Subgroups — #26216601

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243533	C₂	1	2	yes	maximal, complemented
2	#20243508	C₂	1	2	yes	maximal, complemented
3	#20243459	C₂	1	2	yes	maximal
4	#20243372	C₂	1	2	yes	maximal
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
7	#20243134	C₂	1	2	yes	maximal, complemented
8	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	#15688878	C₂²	1	4	yes	—
13	#15688878	C₂²	1	4	yes	—
14	#15688878	C₂²	1	4	yes	—
15	#15688878	C₂²	1	4	yes	—
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
20	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
21	C₄₇×#376	C₂²	1	1	yes	characteristic
22	C₄₇×#376	C₂²	1	1	yes	characteristic
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
26	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
28	C₄×C₄₇⋊C₈	—	2	8	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	#117×C₄₇	—	2	4	no	—
32	#117×C₄₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	—	2	4	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#95×C₄₇	C₂×C₄	1	4	yes	—
48	#95×C₄₇	C₂×C₄	1	4	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₂²⋊C₄×C₄₇	—	2	4	no	—
66	C₂²⋊C₄×C₄₇	—	2	4	no	—
67	C₄²×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	8	no	—
69	C₄²×C₄₇	—	2	8	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	#41	1	4	yes	—
75	C₄²×C₄₇	#41	1	4	yes	—
76	C₂×C₄₇⋊C₈	—	2	8	no	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	#117	1	2	yes	—
83	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	#117	1	4	yes	—
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	#117	1	4	yes	—
100	C₂×C₄×C₄₇	#131	1	8	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
104	C₄₇⋊C₈	—	4	8	no	—
105	#20782	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	2	2	no	—
108	C₂²×C₄₇	—	4	2	no	—
109	C₂²×C₄₇	—	4	2	no	—
110	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#131	1	4	yes	—
114	C₄×C₄₇	—	2	4	no	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	4	2	no	—
117	C₄×C₄₇	—	4	2	no	—
118	C₄×C₄₇	—	4	2	no	—
119	C₄×C₄₇	—	4	2	no	—
120	C₄×C₄₇	#433	1	8	yes	—
121	C₄×C₄₇	#529	1	4	yes	—
122	C₄×#117	D₄₇	1	1	yes	characteristic
123	#1840	—	47	2	no	—
124	#1832	—	47	2	no	—
125	#1818	—	47	2	no	—
126	#1798	—	47	2	no	—
127	C₄×C₄⋊C₈	—	47	2	no	—
128	C₄×C₂²⋊C₈	—	47	2	no	—
129	#1744	—	47	2	no	—
130	C₂×C₄₇	—	4	2	no	—
131	C₂×C₄₇	#2283	1	1	yes	characteristic
132	C₂×C₄₇	C₂×#433	1	2	yes	—
133	C₂×C₄₇	#2922	1	2	yes	—
134	C₂×C₄₇	#2922	1	2	yes	—
135	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
136	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
137	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	47	4	no	—
139	C₂×C₂²⋊C₈	—	47	4	no	—
140	C₂×C₄×C₈	—	47	4	no	—
141	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
142	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
143	#376	C₂×D₄₇	1	1	yes	characteristic
144	#376	C₂×D₄₇	1	1	yes	characteristic
145	C₄³	C₂×D₄₇	1	1	yes	characteristic
146	C₄³	C₂×D₄₇	1	1	yes	characteristic
147	#336	—	47	4	no	—
148	#336	—	47	4	no	—
149	#336	—	47	4	no	—
150	#336	—	47	4	no	—
151	C₄₇	#20782	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
153	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
154	C₂²×C₈	—	47	8	no	—
155	#117	—	2	4	no	—
156	#117	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
163	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
164	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₄⋊C₈	—	94	8	no	—
172	C₂²⋊C₈	—	94	8	no	—
173	C₄×C₈	—	94	8	no	—
174	#95	C₂×Dic₄₇	1	4	yes	—
175	#95	C₂×Dic₄₇	1	4	yes	—
176	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	#10625350	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
189	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
190	C₂×C₈	—	94	8	no	—
191	C₂×C₈	—	94	8	no	—
192	C₂×C₈	—	94	8	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	8	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	#10625350	1	4	yes	—
205	C₄²	#10625350	1	4	yes	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#13093983	1	2	yes	—
210	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	#13093983	1	4	yes	—
225	C₂×C₄	#13093983	1	2	yes	—
226	C₂×C₄	#13093983	1	4	yes	—
227	C₂×C₄	#13094031	1	8	yes	—
228	C₂×C₄	C₂×#10625350	1	2	yes	—
229	C₂×C₄	C₂×#10625350	1	2	yes	—
230	C₂×C₄	C₂×#10625350	1	2	yes	—
231	C₈	—	188	8	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×#13093983	1	1	yes	characteristic
237	C₂²	C₂×#13093983	1	1	yes	characteristic
238	C₂²	C₂×#13093983	1	1	yes	characteristic
239	C₂²	C₂×#13094031	1	4	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#15689184	1	8	yes	—
247	C₄	#15689657	1	4	yes	—
248	C₂	—	4	2	no	—
249	C₂	#20245786	1	1	yes	characteristic
250	C₂	C₂×#15689184	1	2	yes	—
251	C₂	#20252032	1	2	yes	—
252	C₂	#20252032	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20782
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #20243533	C₂	semidirect
2. #20243508	C₂	semidirect
5. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
7. #20243134	C₂	semidirect
151. C₄₇	#20782	semidirect

Maximal subgroups

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

subgroup	index
#20243533	2
#20243508	2
#20243459	2
#20243372	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20243134	2
C₄×#117×C₄₇	2
#20782	47