Subgroups — #27187697

Order 12928.

564 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	C₂²×#13108084	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₀₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#20266142	C₂	1	4	yes	maximal, complemented
4	#20266141	C₂	1	4	yes	maximal, complemented
5	C₄×#13108084	C₂	1	4	yes	maximal, complemented
6	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#13108084	C₄	1	8	yes	complemented
8	C₂×#13108084	C₂²	1	2	yes	—
9	C₂×#13108084	C₂²	1	2	yes	—
10	C₂×#13108084	C₂²	1	2	yes	—
11	D₁₀₁⋊M₁₆	C₄	1	8	yes	complemented
12	C₂×#13108083	C₂²	1	2	yes	—
13	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
14	C₂²×#11717581	C₂²	1	2	yes	—
15	#15692310	C₂²	1	8	yes	complemented
16	C₄×#11717581	C₂²	1	8	yes	complemented
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692214	C₄	1	1	yes	characteristic
19	#15692213	C₂²	1	2	yes	—
20	#15692212	C₄	1	2	yes	—
21	#15692211	C₂²	1	2	yes	—
22	C₄⋊C₄×D₁₀₁	C₄	1	2	yes	—
23	C₂²⋊C₄×D₁₀₁	C₄	1	2	yes	—
24	C₄²×D₁₀₁	C₂²	1	2	yes	—
25	#117×C₁₀₁	C₄	1	1	yes	characteristic
26	#13108084	C₂×C₄	1	8	yes	—
27	#13108084	C₂×C₄	1	8	yes	—
28	#13108084	C₂×C₄	1	8	yes	complemented
29	#13108083	C₂×C₄	1	8	yes	complemented
30	D₁₀₁⋊C₈	C₂×C₄	1	8	yes	complemented
31	C₂×#11717581	C₂×C₄	1	8	yes	complemented
32	C₂×#11717581	C₂³	1	4	yes	—
33	C₂×#11717581	C₂³	1	4	yes	—
34	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
43	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
44	#13108053	C₂×C₄	1	4	yes	—
45	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
46	#13108051	C₂×C₄	1	4	yes	—
47	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
52	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
53	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₁₀₁	—	4	2	no	—
55	C₂²×D₁₀₁	C₄²	1	4	yes	—
56	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₁	C₄²	1	4	yes	—
59	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
67	C₄×D₁₀₁	—	4	4	no	—
68	C₄×D₁₀₁	C₄²	1	4	yes	—
69	C₄×D₁₀₁	C₄²	1	4	yes	—
70	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
71	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
72	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
73	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
74	#11717581	C₂²×C₄	1	8	yes	—
75	#11717581	C₂²×C₄	1	8	yes	—
76	C₂²×C₁₀₁	—	2	2	no	—
77	C₂²×C₁₀₁	C₂×C₄²	1	2	yes	—
78	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×D₁₀₁	—	2	2	no	—
80	C₂×D₁₀₁	—	2	2	no	—
81	C₂×D₁₀₁	—	4	4	no	—
82	C₂×D₁₀₁	C₂×C₄²	1	2	yes	—
83	C₂×D₁₀₁	C₂×C₄²	1	2	yes	—
84	C₄×C₁₀₁	—	2	4	no	—
85	C₄×C₁₀₁	C₂×C₄²	1	2	yes	—
86	C₄×C₁₀₁	C₂×C₄²	1	2	yes	—
87	Dic₁₀₁	—	2	4	no	—
88	Dic₁₀₁	C₂×C₄²	1	2	yes	—
89	Dic₁₀₁	C₂³×C₄	1	1	yes	characteristic
90	Dic₁₀₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₁₀₁	—	2	2	no	—
92	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
93	C₂×C₁₀₁	#568	1	2	yes	—
94	D₁₀₁	—	2	2	no	—
95	D₁₀₁	—	2	2	no	—
96	#2873	—	101	1	no	maximal, Sylow
97	C₁₀₁	#2873	1	1	yes	characteristic, complemented, Sylow
98	C₂²×M₁₆	—	101	1	no	—
99	C₂²×M₁₆	—	101	1	no	—
100	C₂×#117	—	101	1	no	—
101	#405	—	101	4	no	—
102	#405	—	101	4	no	—
