Subgroups — #26213769

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20243140	C₂	1	4	yes	maximal, complemented
4	#20243137	C₂	1	4	yes	maximal, complemented
5	#20243085	C₂	1	4	yes	maximal
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₄×#10625329	C₄	1	8	yes	complemented
10	C₂×#13093935	C₄	1	8	yes	complemented
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688878	C₂²	1	16	yes	—
16	C₂×#117×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
19	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
20	C₄₇×#376	C₂²	1	2	yes	—
21	C₄³×C₄₇	C₂²	1	2	yes	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	C₂×C₄	1	8	yes	—
24	C₂²×C₄₇⋊C₈	—	2	8	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	#13093935	C₂×C₄	1	8	yes	—
29	#13093935	C₂×C₄	1	8	yes	—
30	#13093935	C₂×C₄	1	8	yes	—
31	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	#117×C₄₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
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	4	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#95×C₄₇	C₂×C₄	1	8	yes	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	#41	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	8	no	—
57	C₂²⋊C₄×C₄₇	—	2	8	no	—
58	C₄²×C₄₇	—	2	8	no	—
59	C₄²×C₄₇	—	4	8	no	—
60	C₄²×C₄₇	C₄²	1	8	yes	—
61	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	#10625329	—	2	16	no	—
66	C₂×C₄₇⋊C₈	—	2	16	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
70	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
73	C₂³×C₄₇	#117	1	4	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	2	8	no	—
78	C₂×C₄×C₄₇	—	2	16	no	—
79	C₂×C₄×C₄₇	—	4	8	no	—
80	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₄₇	#117	1	8	yes	—
84	C₂×C₄×C₄₇	#117	1	8	yes	—
85	C₂×C₄×C₄₇	#117	1	4	yes	—
86	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
89	C₄₇⋊C₈	—	2	8	no	—
90	C₄₇⋊C₈	—	2	8	no	—
91	#20367	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	4	no	—
93	C₂²×C₄₇	—	2	8	no	—
94	C₂²×C₄₇	#376	1	4	yes	—
95	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₄₇	C₂×#117	1	2	yes	—
97	C₂²×C₄₇	#568	1	4	yes	—
98	C₄×C₄₇	—	2	4	no	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	4	8	no	—
101	C₄×C₄₇	#376	1	4	yes	—
102	C₄×C₄₇	C₂×#117	1	2	yes	—
103	C₄×C₄₇	C₂×#117	1	2	yes	—
104	C₂×C₄×M₁₆	—	47	1	no	—
105	C₂²×#97	—	47	1	no	—
106	C₄×#117	D₄₇	1	1	yes	characteristic
107	#1746	—	47	4	no	—
108	#1745	—	47	4	no	—
109	#1729	—	47	4	no	—
110	C₂×C₄₇	—	2	4	no	—
111	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
112	C₂×C₄₇	#2873	1	2	yes	—
113	C₂×C₄₇	#2923	1	4	yes	—
114	C₂²×M₁₆	—	47	2	no	—
115	C₂³×C₈	—	47	2	no	—
116	C₂×#117	C₂×D₄₇	1	2	yes	—
117	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	C₄×M₁₆	—	47	8	no	—
119	C₂×#97	—	47	8	no	—
120	C₂×#97	—	47	2	no	—
121	C₂×#97	—	47	2	no	—
122	C₂×#97	—	47	2	no	—
123	C₂×C₄×C₈	—	47	2	no	—
124	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
125	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
126	#376	C₂×D₄₇	1	2	yes	—
127	C₄³	C₂×D₄₇	1	2	yes	—
128	#336	—	47	16	no	—
129	C₄₇	#20367	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
131	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
132	C₂×M₁₆	—	47	8	no	—
133	C₂×M₁₆	—	94	8	no	—
134	C₂²×C₈	—	47	8	no	—
135	C₂²×C₈	—	47	4	no	—
136	C₂²×C₈	—	47	4	no	—
137	C₂²×C₈	—	94	8	no	—
138	#117	—	2	8	no	—
139	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
140	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
141	C₂×C₄²	C₄×D₄₇	1	4	yes	—
142	C₂×C₄²	C₄×D₄₇	1	4	yes	—
143	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
144	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
145	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
148	#97	—	47	8	no	—
149	#97	—	47	8	no	—
150	#97	—	47	8	no	—
151	C₄×C₈	—	47	8	no	—
152	#95	C₂×Dic₄₇	1	8	yes	—
153	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
156	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
158	C₂²×C₄	#10625350	1	8	yes	—
159	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
160	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
164	M₁₆	—	94	16	no	—
165	C₂×C₈	—	47	8	no	—
166	C₂×C₈	—	47	8	no	—
167	C₂×C₈	—	94	16	no	—
168	C₂×C₈	—	94	8	no	—
169	C₂×C₈	—	94	8	no	—
170	C₄⋊C₄	—	2	8	no	—
171	C₂²⋊C₄	—	2	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	—	4	8	no	—
174	C₄²	C₄×Dic₄₇	1	8	yes	—
175	C₄²	C₂×C₄×D₄₇	1	2	yes	—
176	C₄²	C₂×C₄×D₄₇	1	2	yes	—
177	C₄²	C₂×C₄×D₄₇	1	2	yes	—
178	C₄²	C₂×C₄×D₄₇	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
181	C₂³	#13093983	1	4	yes	—
182	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
189	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
190	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
191	C₂×C₄	#13093981	1	8	yes	—
192	C₂×C₄	#13093981	1	8	yes	—
193	C₂×C₄	#13093983	1	4	yes	—
194	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	C₂×#10625350	1	4	yes	—
197	C₈	—	94	8	no	—
198	C₈	—	94	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	4	no	—
201	C₂²	#15688969	1	4	yes	—
202	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
203	C₂²	C₂×#13093983	1	2	yes	—
204	C₂²	#15689890	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	#15688969	1	4	yes	—
209	C₄	C₂×#13093981	1	2	yes	—
210	C₄	C₂×#13093981	1	2	yes	—
211	C₂	—	2	4	no	—
212	C₂	C₂×#15688969	1	1	yes	characteristic
213	C₂	#20251807	1	2	yes	—
214	C₂	#20252039	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20367
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243140	C₂	semidirect
4. #20243137	C₂	semidirect
9. C₄×#10625329	C₄	semidirect
10. C₂×#13093935	C₄	semidirect
129. C₄₇	#20367	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₂²×#13093935	2
#20243140	2
#20243137	2
#20243085	2
C₄×#117×C₄₇	2
#20367	47