Subgroups — #27173724

Order 12928.

880 conjugacy classes of subgroups.

The subgroup lattice has 232 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, complemented
2	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265084	C₂	1	2	yes	maximal, complemented
4	#20265080	C₂	1	2	yes	maximal, complemented
5	#20265057	C₂	1	4	yes	maximal, complemented
6	#20265037	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×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	2	yes	—
12	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₀₁	C₂²	1	8	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
17	#15692162	C₂²	1	8	yes	—
18	C₂×#41×C₁₀₁	—	2	4	no	—
19	C₂×#41×C₁₀₁	—	2	4	no	—
20	C₂×#41×C₁₀₁	C₄	1	8	yes	complemented
21	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₁	—	2	4	no	—
27	C₂²×Dic₁₀₁	—	4	4	no	—
28	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
30	C₄⋊Dic₁₀₁	—	2	4	no	—
31	C₄⋊Dic₁₀₁	—	2	4	no	—
32	C₄⋊Dic₁₀₁	—	2	4	no	—
33	C₂²⋊Dic₁₀₁	—	2	4	no	—
34	C₂²⋊Dic₁₀₁	—	4	8	no	—
35	C₄×Dic₁₀₁	—	2	4	no	—
36	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
37	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
38	#41×C₁₀₁	—	2	32	no	—
39	C₂×Q₈×C₁₀₁	—	2	2	no	—
40	C₂×Q₈×C₁₀₁	—	2	2	no	—
41	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
42	C₂×D₄×C₁₀₁	—	2	4	no	—
43	C₂×D₄×C₁₀₁	—	2	4	no	—
44	C₂×D₄×C₁₀₁	—	2	2	no	—
45	C₂×D₄×C₁₀₁	—	2	2	no	—
46	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	—
47	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
48	C₂²×C₄×C₁₀₁	—	2	4	no	—
49	C₂²×C₄×C₁₀₁	—	2	4	no	—
50	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
52	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
53	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
54	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂³×C₁₀₁	—	2	2	no	—
58	C₂³×C₁₀₁	—	2	2	no	—
59	C₂³×C₁₀₁	—	2	2	no	—
60	C₂³×C₁₀₁	—	2	2	no	—
61	C₂³×C₁₀₁	—	2	4	no	—
62	C₂³×C₁₀₁	—	2	4	no	—
63	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
64	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
66	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
67	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₀₁	—	2	8	no	—
69	D₄×C₁₀₁	—	2	16	no	—
70	D₄×C₁₀₁	—	2	8	no	—
71	C₂×C₄×C₁₀₁	—	2	2	no	—
72	C₂×C₄×C₁₀₁	—	2	2	no	—
73	C₂×C₄×C₁₀₁	—	2	16	no	—
74	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	16	yes	complemented
75	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
82	C₂×Dic₁₀₁	—	2	4	no	—
83	C₂×Dic₁₀₁	—	2	4	no	—
84	C₂×Dic₁₀₁	—	2	4	no	—
85	C₂×Dic₁₀₁	—	2	8	no	—
86	C₂×Dic₁₀₁	—	2	4	no	—
87	C₂×Dic₁₀₁	—	4	8	no	—
88	C₂²×C₁₀₁	—	2	4	no	—
89	C₂²×C₁₀₁	—	2	2	no	—
90	C₂²×C₁₀₁	—	2	2	no	—
91	C₂²×C₁₀₁	—	2	4	no	—
92	C₂²×C₁₀₁	—	2	4	no	—
93	C₂²×C₁₀₁	—	2	8	no	—
94	C₂²×C₁₀₁	—	4	2	no	—
95	C₂²×C₁₀₁	—	4	2	no	—
96	C₂²×C₁₀₁	—	4	4	no	—
97	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
103	C₄×C₁₀₁	—	2	4	no	—
104	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
106	Dic₁₀₁	—	4	4	no	—
107	Dic₁₀₁	—	4	4	no	—
108	C₂×C₁₀₁	—	2	4	no	—
109	C₂×C₁₀₁	—	4	4	no	—
110	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₀₁	#520	1	2	yes	—
112	C₂×C₁₀₁	#535	1	1	yes	characteristic
113	C₂×C₁₀₁	#536	1	1	yes	characteristic
114	C₂×C₁₀₁	#537	1	2	yes	—
115	#2311	—	101	1	no	maximal, Sylow
116	C₁₀₁	#2311	1	1	yes	characteristic, complemented, Sylow
117	C₂²×#41	D₁₀₁	1	1	yes	characteristic
118	C₂×#117	—	101	1	no	—
119	C₂²×C₄⋊C₄	—	101	1	no	—
120	#391	—	101	2	no	—
121	#390	—	101	2	no	—
122	#386	—	101	4	no	—
123	#380	—	101	4	no	—
124	C₂×#41	—	2	4	no	—
125	C₂×#41	—	2	4	no	—
126	C₂×#41	Dic₁₀₁	1	8	yes	complemented
