Subgroups — C₂×#20265073 #27173581

Order 12928.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108053	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265073	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
8	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₁	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
12	C₂×#13108053	—	2	8	no	—
13	C₂×#13108053	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₀₁	—	2	4	no	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
16	#15692162	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₂²×Dic₁₀₁	—	2	4	no	—
24	C₂²×Dic₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	4	no	—
26	C₂²×Dic₁₀₁	—	2	2	no	—
27	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
28	C₂²×Dic₁₀₁	D₄	1	4	yes	—
29	C₂²×Dic₁₀₁	Q₈	1	4	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
31	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
32	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
33	C₄⋊Dic₁₀₁	C₂×C₄	1	16	yes	complemented
34	#13108053	—	2	16	no	—
35	C₄×Dic₁₀₁	—	2	8	no	—
36	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	—	2	4	no	—
38	C₂²×C₄×C₁₀₁	—	2	4	no	—
39	C₂²×C₄×C₁₀₁	—	2	4	no	—
40	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
42	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	—
43	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₀₁	—	2	8	no	—
46	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
47	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
48	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
49	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
50	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
51	C₂³×C₁₀₁	#41	1	2	yes	—
52	C₂³×C₁₀₁	#41	1	2	yes	—
53	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₀₁	—	2	4	no	—
55	C₂×C₄×C₁₀₁	—	2	4	no	—
56	C₂×C₄×C₁₀₁	—	2	4	no	—
57	C₂×C₄×C₁₀₁	—	4	8	no	—
58	C₂×C₄×C₁₀₁	C₄⋊C₄	1	16	yes	complemented
59	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
62	C₂×Dic₁₀₁	—	2	16	no	—
63	C₂×Dic₁₀₁	—	2	8	no	—
64	C₂×Dic₁₀₁	—	2	8	no	—
65	C₂×Dic₁₀₁	—	2	2	no	—
66	C₂×Dic₁₀₁	—	2	2	no	—
67	C₂×Dic₁₀₁	—	2	4	no	—
68	C₂×Dic₁₀₁	—	2	2	no	—
69	C₂×Dic₁₀₁	—	2	4	no	—
70	C₂×Dic₁₀₁	—	2	2	no	—
71	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
72	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
73	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
76	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
77	C₂²×C₁₀₁	#121	1	4	yes	—
78	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₁₀₁	#125	1	4	yes	—
80	C₂²×C₁₀₁	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₀₁	—	4	4	no	—
89	C₄×C₁₀₁	C₂×C₄⋊C₄	1	8	yes	complemented
90	Dic₁₀₁	—	2	8	no	—
91	Dic₁₀₁	—	2	4	no	—
92	Dic₁₀₁	—	2	4	no	—
93	C₂×C₁₀₁	#389	1	8	yes	complemented
94	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
98	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₀₁	C₂×#125	1	1	yes	characteristic
100	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×#389	—	101	1	no	maximal, Sylow
102	C₁₀₁	C₂×#389	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄⋊C₄	—	101	1	no	—
104	C₂²×C₄⋊C₄	—	101	2	no	—
105	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
106	C₂²×C₄²	—	101	1	no	—
107	#389	—	101	8	no	—
108	C₂×#95	—	101	2	no	—
109	C₂³×C₄	—	101	2	no	—
110	C₂³×C₄	—	101	1	no	—
111	C₂³×C₄	—	101	1	no	—
112	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
113	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	101	8	no	—
116	C₂×C₄⋊C₄	—	101	4	no	—
