Subgroups — #26216432

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20243507	C₂	1	8	yes	maximal, complemented
4	#20243081	C₂	1	4	yes	maximal
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	#15689147	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688969	C₄	1	4	yes	—
11	#15688969	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	C₄	1	4	yes	—
13	#15688877	C₂²	1	16	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#423	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
18	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	8	no	—
21	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
22	C₂×C₄×Dic₄₇	—	2	4	no	—
23	C₂×C₄×Dic₄₇	—	2	4	no	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
26	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
27	C₄₇⋊#95	C₂×C₄	1	16	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₄₇	—	2	4	no	—
31	C₂×M₁₆×C₄₇	—	2	8	no	—
32	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	4	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
37	C₄⋊C₈×C₄₇	—	2	8	no	—
38	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	4	4	no	—
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	#41	1	4	yes	—
46	C₂²×C₄×C₄₇	#41	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₇	—	2	8	no	—
49	C₂²×Dic₄₇	—	4	8	no	—
50	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
51	M₁₆×C₄₇	—	4	8	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	—	2	8	no	—
54	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₄²×C₄₇	C₂×D₄	1	4	yes	—
56	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
57	C₄×Dic₄₇	—	2	8	no	—
58	C₄×Dic₄₇	—	2	4	no	—
59	C₄×Dic₄₇	—	2	4	no	—
60	#10625329	—	4	8	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂³×C₄₇	—	2	4	no	—
64	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂³×C₄₇	#117	1	4	yes	—
66	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	4	8	no	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₄₇	#117	1	4	yes	—
72	C₂×C₄×C₄₇	#124	1	4	yes	—
73	C₂×C₄×C₄₇	#124	1	4	yes	—
74	C₂×C₄×C₄₇	#125	1	8	yes	—
75	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
79	C₂×Dic₄₇	—	2	8	no	—
80	C₂×Dic₄₇	—	2	16	no	—
81	C₂×Dic₄₇	—	4	16	no	—
82	C₈×C₄₇	—	4	4	no	—
83	C₄₇⋊C₈	—	4	4	no	—
84	#20753	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	4	no	—
86	C₂²×C₄₇	—	2	8	no	—
87	C₂²×C₄₇	#382	1	4	yes	—
88	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×#117	1	2	yes	—
90	C₂²×C₄₇	#568	1	4	yes	—
91	C₄×C₄₇	—	2	8	no	—
92	C₄×C₄₇	#382	1	4	yes	—
93	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
94	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
95	C₄×C₄₇	C₂×#125	1	2	yes	—
96	Dic₄₇	—	4	8	no	—
97	C₂×#423	—	47	1	no	—
98	C₂×#423	D₄₇	1	1	yes	characteristic
99	C₂×#376	—	47	1	no	—
100	#1832	—	47	8	no	—
101	#1728	—	47	4	no	—
102	C₂×C₄₇	—	2	4	no	—
103	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
104	C₂×C₄₇	#2905	1	2	yes	—
105	C₂×C₄₇	#2921	1	4	yes	—
106	C₂²×M₁₆	—	47	2	no	—
107	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
108	C₂²×C₄²	—	47	2	no	—
109	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
110	#423	—	2	4	no	—
111	#423	—	94	4	no	—
112	C₂×C₄⋊C₈	—	47	4	no	—
113	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
114	#376	—	47	4	no	—
115	#376	—	47	4	no	—
116	C₂×#95	—	47	4	no	—
117	#336	—	47	16	no	—
118	C₄₇	#20753	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	—	47	4	no	—
120	C₂³×C₄	C₄×D₄₇	1	2	yes	—
121	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
122	C₂×M₁₆	—	2	8	no	—
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	94	8	no	—
125	C₂×M₁₆	—	94	4	no	—
126	C₂²×C₈	—	47	4	no	—
127	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
128	C₂×C₄²	—	47	4	no	—
129	C₂×C₄²	—	47	4	no	—
130	C₂×C₄²	—	94	4	no	—
131	C₂×C₄²	—	94	4	no	—
132	C₂×C₄²	Dic₉₄	1	4	yes	—
133	C₂×C₄²	C₄×D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
136	C₄⋊C₈	—	2	8	no	—
137	C₄⋊C₈	—	94	8	no	—
138	#95	—	47	16	no	—
139	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	—	47	8	no	—
142	C₂²×C₄	—	94	8	no	—
143	C₂²×C₄	—	188	8	no	—
144	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
145	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
147	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
148	C₂²×C₄	#10625352	1	4	yes	—
149	C₂²×C₄	#10625354	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
151	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
152	M₁₆	—	4	8	no	—
153	M₁₆	—	188	8	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	94	4	no	—
157	C₂×C₈	—	94	8	no	—
158	C₄²	—	94	8	no	—
159	C₄²	—	94	4	no	—
160	C₄²	—	94	4	no	—
161	C₄²	Dic₄₇⋊C₄	1	8	yes	—
162	C₄²	C₂×Dic₉₄	1	4	yes	—
163	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
166	C₂³	#13093982	1	4	yes	—
167	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	—	94	8	no	—
171	C₂×C₄	—	94	16	no	—
172	C₂×C₄	—	188	16	no	—
173	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
174	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
175	C₂×C₄	#13093982	1	4	yes	—
176	C₂×C₄	#13094011	1	4	yes	—
177	C₂×C₄	#13094013	1	4	yes	—
178	C₂×C₄	#13094015	1	8	yes	—
179	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
181	C₂×C₄	C₂×#10625352	1	2	yes	—
182	C₂×C₄	C₂×#10625354	1	2	yes	—
183	C₈	—	4	4	no	—
184	C₈	—	188	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	8	no	—
187	C₂²	#15688988	1	4	yes	—
188	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
189	C₂²	C₂×#13093982	1	2	yes	—
190	C₂²	#15689889	1	4	yes	—
191	C₄	—	2	8	no	—
192	C₄	—	188	8	no	—
193	C₄	#15688988	1	4	yes	—
194	C₄	C₂×#13094011	1	1	yes	characteristic
195	C₄	C₂×#13094013	1	1	yes	characteristic
196	C₄	C₂×#13094015	1	2	yes	—
197	C₂	—	2	4	no	—
198	C₂	C₂×#15688988	1	1	yes	characteristic
199	C₂	#20251951	1	2	yes	—
200	C₂	#20252025	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#423	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20753
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243507	C₂	semidirect
118. C₄₇	#20753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×#15688969	2
#20243507	2
#20243081	2
C₂×C₄₇×#423	2
#20753	47