Subgroups — #27174010

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108053	C₂	1	1	yes	maximal, characteristic
2	#20265074	C₂	1	2	yes	maximal, complemented
3	#20265064	C₂	1	2	yes	maximal, complemented
4	#20265061	C₂	1	2	yes	maximal, complemented
5	#20265037	C₂	1	1	yes	maximal, characteristic
6	#20265035	C₂	1	1	yes	maximal, characteristic
7	#20265035	C₂	1	1	yes	maximal, characteristic
8	C₄×#13108053	C₂	1	2	yes	maximal, complemented
9	C₂²⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂²⋊C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×#13108053	C₄	1	8	yes	complemented
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	2	yes	—
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	1	yes	characteristic
20	C₂×#13108053	C₂²	1	1	yes	characteristic
21	C₂×#13108053	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
26	#15692162	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
35	#95×C₁₀₁	C₂²	1	2	yes	—
36	C₂²×Dic₁₀₁	—	2	4	no	—
37	C₂²×Dic₁₀₁	—	4	2	no	—
38	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
39	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₁₀₁	—	4	2	no	—
47	#13108053	—	2	4	no	—
48	#13108053	—	4	2	no	—
49	#13108053	C₂×C₄	1	4	yes	—
50	#13108053	C₂×C₄	1	4	yes	—
51	#13108053	C₂×C₄	1	4	yes	—
52	#13108053	C₂×C₄	1	4	yes	—
53	C₂²⋊Dic₁₀₁	—	2	2	no	—
54	C₂²⋊Dic₁₀₁	—	2	2	no	—
55	C₄×Dic₁₀₁	—	2	4	no	—
56	C₄×Dic₁₀₁	—	4	2	no	—
57	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	—	4	2	no	—
59	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₁	Q₈	1	4	yes	—
66	C₄²×C₁₀₁	—	4	2	no	—
67	C₂³×C₁₀₁	—	2	2	no	—
68	C₂³×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
71	C₂³×C₁₀₁	#41	1	2	yes	—
72	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
73	C₂×C₄×C₁₀₁	—	2	2	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	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	—	4	4	no	—
81	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
84	C₂×Dic₁₀₁	—	2	2	no	—
85	C₂×Dic₁₀₁	—	2	2	no	—
86	C₂×Dic₁₀₁	—	2	2	no	—
87	C₂×Dic₁₀₁	—	2	8	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	4	4	no	—
95	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	#41	1	4	yes	—
100	C₂²×C₁₀₁	—	2	2	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	4	no	—
103	C₂²×C₁₀₁	—	2	4	no	—
104	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
105	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
106	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
108	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
110	C₄×C₁₀₁	—	2	2	no	—
111	C₄×C₁₀₁	—	2	2	no	—
112	C₄×C₁₀₁	—	2	4	no	—
113	C₄×C₁₀₁	—	4	2	no	—
114	Dic₁₀₁	—	2	2	no	—
115	Dic₁₀₁	—	2	2	no	—
116	Dic₁₀₁	—	2	2	no	—
117	Dic₁₀₁	—	4	2	no	—
118	Dic₁₀₁	—	4	2	no	—
119	Dic₁₀₁	#117	1	4	yes	—
120	C₂×C₁₀₁	—	2	4	no	—
121	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
122	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₀₁	#518	1	1	yes	characteristic
124	C₂×C₁₀₁	#553	1	2	yes	—
125	C₂×C₁₀₁	#554	1	2	yes	—
126	#2345	—	101	1	no	maximal, Sylow
127	C₁₀₁	#2345	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄⋊C₄	—	101	1	no	—
129	#389	—	101	2	no	—
130	#387	—	101	2	no	—
131	#387	—	101	2	no	—
132	#380	—	101	1	no	—
133	#380	—	101	1	no	—
134	#380	—	101	1	no	—
135	C₄×C₄⋊C₄	—	101	2	no	—
136	C₄×C₂²⋊C₄	—	101	1	no	—
137	C₄×C₂²⋊C₄	—	101	1	no	—