103	C₄×M₁₆	—	101	4	no	—
104	C₂³×C₄	—	101	1	no	—
105	C₂×M₁₆	—	101	2	no	—
106	C₂×M₁₆	—	101	8	no	—
107	C₂×M₁₆	—	101	8	no	—
108	C₂×M₁₆	—	101	2	no	—
109	C₂×M₁₆	—	101	2	no	—
110	C₂×M₁₆	—	101	2	no	—
111	C₂²×C₈	—	101	2	no	—
112	C₂²×C₈	—	101	2	no	—
113	#117	—	101	1	no	—
114	#117	—	101	2	no	—
115	#117	—	101	2	no	—
116	#117	—	101	2	no	—
117	#117	#87938	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	101	2	no	—
119	C₂×C₂²⋊C₄	—	101	2	no	—
120	C₂×C₄²	—	101	2	no	—
121	#97	—	101	8	no	—
122	C₄×C₈	—	101	8	no	—
123	C₂⁴	—	101	1	no	—
124	C₂²×C₄	—	101	1	no	—
125	C₂²×C₄	—	101	4	no	—
126	C₂²×C₄	—	101	4	no	—
127	C₂²×C₄	—	101	1	no	—
128	C₂²×C₄	—	101	1	no	—
129	C₂²×C₄	—	101	1	no	—
130	C₂²×C₄	—	101	1	no	—
131	C₂²×C₄	—	101	4	no	—
132	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
133	M₁₆	—	101	8	no	—
134	M₁₆	—	101	8	no	—
135	M₁₆	—	101	8	no	—
136	M₁₆	—	101	8	no	—
137	C₂×C₈	—	101	4	no	—
138	C₂×C₈	—	101	4	no	—
139	C₂×C₈	—	101	8	no	—
140	C₂×C₈	—	101	8	no	—
141	C₄⋊C₄	—	101	4	no	—
142	C₄⋊C₄	—	101	2	no	—
143	C₄⋊C₄	C₂×#87938	1	2	yes	—
144	C₂²⋊C₄	—	101	2	no	—
145	C₂²⋊C₄	—	101	4	no	—
146	C₂²⋊C₄	C₂×#87938	1	2	yes	—
147	C₄²	—	101	2	no	—
148	C₄²	—	101	4	no	—
149	C₄²	C₂×#87938	1	2	yes	—
150	C₂³	—	101	4	no	—
151	C₂³	—	101	1	no	—
152	C₂³	—	101	1	no	—
153	C₂³	—	404	2	no	—
154	C₂³	C₂²×#87938	1	1	yes	characteristic
155	C₂×C₄	—	101	2	no	—
156	C₂×C₄	—	101	2	no	—
157	C₂×C₄	—	101	1	no	—
158	C₂×C₄	—	101	1	no	—
159	C₂×C₄	—	101	2	no	—
160	C₂×C₄	—	101	2	no	—
161	C₂×C₄	—	101	2	no	—
162	C₂×C₄	—	101	2	no	—
163	C₂×C₄	—	101	4	no	—
164	C₂×C₄	—	101	4	no	—
165	C₂×C₄	—	101	4	no	—
166	C₂×C₄	—	404	4	no	—
167	C₂×C₄	C₄×#87938	1	4	yes	—
168	C₂×C₄	C₂²×#87938	1	4	yes	—
169	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
170	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
171	C₈	—	101	8	no	—
172	C₈	—	101	8	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	101	2	no	—
175	C₂²	—	101	2	no	—
176	C₂²	—	404	4	no	—
177	C₂²	—	202	2	no	—
178	C₂²	—	202	2	no	—
179	C₂²	C₂×C₄×#87938	1	2	yes	—
180	C₂²	C₂³×#87938	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	101	1	no	—
183	C₄	—	101	1	no	—
184	C₄	—	101	2	no	—
185	C₄	—	202	4	no	—
186	C₄	C₂×C₄×#87938	1	2	yes	—
187	C₄	C₂×C₄×#87938	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	202	2	no	—
190	C₂	—	202	2	no	—
191	C₂	C₂²×C₄×#87938	1	1	yes	characteristic
192	C₂	#20266256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator	C₂×C₁₀₁	C₂²×C₄²
Fitting	#117×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂³×C₄
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#117	#87938
hypocenter	C₁₀₁	#2873
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20266142	C₂	semidirect
4. #20266141	C₂	semidirect
5. C₄×#13108084	C₂	semidirect
7. C₂×#13108084	C₄	semidirect
11. D₁₀₁⋊M₁₆	C₄	semidirect
15. #15692310	C₂²	semidirect
16. C₄×#11717581	C₂²	semidirect
28. #13108084	C₂×C₄	semidirect
29. #13108083	C₂×C₄	semidirect
30. D₁₀₁⋊C₈	C₂×C₄	semidirect
31. C₂×#11717581	C₂×C₄	semidirect
97. C₁₀₁	#2873	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108084	2
C₂×D₁₀₁⋊M₁₆	2
#20266142	2
#20266141	2
C₄×#13108084	2
#117×D₁₀₁	2
#2873	101