Subgroups — #27174018

Order 12928.

504 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692212	C₂	1	1	yes	maximal, characteristic
2	#20265074	C₂	1	2	yes	maximal, complemented
3	#20265069	C₂	1	2	yes	maximal
4	#20265068	C₂	1	2	yes	maximal
5	#20265043	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265041	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265036	C₂	1	1	yes	maximal, characteristic
8	C₄×#13108053	C₂	1	2	yes	maximal, complemented
9	#20265024	C₂	1	1	yes	maximal, characteristic
10	#20265022	C₂	1	1	yes	maximal, characteristic, complemented
11	#376×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692212	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×#13108053	C₂²	1	2	yes	—
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	1	yes	characteristic
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108051	C₂²	1	2	yes	—
20	C₂×#13108051	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	1	yes	characteristic
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	1	yes	characteristic
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	1	yes	characteristic
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
33	#95×C₁₀₁	C₂²	1	1	yes	characteristic
34	#95×C₁₀₁	C₂²	1	1	yes	characteristic
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₀₁	—	2	4	no	—
43	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₀₁	—	4	2	no	—
45	#13108053	—	2	4	no	—
46	#13108053	—	4	2	no	—
47	#13108053	C₂×C₄	1	4	yes	—
48	#13108053	C₂×C₄	1	4	yes	—
49	#13108051	—	2	4	no	—
50	#13108051	—	4	2	no	—
51	#13108051	C₂×C₄	1	4	yes	—
52	#13108051	C₂×C₄	1	4	yes	—
53	C₄×Dic₁₀₁	—	2	4	no	—
54	C₄×Dic₁₀₁	—	2	4	no	—
55	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
61	C₄²×C₁₀₁	—	2	4	no	—
62	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
63	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
64	C₂²×D₁₀₁	—	2	2	no	—
65	C₂²×D₁₀₁	—	2	2	no	—
66	C₂²×D₁₀₁	—	2	1	no	—
67	C₂²×D₁₀₁	—	2	1	no	—
68	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₀₁	—	2	2	no	—
70	C₂×C₄×C₁₀₁	—	2	2	no	—
71	C₂×C₄×C₁₀₁	—	2	2	no	—
72	C₂×C₄×C₁₀₁	—	2	1	no	—
73	C₂×C₄×C₁₀₁	—	2	1	no	—
74	C₂×C₄×C₁₀₁	—	2	2	no	—
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₀₁	#41	1	4	yes	—
82	C₂×Dic₁₀₁	—	2	2	no	—
83	C₂×Dic₁₀₁	—	2	1	no	—
84	C₂×Dic₁₀₁	—	2	1	no	—
85	C₂×Dic₁₀₁	—	2	2	no	—
86	C₂×Dic₁₀₁	—	2	2	no	—
87	C₂×Dic₁₀₁	—	2	2	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₀₁	#41	1	4	yes	—
95	C₄×D₁₀₁	—	2	8	no	—
96	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
98	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
99	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×D₁₀₁	—	2	2	no	—
102	C₂×D₁₀₁	—	2	2	no	—
103	C₂×D₁₀₁	—	2	2	no	—
104	C₂×D₁₀₁	—	4	2	no	—
105	C₂×D₁₀₁	—	4	2	no	—
106	C₄×C₁₀₁	—	2	2	no	—
107	C₄×C₁₀₁	—	2	2	no	—
108	C₄×C₁₀₁	—	2	2	no	—
109	C₄×C₁₀₁	—	2	4	no	—
110	C₄×C₁₀₁	—	4	2	no	—
111	Dic₁₀₁	—	2	2	no	—
112	Dic₁₀₁	—	2	2	no	—
113	Dic₁₀₁	—	4	2	no	—
114	Dic₁₀₁	—	4	2	no	—
115	Dic₁₀₁	#117	1	4	yes	—
116	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
117	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
118	C₂×C₁₀₁	#520	1	1	yes	characteristic
119	C₂×C₁₀₁	#552	1	2	yes	—
120	C₂×C₁₀₁	#555	1	2	yes	—
121	D₁₀₁	—	4	2	no	—
122	#2346	—	101	1	no	maximal, Sylow
123	C₁₀₁	#2346	1	1	yes	characteristic, complemented, Sylow
124	C₂×#117	—	101	1	no	—
125	#389	—	101	2	no	—
126	#388	—	101	2	no	—
127	#388	—	101	2	no	—
128	#382	—	101	1	no	—
129	#382	—	101	1	no	—
130	#380	—	101	1	no	—
