Subgroups — #26121534

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	C₂×#15688888	C₂	1	4	yes	maximal, complemented
3	C₂⁴×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689087	—	2	4	no	—
6	#15689087	C₄	1	8	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
8	#15688888	—	2	8	no	—
9	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₇	—	2	4	no	—
11	C₂³×C₈×C₄₇	—	2	8	no	—
12	C₂³×C₈×C₄₇	C₄	1	4	yes	—
13	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×#10625329	—	2	4	no	—
15	C₂×#10625329	—	2	8	no	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	C₂×C₄	1	8	yes	—
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
20	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	—	2	4	no	—
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	D₄	1	2	yes	—
26	C₂³×C₄×C₄₇	D₄	1	4	yes	—
27	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	—	2	32	no	—
30	C₂²×C₈×C₄₇	—	2	8	no	—
31	C₂²×C₈×C₄₇	—	2	4	no	—
32	C₂²×C₈×C₄₇	—	2	4	no	—
33	C₂²×C₈×C₄₇	—	2	16	no	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₇	D₄	1	8	yes	—
36	C₂⁴×C₄₇	—	2	8	no	—
37	C₂⁴×C₄₇	—	2	4	no	—
38	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
39	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
40	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	16	no	—
45	C₂²×C₄×C₄₇	—	2	8	no	—
46	C₂²×C₄×C₄₇	—	2	8	no	—
47	C₂²×C₄×C₄₇	—	2	8	no	—
48	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂×C₈×C₄₇	—	2	32	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	—	2	16	no	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
64	#10625329	—	2	16	no	—
65	#10625329	—	4	4	no	—
66	#10625329	—	4	4	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	4	8	no	—
69	C₂³×C₄₇	—	2	32	no	—
70	C₂³×C₄₇	—	2	16	no	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	—	2	8	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	—	2	4	no	—
75	C₂³×C₄₇	#95	1	4	yes	—
76	C₂³×C₄₇	#108	1	8	yes	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	16	no	—
86	C₂×C₄×C₄₇	#95	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#117	1	2	yes	—
92	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
93	C₂×C₄×C₄₇	#120	1	4	yes	—
94	C₂×C₄×C₄₇	#121	1	4	yes	—
95	C₂×C₄×C₄₇	#123	1	4	yes	—
96	C₈×C₄₇	—	2	8	no	—
97	C₄₇⋊C₈	—	4	8	no	—
98	#10723	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	32	no	—
100	C₂²×C₄₇	—	2	16	no	—
101	C₂²×C₄₇	—	2	8	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	#341	1	8	yes	complemented
106	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
107	C₂²×C₄₇	#380	1	2	yes	—
108	C₂²×C₄₇	#385	1	4	yes	—
109	C₂²×C₄₇	C₂×#108	1	4	yes	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
112	C₄×C₄₇	#379	1	1	yes	characteristic
113	C₄×C₄₇	#381	1	1	yes	characteristic
114	C₄×C₄₇	#386	1	4	yes	—
115	C₂⁴×C₈	D₄₇	1	1	yes	characteristic
116	C₂×#407	—	47	2	no	—
117	C₂×#341	—	47	4	no	—
118	C₂×C₄₇	—	2	8	no	—
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
121	C₂×C₄₇	C₂×#341	1	2	yes	—
122	C₂×C₄₇	#1810	1	4	yes	—
123	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
124	C₂²×M₁₆	—	47	4	no	—
125	C₂³×C₈	—	2	4	no	—
126	C₂³×C₈	—	2	8	no	—
127	C₂³×C₈	Dic₄₇	1	4	yes	—
128	C₂³×C₈	C₂×D₄₇	1	2	yes	—
129	#407	—	47	8	no	—
130	#407	—	94	4	no	—
131	C₂×C₂²⋊C₈	—	94	4	no	—
132	#341	—	94	8	no	—
133	C₄₇	#10723	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	Dic₉₄	1	1	yes	characteristic
135	C₂³×C₄	—	2	4	no	—
136	C₂³×C₄	—	2	4	no	—
137	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
138	C₂³×C₄	C₄×D₄₇	1	4	yes	—
139	C₂³×C₄	D₁₈₈	1	2	yes	—
140	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
141	C₂³×C₄	C₂²⋊D₄₇	1	4	yes	—
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂×M₁₆	—	47	8	no	—
144	C₂×M₁₆	—	94	4	no	—
145	C₂×M₁₆	—	94	8	no	—
146	C₂×M₁₆	—	94	8	no	—
147	C₂²×C₈	—	2	32	no	—
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	—	2	16	no	—
152	C₂²×C₈	—	94	4	no	—
153	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
154	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
155	C₂²⋊C₈	—	94	16	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	Dic₄₇⋊C₄	1	4	yes	—
159	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
160	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
169	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
170	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
171	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
172	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
175	C₂²×C₄	#10625350	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
179	M₁₆	—	94	16	no	—
180	M₁₆	—	188	4	no	—
181	M₁₆	—	188	4	no	—
182	C₂×C₈	—	2	32	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	16	no	—
186	C₂×C₈	—	94	8	no	—
187	C₂×C₈	—	188	8	no	—
188	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
189	C₂³	—	2	32	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	8	no	—
192	C₂³	—	2	16	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	C₄₇⋊#95	1	4	yes	—
196	C₂³	#13093960	1	8	yes	—
197	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
198	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
199	C₂³	#13094003	1	4	yes	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	C₄₇⋊#95	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	#13093983	1	2	yes	—
212	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
213	C₂×C₄	#13093993	1	4	yes	—
214	C₂×C₄	#13093998	1	4	yes	—
215	C₂×C₄	#13094007	1	4	yes	—
216	C₈	—	2	8	no	—
217	C₈	—	188	8	no	—
218	C₂²	—	2	32	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	2	8	no	—
224	C₂²	#15688888	1	8	yes	complemented
225	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
226	C₂²	#15688983	1	2	yes	—
227	C₂²	#15688996	1	4	yes	—
228	C₂²	C₂×#13093960	1	4	yes	—
229	C₄	—	2	4	no	—
230	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
231	C₄	#15688979	1	1	yes	characteristic
232	C₄	#15688986	1	1	yes	characteristic
233	C₄	#15689002	1	4	yes	—
234	C₂	—	2	4	no	—
235	C₂	—	2	8	no	—
236	C₂	#20241611	1	1	yes	characteristic
237	C₂	C₂×#15688888	1	2	yes	—
238	C₂	#20243426	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂⁴×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂⁴×C₈	D₄₇
hypocenter	C₄₇	#10723
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15688888	C₂	semidirect
105. C₂²×C₄₇	#341	semidirect
133. C₄₇	#10723	semidirect
224. C₂²	#15688888	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
C₂×#15688888	2
C₂⁴×C₈×C₄₇	2
#10723	47