Subgroups — #26114821

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244107	C₂	1	2	yes	maximal
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20241229	C₂	1	2	yes	maximal
5	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688878	—	2	2	no	—
11	#15688876	—	2	2	no	—
12	C₈×C₄₇⋊C₈	—	2	2	no	—
13	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
14	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#340	C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#107×C₄₇	—	2	2	no	—
36	#107×C₄₇	C₈	1	4	yes	complemented
37	C₄×C₈×C₄₇	D₄	1	2	yes	—
38	C₄×C₈×C₄₇	Q₈	1	2	yes	—
39	#95×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
51	C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
55	C₄²×C₄₇	—	4	2	no	—
56	C₄²×C₄₇	—	4	2	no	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₄₇	#41	1	1	yes	characteristic
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	2	1	no	—
64	C₂×C₄₇⋊C₈	—	2	1	no	—
65	C₂×C₄₇⋊C₈	—	2	2	no	—
66	C₂×C₄₇⋊C₈	—	2	1	no	—
67	C₂×C₄₇⋊C₈	—	2	1	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
72	C₂×C₄₇⋊C₈	Q₁₆	1	2	yes	—
73	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
96	C₈×C₄₇	—	2	2	no	—
97	C₈×C₄₇	C₄⋊C₈	1	4	yes	complemented
98	C₄₇⋊C₈	—	2	4	no	—
99	C₄₇⋊C₈	—	2	2	no	—
100	C₄₇⋊C₈	—	2	2	no	—
101	C₄₇⋊C₈	—	8	2	no	—
102	#9594	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	#389	1	1	yes	characteristic
104	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
105	C₂²×C₄₇	#436	1	1	yes	characteristic
106	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
107	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
108	C₂²×C₄₇	#443	1	1	yes	characteristic
109	C₂²×C₄₇	#443	1	1	yes	characteristic
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	4	4	no	—
113	C₄×C₄₇	—	8	2	no	—
114	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₄₇	#424	1	1	yes	characteristic
116	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
117	C₄×C₄₇	#446	1	1	yes	characteristic
118	C₄×C₄₇	#466	1	1	yes	characteristic
119	C₄×C₄₇	#467	1	1	yes	characteristic
120	C₄×C₄₇	#499	1	1	yes	characteristic
121	C₄×C₄₇	#500	1	1	yes	characteristic
122	#1939	—	47	2	no	—
123	C₄×#107	D₄₇	1	1	yes	characteristic
124	C₂×#335	—	47	1	no	—
125	C₂×C₈²	—	47	1	no	—
126	#1279	—	47	2	no	—
127	C₂×C₄₇	C₈×D₈	1	1	yes	characteristic
128	C₂×C₄₇	C₈×Q₁₆	1	1	yes	characteristic
129	C₂×C₄₇	#1589	1	1	yes	characteristic
130	C₂×C₄₇	#1591	1	1	yes	characteristic
131	C₂×C₄₇	#1939	1	1	yes	characteristic
132	C₂×C₄₇	#1942	1	1	yes	characteristic
133	C₂×C₄₇	#1945	1	1	yes	characteristic
134	C₂×#107	Dic₄₇	1	1	yes	characteristic
135	C₂×#107	Dic₄₇	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	47	2	no	—
137	C₂×C₄⋊C₈	—	94	2	no	—
138	C₂×C₄×C₈	—	47	1	no	—
139	C₂×C₄×C₈	—	47	1	no	—
140	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
142	#340	Dic₄₇	1	2	yes	—
143	#336	—	94	2	no	—
144	#335	—	94	2	no	—
145	C₈²	—	94	2	no	—
146	C₄₇	#9594	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	47	1	no	—
148	C₂²×C₈	—	47	1	no	—
149	C₂²×C₈	—	94	1	no	—
150	C₂²×C₈	—	94	2	no	—
151	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	#107	C₄₇⋊C₈	1	4	yes	complemented
159	C₄⋊C₈	—	94	2	no	—
160	C₄⋊C₈	—	94	2	no	—
161	C₄⋊C₈	—	188	4	no	—
162	C₄×C₈	—	47	2	no	—
163	C₄×C₈	—	94	1	no	—
164	C₄×C₈	—	94	1	no	—
165	C₄×C₈	—	94	1	no	—
166	C₄×C₈	Dic₉₄	1	2	yes	—
167	C₄×C₈	D₁₈₈	1	2	yes	—
168	#95	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₄	#10625352	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	47	2	no	—
177	C₂×C₈	—	47	2	no	—
178	C₂×C₈	—	94	2	no	—
179	C₂×C₈	—	94	2	no	—
180	C₂×C₈	—	94	1	no	—
181	C₂×C₈	—	94	1	no	—
182	C₂×C₈	—	94	2	no	—
183	C₂×C₈	—	94	1	no	—
184	C₂×C₈	—	94	1	no	—
185	C₂×C₈	—	188	2	no	—
186	C₂×C₈	—	188	2	no	—
187	C₂×C₈	—	188	2	no	—
188	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
189	C₂×C₈	#10625329	1	2	yes	—
190	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
191	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
199	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
200	C₄²	Q₈×D₄₇	1	1	yes	characteristic
201	C₄²	#10625354	1	1	yes	characteristic
202	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
213	C₂×C₄	#13093999	1	1	yes	characteristic
214	C₂×C₄	#13094004	1	1	yes	characteristic
215	C₂×C₄	#13094015	1	1	yes	characteristic
216	C₂×C₄	#13094023	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
218	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
219	C₂×C₄	#13094031	1	1	yes	characteristic
220	C₂×C₄	#13094031	1	1	yes	characteristic
221	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
223	C₂×C₄	#13094043	1	1	yes	characteristic
224	C₂×C₄	#13094047	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	94	4	no	—
227	C₈	—	94	2	no	—
228	C₈	—	94	2	no	—
229	C₈	—	376	2	no	—
230	C₈	C₄⋊(C₄₇⋊C₈)	1	4	yes	complemented
231	C₂²	#15689022	1	1	yes	characteristic
232	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
233	C₂²	#15689201	1	1	yes	characteristic
234	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
235	C₂²	Q₁₆×Dic₄₇	1	1	yes	characteristic
236	C₂²	#15689240	1	1	yes	characteristic
237	C₂²	#15689240	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
243	C₄	#15689150	1	1	yes	characteristic
244	C₄	Q₈×C₄₇⋊C₈	1	1	yes	characteristic
245	C₄	#15689254	1	1	yes	characteristic
246	C₄	#15689360	1	1	yes	characteristic
247	C₄	#15689365	1	1	yes	characteristic
248	C₄	#15689524	1	1	yes	characteristic
249	C₄	#15689527	1	1	yes	characteristic
250	C₂	D₈×C₄₇⋊C₈	1	1	yes	characteristic
251	C₂	Q₁₆×C₄₇⋊C₈	1	1	yes	characteristic
252	C₂	#20242273	1	1	yes	characteristic
253	C₂	#20242285	1	1	yes	characteristic
254	C₂	#20244107	1	1	yes	characteristic
255	C₂	#20244122	1	1	yes	characteristic
256	C₂	#20244137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	#9594
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
36. #107×C₄₇	C₈	semidirect
97. C₈×C₄₇	C₄⋊C₈	semidirect
146. C₄₇	#9594	semidirect
158. #107	C₄₇⋊C₈	semidirect
230. C₈	C₄⋊(C₄₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#20244107	2
C₂×#15688876	2
C₂×C₈×C₄₇⋊C₈	2
#20241229	2
C₄×#107×C₄₇	2
#9594	47