Subgroups — #26213728

Order 12032.

2046 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689890	C₂	1	2	yes	maximal
2	#20243127	C₂	1	6	yes	maximal, complemented
3	C₂×#15688892	C₂	1	6	yes	maximal, complemented
4	C₂×C₄₇×#520	C₂	1	1	yes	maximal, characteristic
5	#15689890	—	2	8	no	—
6	#15689890	C₄	1	16	yes	complemented
7	C₂×#13094031	—	2	8	no	—
8	C₂²×#10625329	C₂²	1	12	yes	—
9	#15688892	C₂²	1	12	yes	—
10	#15688892	C₂²	1	4	yes	—
11	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#520	—	2	8	no	—
13	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₄₇	C₄	1	6	yes	—
15	C₂×#117×C₄₇	C₂²	1	3	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
17	#13094031	—	2	64	no	—
18	C₂×#10625329	—	2	12	no	—
19	C₂×#10625329	—	2	24	no	—
20	C₂×#10625329	C₂×C₄	1	48	yes	complemented
21	C₂×#10625329	C₂³	1	8	yes	—
22	C₂²×C₄₇⋊C₈	—	2	12	no	—
23	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
24	C₂×#41×C₄₇	D₄	1	8	yes	—
25	C₂×#41×C₄₇	D₄	1	4	yes	—
26	C₂×#41×C₄₇	Q₈	1	4	yes	—
27	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₄×C₄₇	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
31	C₄×Q₈×C₄₇	—	2	8	no	—
32	C₄×D₄×C₄₇	—	2	24	no	—
33	#117×C₄₇	—	2	12	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
36	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
40	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
41	#41×C₄₇	C₂²⋊C₄	1	32	yes	—
42	#41×C₄₇	C₄⋊C₄	1	32	yes	—
43	C₂×Q₈×C₄₇	C₄²	1	4	yes	—
44	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
45	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂×D₄×C₄₇	C₄²	1	12	yes	—
47	C₂×D₄×C₄₇	C₂²⋊C₄	1	12	yes	—
48	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
49	C₂²×C₄×C₄₇	—	2	12	no	—
50	C₂²×C₄×C₄₇	—	8	6	no	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	12	no	—
60	C₄⋊C₄×C₄₇	—	2	6	no	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂²⋊C₄×C₄₇	—	2	12	no	—
63	C₄²×C₄₇	—	2	12	no	—
64	#10625329	—	2	48	no	—
65	#10625329	—	2	8	no	—
66	#10625329	C₂²×C₄	1	16	yes	—
67	C₂×C₄₇⋊C₈	—	2	48	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂³×C₄₇	—	4	3	no	—
70	C₂³×C₄₇	—	4	3	no	—
71	C₂³×C₄₇	C₂×C₄²	1	6	yes	—
72	C₂³×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
73	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₄₇	#95	1	16	yes	—
76	D₄×C₄₇	#95	1	48	yes	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	6	no	—
79	C₂×C₄×C₄₇	—	4	6	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	8	12	no	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
84	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
91	C₄₇⋊C₈	—	2	16	no	—
92	#20358	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	1	no	—
94	C₂²×C₄₇	—	2	6	no	—
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	—	2	6	no	—
97	C₂²×C₄₇	—	4	6	no	—
98	C₂²×C₄₇	—	4	6	no	—
99	C₂²×C₄₇	C₂×#95	1	12	yes	—
100	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
101	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₄₇	—	4	6	no	—
104	C₄×C₄₇	—	4	2	no	—
105	C₄×C₄₇	C₂×#95	1	12	yes	—
106	C₄×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
107	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
108	C₂×#568	—	47	2	no	—
109	C₂×#520	D₄₇	1	1	yes	characteristic
110	#1742	—	47	6	no	—
111	C₂×#343	—	47	6	no	—
112	C₂×C₄₇	—	2	1	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	—	2	12	no	—
115	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
116	C₂×C₄₇	#2888	1	1	yes	characteristic
117	C₂×C₄₇	#2889	1	1	yes	characteristic
118	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
119	#568	—	47	16	no	—
120	#568	—	94	8	no	—
121	C₂×#131	—	94	8	no	—
122	C₂²×M₁₆	—	47	12	no	—
123	#520	—	2	8	no	—
124	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
125	C₂×C₄×D₄	Dic₄₇	1	6	yes	—
126	C₂×#117	C₂×D₄₇	1	3	yes	—
127	C₂²×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
128	#343	—	47	12	no	—