131	C₄×C₄⋊C₄	—	101	2	no	—
132	C₄×C₂²⋊C₄	—	101	1	no	—
133	#376	—	101	1	no	—
134	#376	D₁₀₁	1	1	yes	characteristic, complemented
135	C₂³×C₄	—	101	1	no	—
136	#117	—	101	8	no	—
137	C₂×C₄⋊C₄	—	101	1	no	—
138	C₂×C₄⋊C₄	—	101	1	no	—
139	C₂×C₄⋊C₄	—	101	2	no	—
140	C₂×C₄⋊C₄	—	101	2	no	—
141	C₂×C₄⋊C₄	—	101	2	no	—
142	C₂×C₂²⋊C₄	—	101	1	no	—
143	C₂×C₂²⋊C₄	—	101	1	no	—
144	C₂×C₂²⋊C₄	—	101	2	no	—
145	C₂×C₂²⋊C₄	—	101	2	no	—
146	C₂×C₄²	—	101	1	no	—
147	C₂×C₄²	—	101	2	no	—
148	C₂×C₄²	—	101	2	no	—
149	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
150	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
151	#95	—	101	1	no	—
152	#95	—	101	2	no	—
153	#95	—	101	1	no	—
154	#95	—	101	2	no	—
155	#95	—	101	2	no	—
156	#95	C₂×D₁₀₁	1	1	yes	characteristic
157	#95	C₂×D₁₀₁	1	1	yes	characteristic
158	#95	C₂×D₁₀₁	1	2	yes	—
159	C₂⁴	—	101	1	no	—
160	C₂²×C₄	—	101	1	no	—
161	C₂²×C₄	—	101	1	no	—
162	C₂²×C₄	—	101	1	no	—
163	C₂²×C₄	—	101	4	no	—
164	C₂²×C₄	—	101	2	no	—
165	C₂²×C₄	—	101	2	no	—
166	C₂²×C₄	—	202	4	no	—
167	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
172	C₄⋊C₄	—	101	4	no	—
173	C₄⋊C₄	—	101	4	no	—
174	C₄⋊C₄	—	404	2	no	—
175	C₄⋊C₄	—	404	2	no	—
176	C₄⋊C₄	—	202	4	no	—
177	C₂²⋊C₄	—	101	4	no	—
178	C₂²⋊C₄	—	101	4	no	—
179	C₂²⋊C₄	—	404	2	no	—
180	C₂²⋊C₄	—	202	4	no	—
181	C₄²	—	2	4	no	—
182	C₄²	—	101	4	no	—
183	C₄²	—	202	4	no	—
184	C₄²	—	202	4	no	—
185	C₄²	C₄×D₁₀₁	1	4	yes	—
186	C₂³	—	101	2	no	—
187	C₂³	—	202	2	no	—
188	C₂³	—	202	2	no	—
189	C₂³	—	202	1	no	—
190	C₂³	—	202	1	no	—
191	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	101	2	no	—
202	C₂×C₄	—	101	4	no	—
203	C₂×C₄	—	101	2	no	—
204	C₂×C₄	—	101	2	no	—
205	C₂×C₄	—	202	8	no	—
206	C₂×C₄	—	202	2	no	—
207	C₂×C₄	—	202	2	no	—
208	C₂×C₄	—	202	2	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	2	no	—
211	C₂×C₄	—	202	2	no	—
212	C₂×C₄	—	202	2	no	—
213	C₂×C₄	—	202	1	no	—
214	C₂×C₄	—	202	1	no	—
215	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
216	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
217	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
218	C₂×C₄	#13108074	1	4	yes	—
219	C₂²	—	404	2	no	—
220	C₂²	—	404	2	no	—
221	C₂²	—	202	2	no	—
222	C₂²	—	202	2	no	—
223	C₂²	—	202	2	no	—
224	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
225	C₂²	#41×D₁₀₁	1	2	yes	—
226	C₂²	C₂×#13108074	1	2	yes	—
227	C₂²	#15692301	1	1	yes	characteristic
228	C₂²	#15692302	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	101	4	no	—
235	C₄	—	404	2	no	—
236	C₄	—	404	2	no	—
237	C₄	—	202	2	no	—
238	C₄	—	202	2	no	—
239	C₂	—	404	2	no	—
240	C₂	#117×D₁₀₁	1	1	yes	characteristic
241	C₂	C₄×#13108074	1	1	yes	characteristic
242	C₂	#20265657	1	1	yes	characteristic
243	C₂	#20265860	1	2	yes	—
244	C₂	#20265890	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#376×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	#376	D₁₀₁
hypocenter	C₁₀₁	#2346
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #376

Complements

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

N	H ≅ G/N	type
2. #20265074	C₂	semidirect
5. #20265043	C₂	semidirect
6. #20265041	C₂	semidirect
8. C₄×#13108053	C₂	semidirect
10. #20265022	C₂	semidirect
11. #376×C₁₀₁	C₂	semidirect
13. #15692212	C₄	semidirect
123. C₁₀₁	#2346	semidirect
134. #376	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692212	2
#20265074	2
#20265069	2
#20265068	2
#20265043	2
#20265041	2
#20265036	2
C₄×#13108053	2
#20265024	2
#20265022	2
#376×C₁₀₁	2
#2346	101