Subgroups — #26216701

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	4	yes	maximal
2	#20245795	C₂	1	2	yes	maximal
3	#20243556	C₂	1	8	yes	maximal, complemented
4	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	#15689161	—	2	16	no	—
7	C₂×#13093958	C₂²	1	8	yes	—
8	C₂×#13093956	C₂²	1	8	yes	—
9	#15688993	C₂²	1	4	yes	—
10	#15688989	C₂²	1	4	yes	—
11	#15688983	—	2	4	no	—
12	C₄×C₄⋊Dic₄₇	—	2	4	no	—
13	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#97×C₄₇	C₄	1	8	yes	complemented
16	C₂×#97×C₄₇	C₂²	1	4	yes	—
17	#13093983	—	2	16	no	—
18	C₂×C₄⋊Dic₄₇	—	2	8	no	—
19	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	—	2	4	no	—
22	#13093958	—	2	16	no	—
23	#13093956	—	2	16	no	—
24	C₄₇⋊#95	—	2	8	no	—
25	C₂³×C₄×C₄₇	D₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₄₇	D₄	1	8	yes	—
29	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
35	#97×C₄₇	C₂×C₄	1	16	yes	—
36	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	—	4	4	no	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₄₇	—	2	8	no	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
51	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
52	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
53	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
54	C₂²⋊Dic₄₇	—	4	8	no	—
55	C₄⋊Dic₄₇	—	2	16	no	—
56	C₄⋊Dic₄₇	—	4	16	no	—
57	C₄×Dic₄₇	—	4	8	no	—
58	C₂³×C₄₇	—	2	4	no	—
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₄₇	#127	1	2	yes	—
61	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
62	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	8	no	—
64	C₂×C₄×C₄₇	—	4	8	no	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₄₇	#127	1	2	yes	—
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
70	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂×Dic₄₇	—	4	8	no	—
76	C₂×Dic₄₇	—	4	16	no	—
77	C₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
78	#20799	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	—	2	4	no	—
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	#384	1	4	yes	—
83	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₄₇	#575	1	4	yes	—
87	C₄×C₄₇	—	2	8	no	—
88	C₄×C₄₇	#384	1	4	yes	—
89	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
90	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
91	Dic₄₇	—	8	8	no	—
92	C₂×#427	—	47	4	no	—
93	C₂²×#97	D₄₇	1	1	yes	characteristic
94	#2285	—	47	2	no	—
95	#1847	—	47	8	no	—
96	C₂×C₄₇	—	2	4	no	—
97	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
98	C₂×C₄₇	#2911	1	2	yes	—
99	C₂×C₄₇	#3150	1	2	yes	—
100	C₂×C₄₇	#3162	1	2	yes	—
101	C₂³×C₈	C₂×D₄₇	1	2	yes	—
102	C₂×#117	—	47	4	no	—
103	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
104	#427	—	94	16	no	—
105	C₂×#107	—	47	8	no	—
106	C₂×#106	—	47	8	no	—
107	C₂×#97	Dic₄₇	1	8	yes	complemented
108	C₂×#97	C₂×D₄₇	1	4	yes	—
109	#384	—	47	4	no	—
110	#382	—	47	4	no	—
111	#380	—	94	4	no	—
112	C₄×C₄⋊C₄	—	94	4	no	—
113	C₄₇	#20799	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	D₁₈₈	1	2	yes	—
115	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	C₂²×C₈	Dic₉₄	1	8	yes	—
117	C₂²×C₈	D₁₈₈	1	8	yes	—
118	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
119	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
120	#117	—	94	16	no	—
121	C₂×C₄⋊C₄	—	47	8	no	—
122	C₂×C₄⋊C₄	—	94	8	no	—
123	C₂×C₂²⋊C₄	—	94	4	no	—
124	C₂×C₄²	—	94	4	no	—
125	C₂×C₄²	Dic₉₄	1	4	yes	—
126	C₂×C₄²	D₁₈₈	1	2	yes	—
127	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
128	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
129	#107	—	94	16	no	—
130	#106	—	94	16	no	—
131	#97	C₂×Dic₄₇	1	16	yes	—
132	#95	—	94	8	no	—
133	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
134	C₂²×C₄	—	4	4	no	—
135	C₂²×C₄	—	94	8	no	—
136	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
137	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
138	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
139	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
140	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
141	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
142	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
143	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₄₇	1	32	yes	complemented
145	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
146	C₂×C₈	C₂×D₁₈₈	1	8	yes	—
147	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
148	C₄⋊C₄	—	94	16	no	—
149	C₄⋊C₄	—	188	16	no	—
150	C₂²⋊C₄	—	188	8	no	—
151	C₄²	—	188	8	no	—
152	C₄²	C₄⋊Dic₄₇	1	8	yes	—
153	C₄²	C₂×Dic₉₄	1	4	yes	—
154	C₄²	C₂²×Dic₄₇	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
157	C₂³	#13094021	1	2	yes	—
158	C₂³	#13094024	1	2	yes	—
159	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	188	8	no	—
163	C₂×C₄	—	188	16	no	—
164	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
167	C₂×C₄	#13094021	1	2	yes	—
168	C₂×C₄	#13094024	1	2	yes	—
169	C₂×C₄	#13094024	1	8	yes	—
170	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
172	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
173	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
174	C₈	C₂×C₄⋊Dic₄₇	1	16	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	#15688993	1	4	yes	—
179	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
180	C₂²	C₂×#13094021	1	1	yes	characteristic
181	C₂²	C₂×#13094024	1	1	yes	characteristic
182	C₂²	#15689923	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	376	8	no	—
185	C₄	#15688993	1	4	yes	—
186	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
187	C₄	C₂×#13094024	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×#15688993	1	1	yes	characteristic
190	C₂	#20251978	1	2	yes	—
191	C₂	#20253834	1	2	yes	—
192	C₂	#20253912	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#20799
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #20243556	C₂	semidirect
15. C₂×#97×C₄₇	C₄	semidirect
47. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
107. C₂×#97	Dic₄₇	semidirect
113. C₄₇	#20799	semidirect
144. C₂×C₈	C₄⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20245795	2
#20243556	2
C₂²×#97×C₄₇	2
#20799	47