117	C₂×C₄⋊C₄	—	101	8	no	—
118	C₂×C₄⋊C₄	—	202	8	no	—
119	C₂×C₄⋊C₄	C₂×D₁₀₁	1	4	yes	—
120	C₂×C₄²	—	101	4	no	—
121	C₂×C₄²	—	202	4	no	—
122	#95	—	101	8	no	—
123	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	101	8	no	—
128	C₂²×C₄	—	101	4	no	—
129	C₂²×C₄	—	101	4	no	—
130	C₂²×C₄	—	101	2	no	—
131	C₂²×C₄	—	101	4	no	—
132	C₂²×C₄	—	101	2	no	—
133	C₂²×C₄	—	202	4	no	—
134	C₂²×C₄	—	202	2	no	—
135	C₂²×C₄	—	202	2	no	—
136	C₂²×C₄	—	202	2	no	—
137	C₂²×C₄	—	202	4	no	—
138	C₂²×C₄	—	202	2	no	—
139	C₂²×C₄	Dic₂₀₂	1	4	yes	—
140	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
141	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
142	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
143	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	101	16	no	—
146	C₄⋊C₄	—	202	16	no	—
147	C₄²	—	202	8	no	—
148	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
149	C₂³	D₄×D₁₀₁	1	2	yes	—
150	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
151	C₂³	Q₈×D₁₀₁	1	2	yes	—
152	C₂³	C₂×Dic₂₀₂	1	2	yes	—
153	C₂³	#13108073	1	2	yes	—
154	C₂³	#13108075	1	2	yes	—
155	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	101	8	no	—
161	C₂×C₄	—	101	4	no	—
162	C₂×C₄	—	101	4	no	—
163	C₂×C₄	—	202	4	no	—
164	C₂×C₄	—	202	2	no	—
165	C₂×C₄	—	202	2	no	—
166	C₂×C₄	—	202	2	no	—
167	C₂×C₄	—	202	4	no	—
168	C₂×C₄	—	202	2	no	—
169	C₂×C₄	—	202	16	no	—
170	C₂×C₄	—	202	8	no	—
171	C₂×C₄	—	202	8	no	—
172	C₂×C₄	#13108053	1	16	yes	complemented
173	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
174	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
175	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	—
176	C₂²	C₂×#13108053	1	4	yes	—
177	C₂²	#15692217	1	4	yes	—
178	C₂²	#15692220	1	4	yes	—
179	C₂²	#15692226	1	4	yes	—
180	C₂²	#15692231	1	4	yes	—
181	C₂²	#15692246	1	4	yes	—
182	C₂²	#15692254	1	4	yes	—
183	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
184	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
185	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
186	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
187	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
188	C₂²	C₂×#13108073	1	1	yes	characteristic
189	C₂²	C₂×#13108075	1	1	yes	characteristic
190	C₄	—	4	4	no	—
191	C₄	—	202	8	no	—
192	C₄	—	202	4	no	—
193	C₄	—	202	4	no	—
194	C₄	C₂×#13108053	1	8	yes	complemented
195	C₂	#20265073	1	8	yes	direct product factor
196	C₂	C₂²×#13108053	1	1	yes	characteristic
197	C₂	C₂×#15692217	1	1	yes	characteristic
198	C₂	C₂×#15692220	1	1	yes	characteristic
199	C₂	C₂×#15692226	1	1	yes	characteristic
200	C₂	C₂×#15692231	1	1	yes	characteristic
201	C₂	C₂×#15692246	1	1	yes	characteristic
202	C₂	C₂×#15692254	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical	C₂⁴×C₁₀₁	C₂³
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#389
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #20265073	C₂	direct (N × H)
10. C₂×C₄⋊Dic₁₀₁	C₄	semidirect
33. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
58. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
89. C₄×C₁₀₁	C₂×C₄⋊C₄	semidirect
93. C₂×C₁₀₁	#389	semidirect
102. C₁₀₁	C₂×#389	semidirect
172. C₂×C₄	#13108053	semidirect
194. C₄	C₂×#13108053	semidirect
195. C₂	#20265073	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₀₁	2
C₂²×#13108053	2
C₂²×C₄×Dic₁₀₁	2
#20265073	2
C₂×#15692162	2
C₂²×C₄⋊C₄×C₁₀₁	2
C₂×#389	101

Subgroups — C2×#20265073 #27173581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265073 #27173581