138	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
139	C₂³×C₄	—	101	1	no	—
140	C₂³×C₄	—	101	1	no	—
141	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	101	8	no	—
143	C₂×C₄⋊C₄	—	101	1	no	—
144	C₂×C₄⋊C₄	—	101	1	no	—
145	C₂×C₄⋊C₄	—	101	1	no	—
146	C₂×C₄⋊C₄	—	101	1	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	2	no	—
149	C₂×C₄⋊C₄	—	101	2	no	—
150	C₂×C₂²⋊C₄	—	101	1	no	—
151	C₂×C₂²⋊C₄	—	101	1	no	—
152	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
154	C₂×C₄²	—	101	2	no	—
155	C₂×C₄²	—	101	2	no	—
156	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
157	#95	—	101	2	no	—
158	#95	—	101	2	no	—
159	#95	—	101	2	no	—
160	#95	—	101	2	no	—
161	#95	—	101	2	no	—
162	#95	C₂×D₁₀₁	1	2	yes	—
163	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	101	1	no	—
166	C₂²×C₄	—	101	4	no	—
167	C₂²×C₄	—	101	4	no	—
168	C₂²×C₄	—	101	1	no	—
169	C₂²×C₄	—	101	1	no	—
170	C₂²×C₄	—	101	1	no	—
171	C₂²×C₄	—	101	2	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	404	2	no	—
174	C₂²×C₄	—	202	4	no	—
175	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
176	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
177	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
180	C₄⋊C₄	—	101	4	no	—
181	C₄⋊C₄	—	101	4	no	—
182	C₄⋊C₄	—	101	4	no	—
183	C₄⋊C₄	—	101	4	no	—
184	C₄⋊C₄	—	404	2	no	—
185	C₄⋊C₄	—	404	2	no	—
186	C₄⋊C₄	—	202	4	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	202	2	no	—
189	C₂²⋊C₄	—	202	2	no	—
190	C₂²⋊C₄	Dic₂₀₂	1	4	yes	—
191	C₄²	—	4	2	no	—
192	C₄²	—	404	2	no	—
193	C₄²	—	202	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
197	C₂³	C₂×Dic₂₀₂	1	2	yes	—
198	C₂³	#13108074	1	2	yes	—
199	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	101	2	no	—
209	C₂×C₄	—	101	4	no	—
210	C₂×C₄	—	101	2	no	—
211	C₂×C₄	—	101	2	no	—
212	C₂×C₄	—	101	2	no	—
213	C₂×C₄	—	404	4	no	—
214	C₂×C₄	—	202	8	no	—
215	C₂×C₄	—	202	2	no	—
216	C₂×C₄	—	202	2	no	—
217	C₂×C₄	—	202	2	no	—
218	C₂×C₄	—	202	2	no	—
219	C₂×C₄	—	202	2	no	—
220	C₂×C₄	—	202	2	no	—
221	C₂×C₄	—	202	2	no	—
222	C₂×C₄	—	202	2	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
225	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
226	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	2	no	—
231	C₂²	C₄×Dic₂₀₂	1	4	yes	—
232	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
233	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
234	C₂²	#41×D₁₀₁	1	2	yes	—
235	C₂²	C₂×#13108074	1	1	yes	characteristic
236	C₂²	#15692300	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	101	4	no	—
242	C₄	—	404	2	no	—
243	C₄	—	404	2	no	—
244	C₄	—	202	2	no	—
245	C₄	—	202	2	no	—
246	C₄	—	202	2	no	—
247	C₂	—	2	4	no	—
248	C₂	#117×D₁₀₁	1	1	yes	characteristic
249	C₂	C₂×C₄×Dic₂₀₂	1	1	yes	characteristic
250	C₂	#20265650	1	1	yes	characteristic
251	C₂	#20265876	1	2	yes	—
252	C₂	#20265886	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#2345
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20265074	C₂	semidirect
3. #20265064	C₂	semidirect
4. #20265061	C₂	semidirect
8. C₄×#13108053	C₂	semidirect
15. C₂×#13108053	C₄	semidirect
127. C₁₀₁	#2345	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108053	2
#20265074	2
#20265064	2
#20265061	2
#20265037	2
#20265035	2
C₄×#13108053	2
C₂²⋊C₄×Dic₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
#2345	101