Subgroups — #27173755

Order 12928.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692214	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20265076	C₂	1	4	yes	maximal, complemented
4	#20265052	C₂	1	4	yes	maximal
5	#20265047	C₂	1	2	yes	maximal, complemented
6	#20265043	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
9	#15692214	—	2	4	no	—
10	C₂×C₄⋊Dic₁₀₁	—	2	4	no	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
18	#15692162	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
20	#117×C₁₀₁	C₄	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
24	C₂²×Dic₁₀₁	—	2	4	no	—
25	C₂²×Dic₁₀₁	—	4	4	no	—
26	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
27	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
29	C₄⋊Dic₁₀₁	—	2	4	no	—
30	C₄⋊Dic₁₀₁	—	2	4	no	—
31	C₄⋊Dic₁₀₁	—	2	2	no	—
32	C₄⋊Dic₁₀₁	—	2	2	no	—
33	C₄⋊Dic₁₀₁	—	4	8	no	—
34	C₂²⋊Dic₁₀₁	—	2	4	no	—
35	C₄×Dic₁₀₁	—	2	4	no	—
36	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
39	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
44	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
45	C₄²×C₁₀₁	C₂×C₄	1	8	yes	complemented
46	C₂³×C₁₀₁	—	2	2	no	—
47	C₂³×C₁₀₁	—	2	2	no	—
48	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
50	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
51	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	—	2	4	no	—
54	C₂×C₄×C₁₀₁	C₄⋊C₄	1	16	yes	complemented
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
62	C₂×Dic₁₀₁	—	2	2	no	—
63	C₂×Dic₁₀₁	—	2	2	no	—
64	C₂×Dic₁₀₁	—	2	4	no	—
65	C₂×Dic₁₀₁	—	2	4	no	—
66	C₂×Dic₁₀₁	—	2	8	no	—
67	C₂×Dic₁₀₁	—	2	4	no	—
68	C₂×Dic₁₀₁	—	4	8	no	—
69	C₂²×C₁₀₁	—	2	4	no	—
70	C₂²×C₁₀₁	—	2	2	no	—
71	C₂²×C₁₀₁	—	2	2	no	—
72	C₂²×C₁₀₁	—	2	4	no	—
73	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
79	C₄×C₁₀₁	—	2	8	no	—
80	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
82	Dic₁₀₁	—	4	2	no	—
83	Dic₁₀₁	—	4	4	no	—
84	Dic₁₀₁	—	4	2	no	—
85	C₂×C₁₀₁	—	2	4	no	—
86	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₁₀₁	#520	1	2	yes	—
88	C₂×C₁₀₁	#535	1	1	yes	characteristic
89	C₂×C₁₀₁	#536	1	1	yes	characteristic
90	C₂×C₁₀₁	#544	1	2	yes	—
91	#2317	—	101	1	no	maximal, Sylow
92	C₁₀₁	#2317	1	1	yes	characteristic, complemented, Sylow
93	C₂×#117	—	101	1	no	—
94	C₂×#117	D₁₀₁	1	1	yes	characteristic
95	C₂²×C₄⋊C₄	—	101	1	no	—
96	#389	—	101	4	no	—
97	#385	—	101	4	no	—
98	#384	—	101	2	no	—
99	#382	—	101	2	no	—
100	C₂³×C₄	—	101	2	no	—
101	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
102	#117	—	202	4	no	—
103	#117	Dic₁₀₁	1	8	yes	complemented
104	C₂×C₄⋊C₄	—	101	8	no	—
105	C₂×C₄⋊C₄	—	101	2	no	—
106	C₂×C₄⋊C₄	—	101	2	no	—
107	C₂×C₄⋊C₄	—	101	1	no	—
108	C₂×C₄⋊C₄	—	101	1	no	—
109	C₂×C₄⋊C₄	—	202	4	no	—
110	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
111	C₂×C₂²⋊C₄	—	101	2	no	—
112	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
113	C₂×C₄²	—	101	2	no	—
114	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
115	#95	—	101	8	no	—
116	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
117	C₂²×C₄	—	101	2	no	—
118	C₂²×C₄	—	101	4	no	—
119	C₂²×C₄	—	101	2	no	—
120	C₂²×C₄	—	404	4	no	—
121	C₂²×C₄	—	202	4	no	—
122	C₂²×C₄	Dic₂₀₂	1	4	yes	—
123	C₂²×C₄	D₄₀₄	1	4	yes	—
124	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
126	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
128	C₄⋊C₄	—	404	8	no	—
129	C₄⋊C₄	—	202	4	no	—
130	C₄⋊C₄	—	202	4	no	—
131	C₄⋊C₄	—	202	2	no	—
132	C₄⋊C₄	—	202	2	no	—
133	C₄⋊C₄	C₂×Dic₁₀₁	1	8	yes	complemented
134	C₂²⋊C₄	—	202	4	no	—
135	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	—
136	C₄²	—	202	4	no	—
137	C₄²	C₂×Dic₁₀₁	1	8	yes	complemented
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
141	C₂³	C₂×D₄₀₄	1	2	yes	—
142	C₂³	C₂×Dic₂₀₂	1	2	yes	—
143	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	404	8	no	—
147	C₂×C₄	—	202	2	no	—
148	C₂×C₄	—	202	2	no	—
149	C₂×C₄	—	202	4	no	—
150	C₂×C₄	—	202	4	no	—
151	C₂×C₄	—	202	8	no	—
152	C₂×C₄	—	202	4	no	—
153	C₂×C₄	C₄⋊Dic₁₀₁	1	16	yes	complemented
154	C₂×C₄	C₂²×Dic₁₀₁	1	8	yes	—
155	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
156	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
157	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
158	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
159	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
160	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×C₄⋊Dic₁₀₁	1	4	yes	—
166	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
167	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
168	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
169	C₂²	#15692301	1	2	yes	—
170	C₂²	#15692302	1	2	yes	—
171	C₄	—	2	8	no	—
172	C₄	—	404	4	no	—
173	C₄	—	404	2	no	—
174	C₄	—	404	2	no	—
175	C₄	C₂×C₄⋊Dic₁₀₁	1	4	yes	—
176	C₄	C₂×C₄⋊Dic₁₀₁	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₄⋊Dic₁₀₁	1	1	yes	characteristic
179	C₂	#20265660	1	2	yes	—
180	C₂	#20265746	1	1	yes	characteristic
181	C₂	#20265753	1	1	yes	characteristic
182	C₂	#20265805	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#117×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	#2317
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20265076	C₂	semidirect
5. #20265047	C₂	semidirect
6. #20265043	C₂	semidirect
20. #117×C₁₀₁	C₄	semidirect
43. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
45. C₄²×C₁₀₁	C₂×C₄	semidirect
54. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
92. C₁₀₁	#2317	semidirect
103. #117	Dic₁₀₁	semidirect
133. C₄⋊C₄	C₂×Dic₁₀₁	semidirect
137. C₄²	C₂×Dic₁₀₁	semidirect
153. C₂×C₄	C₄⋊Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692214	2
C₂²×C₄⋊Dic₁₀₁	2
#20265076	2
#20265052	2
#20265047	2
#20265043	2
C₂×#117×C₁₀₁	2
#2317	101