127	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
128	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
129	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
130	C₂³×C₄	—	101	2	no	—
131	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
132	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
133	#117	—	202	4	no	—
134	C₂×C₄⋊C₄	—	101	2	no	—
135	C₂×C₄⋊C₄	—	101	2	no	—
136	C₂×C₄⋊C₄	—	101	2	no	—
137	C₂×C₄⋊C₄	—	202	4	no	—
138	C₂×C₂²⋊C₄	—	101	8	no	—
139	C₂×C₂²⋊C₄	—	101	2	no	—
140	C₂×C₄²	—	101	2	no	—
141	#95	—	101	8	no	—
142	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
143	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
144	#41	—	2	32	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	C₂×Dic₁₀₁	1	4	yes	complemented
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	C₂×Dic₁₀₁	1	8	yes	—
153	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	101	4	no	—
157	C₂²×C₄	—	101	4	no	—
158	C₂²×C₄	—	404	4	no	—
159	C₂²×C₄	—	202	4	no	—
160	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
161	C₂²×C₄	C₂×Dic₁₀₁	1	8	yes	complemented
162	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
163	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
164	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
167	C₄⋊C₄	—	202	4	no	—
168	C₄⋊C₄	—	202	4	no	—
169	C₄⋊C₄	—	202	4	no	—
170	C₂²⋊C₄	—	404	8	no	—
171	C₂²⋊C₄	—	202	4	no	—
172	C₄²	—	202	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
180	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
181	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
182	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
183	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	2	16	no	—
186	D₄	—	2	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	404	8	no	—
191	C₂×C₄	—	202	4	no	—
192	C₂×C₄	—	202	4	no	—
193	C₂×C₄	—	202	4	no	—
194	C₂×C₄	—	202	8	no	—
195	C₂×C₄	—	202	4	no	—
196	C₂×C₄	C₂²⋊Dic₁₀₁	1	16	yes	complemented
197	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
198	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
199	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	4	no	—
213	C₂²	C₂×C₂²⋊Dic₁₀₁	1	4	yes	—
214	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
215	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
216	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
217	C₂²	#15692301	1	2	yes	—
218	C₂²	#15692302	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	404	4	no	—
221	C₄	—	404	4	no	—
222	C₄	C₂×C₂²⋊Dic₁₀₁	1	4	yes	—
223	C₄	C₂×C₂²⋊Dic₁₀₁	1	4	yes	—
224	C₂	—	2	4	no	—
225	C₂	—	4	4	no	—
226	C₂	C₂²×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
227	C₂	#20265660	1	2	yes	—
228	C₂	#20265741	1	1	yes	characteristic
229	C₂	#20265748	1	1	yes	characteristic
230	C₂	#20265755	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	#2311
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#15692214	C₂	semidirect
2. C₂²×C₄⋊Dic₁₀₁	C₂	semidirect
3. #20265084	C₂	semidirect
4. #20265080	C₂	semidirect
5. #20265057	C₂	semidirect
6. #20265037	C₂	semidirect
20. C₂×#41×C₁₀₁	C₄	semidirect
41. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
47. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
51. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
74. C₂×C₄×C₁₀₁	C₂²⋊C₄	semidirect
116. C₁₀₁	#2311	semidirect
126. C₂×#41	Dic₁₀₁	semidirect
147. C₂×Q₈	C₂×Dic₁₀₁	semidirect
153. C₂×D₄	C₂×Dic₁₀₁	semidirect
161. C₂²×C₄	C₂×Dic₁₀₁	semidirect
196. C₂×C₄	C₂²⋊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
#20265084	2
#20265080	2
#20265057	2
#20265037	2
C₂²×#41×C₁₀₁	2
#2311	101