Subgroups — #27179695

Order 12928.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108058	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692214	C₂	1	1	yes	maximal, characteristic
3	#20265205	C₂	1	4	yes	maximal, complemented
4	#20265203	C₂	1	4	yes	maximal, complemented
5	#20265200	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#13108058	—	2	4	no	—
8	C₂×#13108058	C₂²	1	2	yes	—
9	C₂×#13108058	C₂²	1	2	yes	—
10	C₂²×#11717579	C₂²	1	2	yes	—
11	#15692214	C₄	1	2	yes	—
12	#15692214	C₄	1	2	yes	—
13	#15692214	C₂²	1	2	yes	—
14	#15692214	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₁	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
17	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
18	#15692184	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₁	—	2	4	no	—
21	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
22	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
24	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
25	#13108058	—	2	4	no	—
26	#13108058	—	2	4	no	—
27	C₂×#11717579	—	2	4	no	—
28	C₂×#11717579	C₂³	1	4	yes	—
29	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
31	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₀₁	C₂³	1	4	yes	—
35	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
40	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₀₁	—	2	4	no	—
45	M₁₆×C₁₀₁	—	2	4	no	—
46	C₂×C₈×C₁₀₁	—	2	4	no	—
47	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
48	C₂³×C₁₀₁	—	4	2	no	—
49	C₂³×C₁₀₁	C₄⋊C₄	1	4	yes	—
50	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₁₀₁	—	4	4	no	—
67	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
68	C₈×C₁₀₁	—	2	4	no	—
69	#11717579	—	2	4	no	—
70	C₂²×C₁₀₁	—	2	2	no	—
71	C₂²×C₁₀₁	—	2	2	no	—
72	C₂²×C₁₀₁	—	2	2	no	—
73	C₂²×C₁₀₁	—	4	4	no	—
74	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
83	Dic₁₀₁	—	2	8	no	—
84	C₂×C₁₀₁	—	2	2	no	—
85	C₂×C₁₀₁	—	2	2	no	—
86	C₂×C₁₀₁	—	2	2	no	—
87	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₀₁	#568	1	2	yes	—
89	#2905	—	101	1	no	maximal, Sylow
90	C₁₀₁	#2905	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	101	1	no	—
92	C₂²×M₁₆	D₁₀₁	1	1	yes	characteristic
93	C₂×#117	—	101	1	no	—
94	#424	—	101	4	no	—
95	#424	—	101	4	no	—
96	#423	—	101	4	no	—
97	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	101	2	no	—
100	C₂×M₁₆	—	101	2	no	—
101	C₂×M₁₆	—	202	4	no	—
102	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
103	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
104	C₂²×C₈	—	101	2	no	—
105	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
106	#117	—	101	2	no	—
107	#117	—	101	2	no	—
108	#117	—	101	2	no	—
109	#117	—	101	2	no	—
110	C₂×C₄⋊C₄	—	101	2	no	—
111	C₂×C₂²⋊C₄	—	101	2	no	—
112	C₂×C₄²	—	101	2	no	—
113	C₄⋊C₈	—	101	16	no	—
114	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
115	C₂²×C₄	—	101	4	no	—
116	C₂²×C₄	Dic₂₀₂	1	4	yes	—
117	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	202	4	no	—
127	M₁₆	—	202	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	101	4	no	—
130	C₂×C₈	—	202	4	no	—
131	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
132	C₄⋊C₄	—	101	4	no	—
133	C₄⋊C₄	—	101	4	no	—
134	C₂²⋊C₄	—	101	4	no	—
135	C₂²⋊C₄	—	101	4	no	—
136	C₄²	—	101	4	no	—
137	C₄²	—	101	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	#13108053	1	4	yes	—
140	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
143	C₂×C₄	—	101	8	no	—
144	C₂×C₄	—	404	4	no	—
145	C₂×C₄	#13108053	1	4	yes	—
146	C₂×C₄	#13108053	1	4	yes	—
147	C₂×C₄	#13108053	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
154	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
155	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
156	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
157	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	202	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×#13108053	1	2	yes	—
165	C₂²	C₂×#13108053	1	2	yes	—
166	C₂²	C₂×#13108053	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
168	C₄	—	202	8	no	—
169	C₄	C₂×#13108053	1	2	yes	—
170	C₄	C₂×#13108053	1	2	yes	—
171	C₄	C₂×#13108053	1	2	yes	—
172	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
173	C₄	C₂²×Dic₂₀₂	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×#13108053	1	1	yes	characteristic
178	C₂	#20265943	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₀₁	C₂
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#2905
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20265205	C₂	semidirect
4. #20265203	C₂	semidirect
5. #20265200	C₂	semidirect
18. #15692184	C₂²	semidirect
90. C₁₀₁	#2905	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108058	2
C₂×#15692214	2
#20265205	2
#20265203	2
#20265200	2
C₂²×M₁₆×C₁₀₁	2
#2905	101