129	#343	—	47	4	no	—
130	C₄₇	#20358	1	1	yes	characteristic, complemented, Sylow
131	C₂×#41	Dic₉₄	1	4	yes	—
132	C₂×#41	C₄×D₄₇	1	8	yes	—
133	C₂×#41	D₁₈₈	1	4	yes	—
134	C₂×#41	C₂²⋊D₄₇	1	8	yes	—
135	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
136	C₂²×D₄	C₂×Dic₄₇	1	3	yes	—
137	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
138	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
139	#131	—	94	64	no	—
140	C₂×M₁₆	—	47	48	no	—
141	C₂×M₁₆	—	47	8	no	—
142	C₂×M₁₆	—	94	24	no	—
143	C₂×M₁₆	—	94	12	no	—
144	C₂²×C₈	—	94	12	no	—
145	C₄×Q₈	—	2	8	no	—
146	C₄×D₄	—	2	24	no	—
147	#117	—	2	12	no	—
148	C₂×C₄⋊C₄	—	2	12	no	—
149	C₂×C₄⋊C₄	C₂×Dic₄₇	1	6	yes	—
150	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
151	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
153	C₂×C₄²	C₂×Dic₄₇	1	6	yes	—
154	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
155	#41	Dic₄₇⋊C₄	1	32	yes	—
156	#41	(C₂×D₄₇)⋊C₄	1	32	yes	—
157	C₂×Q₈	C₄×Dic₄₇	1	4	yes	—
158	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
159	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
160	C₂×D₄	C₄×Dic₄₇	1	12	yes	—
161	C₂×D₄	C₄⋊Dic₄₇	1	12	yes	—
162	C₂×D₄	C₂²⋊Dic₄₇	1	12	yes	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	8	6	no	—
165	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
167	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
168	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
170	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
171	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	M₁₆	—	47	16	no	—
174	M₁₆	—	94	48	no	—
175	M₁₆	—	94	8	no	—
176	C₂×C₈	—	94	48	no	—
177	C₂×C₈	—	94	8	no	—
178	C₄⋊C₄	—	2	12	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	12	no	—
182	C₄²	—	2	12	no	—
183	C₂³	—	4	3	no	—
184	C₂³	—	4	3	no	—
185	C₂³	C₂×C₄×Dic₄₇	1	6	yes	—
186	C₂³	C₂×C₂²⋊Dic₄₇	1	6	yes	—
187	C₂³	C₂×C₄⋊Dic₄₇	1	6	yes	—
188	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
189	Q₈	C₄₇⋊#95	1	16	yes	—
190	D₄	C₄₇⋊#95	1	48	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	6	no	—
193	C₂×C₄	—	4	6	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	12	no	—
196	C₂×C₄	C₂×C₄×Dic₄₇	1	6	yes	—
197	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	24	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
199	C₂×C₄	C₂×Dic₄₇⋊C₄	1	24	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
201	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
202	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
203	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
204	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
205	C₈	—	94	16	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	6	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	6	no	—
210	C₂²	—	4	6	no	—
211	C₂²	—	4	6	no	—
212	C₂²	C₂×C₄₇⋊#95	1	12	yes	—
213	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
214	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
215	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₄	—	4	6	no	—
217	C₄	—	4	2	no	—
218	C₄	C₂×C₄₇⋊#95	1	12	yes	—
219	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
220	C₄	C₂²×Dic₄₇⋊C₄	1	2	yes	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	—	2	12	no	—
224	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
225	C₂	#20251867	1	1	yes	characteristic
226	C₂	#20251871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#520	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#520	D₄₇
hypocenter	C₄₇	#20358
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂×#520

Complements

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

N	H ≅ G/N	type
2. #20243127	C₂	semidirect
3. C₂×#15688892	C₂	semidirect
6. #15689890	C₄	semidirect
20. C₂×#10625329	C₂×C₄	semidirect
130. C₄₇	#20358	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
#20243127	2
C₂×#15688892	2
C₂×C₄₇×#520	2
#20358	47