₈	—	188	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	#15689018	1	2	yes	—
288	C₂²	#15689019	1	1	yes	characteristic
289	C₂²	#15689031	1	1	yes	characteristic
290	C₂²	#15689054	1	1	yes	characteristic
291	C₂²	#15689056	1	2	yes	—
292	C₂²	#15689383	1	4	yes	—
293	C₄	—	8	4	no	—
294	C₄	—	376	2	no	—
295	C₄	#15688992	1	1	yes	characteristic
296	C₄	#15688994	1	2	yes	—
297	C₄	#15688996	1	1	yes	characteristic
298	C₄	#15689042	1	1	yes	characteristic
299	C₄	#15689043	1	2	yes	—
300	C₄	#15689058	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	#20241651	1	1	yes	characteristic
304	C₂	#20244166	1	2	yes	—
305	C₂	#20244500	1	2	yes	—
306	C₂	#20244899	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093995
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#106×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#106	D₄₇
hypocenter	C₄₇	#11330
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
14. #15688888	C₄	semidirect
173. C₄₇	#11330	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689163	2
C₂×#15688892	2
C₂×#15688888	2
#20243084	2
C₂²×#106×C₄₇	2
#11330	47