Subgroups — #26215048

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	2	yes	maximal
2	#20243263	C₂	1	4	yes	maximal, complemented
3	C₄⋊C₄×C₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	#20243154	C₂	1	2	yes	maximal
5	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
6	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
7	#15689184	C₄	1	16	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688878	C₂²	1	2	yes	—
16	#15688878	C₂²	1	4	yes	—
17	#15688878	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#389	C₄	1	4	yes	—
22	C₄₇×#385	C₄	1	4	yes	—
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
27	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
28	#13093935	—	4	2	no	—
29	#13093935	C₂×C₄	1	8	yes	complemented
30	C₄×C₄₇⋊C₈	—	4	2	no	—
31	C₄×C₄₇⋊C₈	—	4	4	no	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	#117×C₄₇	—	4	2	no	—
36	#117×C₄₇	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	#95×C₄₇	C₂×C₄	1	8	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	16	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
62	C₂²⋊C₄×C₄₇	—	4	2	no	—
63	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
64	C₄²×C₄₇	—	2	4	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂³×C₄₇	—	2	1	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
80	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₄₇	#131	1	8	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
95	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
96	C₄₇⋊C₈	—	2	8	no	—
97	C₄₇⋊C₈	—	2	8	no	—
98	#20540	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	4	2	no	—
102	C₂²×C₄₇	—	4	2	no	—
103	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
104	C₂²×C₄₇	#518	1	1	yes	characteristic
105	C₂²×C₄₇	#519	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×#131	1	4	yes	—
107	C₄×C₄₇	—	2	8	no	—
108	C₄×C₄₇	—	2	8	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	4	4	no	—
111	C₄×C₄₇	#520	1	4	yes	—
112	C₂×#433	—	47	2	no	—
113	#2296	D₄₇	1	1	yes	characteristic
114	#1771	—	47	4	no	—
115	C₈×C₄⋊C₄	—	47	4	no	—
116	#1750	—	47	2	no	—
117	C₄×C₂²⋊C₈	—	47	2	no	—
118	C₂×C₄₇	—	4	2	no	—
119	C₂×C₄₇	#2274	1	1	yes	characteristic
120	C₂×C₄₇	C₄×#131	1	2	yes	—
121	C₂×C₄₇	#2977	1	2	yes	—
122	C₂×C₄₇	#2986	1	2	yes	—
123	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
124	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	#433	—	47	16	no	—
127	C₂×C₄⋊C₈	—	47	4	no	—
128	C₂×C₂²⋊C₈	—	47	4	no	—
129	C₂×#97	—	47	2	no	—
130	C₂×#97	—	47	2	no	—
131	C₂×C₄×C₈	—	47	2	no	—
132	C₂×C₄×C₈	—	47	4	no	—
133	C₂×C₄×C₈	—	47	2	no	—
134	#389	Dic₄₇	1	4	yes	—
135	#385	Dic₄₇	1	4	yes	—
136	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
137	#336	—	47	2	no	—
138	#336	—	47	2	no	—
139	#336	—	47	4	no	—
140	C₄₇	#20540	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂²×C₈	—	47	4	no	—
144	C₂²×C₈	—	47	4	no	—
145	#117	—	4	2	no	—
146	#117	C₄×D₄₇	1	8	yes	—
147	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
152	C₂×C₄²	—	4	2	no	—
153	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
156	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
157	C₄⋊C₈	—	47	16	no	—
158	C₂²⋊C₈	—	47	16	no	—
159	#97	—	47	8	no	—
160	#97	—	188	2	no	—
161	C₄×C₈	—	47	8	no	—
162	C₄×C₈	—	188	4	no	—
163	C₄×C₈	—	188	2	no	—
164	#95	C₂×Dic₄₇	1	8	yes	—
165	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
173	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
175	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
176	C₂×C₈	—	47	8	no	—
177	C₂×C₈	—	47	8	no	—
178	C₂×C₈	—	94	4	no	—
179	C₂×C₈	—	94	4	no	—
180	C₂×C₈	—	94	4	no	—
181	C₂×C₈	—	94	4	no	—
182	C₄⋊C₄	—	2	16	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂×C₄×D₄₇	1	4	yes	—
192	C₄²	C₂×C₄×D₄₇	1	4	yes	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
197	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
208	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
209	C₂×C₄	#13094031	1	8	yes	—
210	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
211	C₂×C₄	#13094069	1	2	yes	—
212	C₂×C₄	#13094072	1	2	yes	—
213	C₈	—	94	8	no	—
214	C₈	—	94	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
220	C₂²	#15689598	1	1	yes	characteristic
221	C₂²	#15689601	1	1	yes	characteristic
222	C₂²	C₂×#13094031	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	2	8	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	4	no	—
227	C₄	#15689604	1	4	yes	—
228	C₂	—	4	2	no	—
229	C₂	#20245745	1	1	yes	characteristic
230	C₂	C₄×#13094031	1	2	yes	—
231	C₂	#20252444	1	2	yes	—
232	C₂	#20252490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2296	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#20540
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
2. #20243263	C₂	semidirect
3. C₄⋊C₄×C₄₇⋊C₈	C₂	semidirect
7. #15689184	C₄	semidirect
26. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
29. #13093935	C₂×C₄	semidirect
32. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
140. C₄₇	#20540	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#20243263	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243154	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₄₇×#2296	2
#20540	47