Subgroups — #27173610

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265084	C₂	1	2	yes	maximal
2	#20265076	C₂	1	2	yes	maximal
3	#20265063	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊Dic₁₀₁	C₂	1	2	yes	maximal
6	#20265023	C₂	1	2	yes	maximal
7	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
16	#15692162	C₂²	1	4	yes	—
17	#15692162	C₂²	1	4	yes	—
18	#15692162	C₂²	1	2	yes	—
19	#15692162	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₀₁	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
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	8	no	—
31	C₄×Dic₁₀₁	—	2	4	no	—
32	C₄×Dic₁₀₁	—	2	2	no	—
33	C₄×Dic₁₀₁	—	2	4	no	—
34	C₄×Dic₁₀₁	—	2	4	no	—
35	C₄×Dic₁₀₁	Q₈	1	4	yes	complemented
36	C₂×Q₈×C₁₀₁	—	2	4	no	—
37	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	2	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	—
44	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
45	C₄²×C₁₀₁	C₂×C₄	1	8	yes	—
46	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
47	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₀₁	—	2	8	no	—
49	C₂×C₄×C₁₀₁	—	2	4	no	—
50	C₂×C₄×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	—	2	4	no	—
52	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₀₁	#41	1	4	yes	—
59	C₂×C₄×C₁₀₁	#41	1	2	yes	—
60	C₂×Dic₁₀₁	—	2	2	no	—
61	C₂×Dic₁₀₁	—	2	2	no	—
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	4	no	—
67	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
68	C₂×Dic₁₀₁	#41	1	4	yes	—
69	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
72	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
73	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
74	C₄×C₁₀₁	—	2	4	no	—
75	C₄×C₁₀₁	—	2	2	no	—
76	C₄×C₁₀₁	—	2	4	no	—
77	C₄×C₁₀₁	#117	1	4	yes	—
78	C₄×C₁₀₁	C₄×Q₈	1	4	yes	—
79	Dic₁₀₁	—	2	4	no	—
80	Dic₁₀₁	—	2	4	no	—
81	Dic₁₀₁	—	4	4	no	—
82	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
83	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
84	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
85	C₂×C₁₀₁	#529	1	2	yes	—
86	C₂×C₁₀₁	#533	1	2	yes	—
87	#2295	—	101	1	no	maximal, Sylow
88	C₁₀₁	#2295	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic
90	#391	—	101	2	no	—
91	#389	—	101	2	no	—
92	#387	—	101	4	no	—
93	C₄×C₄⋊C₄	—	101	1	no	—
94	C₄×C₄⋊C₄	—	101	2	no	—
95	#376	—	101	2	no	—
96	C₄³	—	101	1	no	—
97	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
98	C₄×Q₈	Dic₁₀₁	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	101	4	no	—
100	C₂×C₄⋊C₄	—	101	2	no	—
101	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
103	C₂×C₄²	—	101	1	no	—
104	C₂×C₄²	—	101	2	no	—
105	C₂×C₄²	—	101	2	no	—
106	C₂×C₄²	—	101	1	no	—
107	C₂×C₄²	—	101	2	no	—
108	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
110	#95	—	101	4	no	—
111	#95	—	101	4	no	—
112	#95	—	101	2	no	—
113	#95	—	101	4	no	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₁₀₁	1	4	yes	—
116	C₂²×C₄	—	101	2	no	—
117	C₂²×C₄	—	101	2	no	—
118	C₂²×C₄	—	101	4	no	—
119	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
124	C₄⋊C₄	—	202	8	no	—
125	C₄⋊C₄	—	202	4	no	—
126	C₄⋊C₄	C₂×Dic₁₀₁	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
128	C₄²	—	101	4	no	—
129	C₄²	—	202	4	no	—
130	C₄²	—	202	4	no	—
131	C₄²	—	202	4	no	—
132	C₄²	—	202	2	no	—
133	C₄²	C₂×Dic₁₀₁	1	8	yes	—
134	C₄²	C₂×Dic₁₀₁	1	4	yes	—
135	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	101	4	no	—
141	C₂×C₄	—	101	4	no	—
142	C₂×C₄	—	202	4	no	—
143	C₂×C₄	—	202	4	no	—
144	C₂×C₄	—	202	4	no	—
145	C₂×C₄	—	202	2	no	—
146	C₂×C₄	—	202	2	no	—
147	C₂×C₄	—	202	2	no	—
148	C₂×C₄	—	202	2	no	—
149	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
150	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
151	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
152	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
153	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
154	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
155	C₂×C₄	#13108074	1	4	yes	—
156	C₂×C₄	#13108075	1	2	yes	—
157	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
158	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
159	C₂²	#41×D₁₀₁	1	2	yes	—
160	C₂²	C₂×#13108074	1	2	yes	—
161	C₂²	C₂×#13108075	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	404	4	no	—
166	C₄	—	202	4	no	—
167	C₄	—	202	4	no	—
168	C₄	#15692214	1	4	yes	—
169	C₄	Q₈×Dic₁₀₁	1	4	yes	—
170	C₂	C₂×#15692214	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₁₀₁	1	1	yes	characteristic
172	C₂	#41×Dic₁₀₁	1	1	yes	characteristic
173	C₂	#20265714	1	2	yes	—
174	C₂	#20265731	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	#2295
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×Q₈×C₁₀₁	C₄	semidirect
35. C₄×Dic₁₀₁	Q₈	semidirect
88. C₁₀₁	#2295	semidirect
98. C₄×Q₈	Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265084	2
#20265076	2
#20265063	2
C₄×C₄⋊Dic₁₀₁	2
#20265023	2
C₄²×Dic₁₀₁	2
C₂×C₄×Q₈×C₁₀₁	2
#2295	101