Subgroups — #27174081

Order 12928.

632 conjugacy classes of subgroups.

The subgroup lattice has 228 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	2	yes	maximal, complemented
2	#20265080	C₂	1	2	yes	maximal
3	#20265070	C₂	1	4	yes	maximal, complemented
4	#20265047	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265037	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
9	#15692214	—	2	8	no	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
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×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
19	#15692162	C₂²	1	2	yes	—
20	#15692162	C₂²	1	4	yes	—
21	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₄×D₄×C₁₀₁	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
30	C₄⋊Dic₁₀₁	—	2	4	no	—
31	C₄⋊Dic₁₀₁	—	2	2	no	—
32	C₄⋊Dic₁₀₁	—	2	4	no	—
33	C₄⋊Dic₁₀₁	—	4	2	no	—
34	C₄⋊Dic₁₀₁	D₄	1	4	yes	complemented
35	C₂²⋊Dic₁₀₁	—	2	8	no	—
36	C₂²⋊Dic₁₀₁	—	4	4	no	—
37	C₄×Dic₁₀₁	—	2	4	no	—
38	C₄×Dic₁₀₁	—	2	8	no	—
39	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
40	C₂×D₄×C₁₀₁	—	2	2	no	—
41	C₂×D₄×C₁₀₁	—	2	2	no	—
42	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₀₁	—	2	4	no	—
44	C₂²×C₄×C₁₀₁	—	2	4	no	—
45	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	—
46	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
51	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
52	C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
53	C₂³×C₁₀₁	—	2	2	no	—
54	C₂³×C₁₀₁	—	2	2	no	—
55	C₂³×C₁₀₁	—	2	2	no	—
56	C₂³×C₁₀₁	—	2	2	no	—
57	C₂³×C₁₀₁	—	2	2	no	—
58	C₂³×C₁₀₁	—	2	2	no	—
59	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
60	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
61	D₄×C₁₀₁	—	2	8	no	—
62	C₂×C₄×C₁₀₁	—	2	2	no	—
63	C₂×C₄×C₁₀₁	—	2	2	no	—
64	C₂×C₄×C₁₀₁	—	2	1	no	—
65	C₂×C₄×C₁₀₁	—	2	1	no	—
66	C₂×C₄×C₁₀₁	—	2	16	no	—
67	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₀₁	#41	1	2	yes	—
73	C₂×C₄×C₁₀₁	#41	1	2	yes	—
74	C₂×C₄×C₁₀₁	#41	1	2	yes	—
75	C₂×Dic₁₀₁	—	2	2	no	—
76	C₂×Dic₁₀₁	—	2	2	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	—	2	2	no	—
79	C₂×Dic₁₀₁	—	2	2	no	—
80	C₂×Dic₁₀₁	—	2	4	no	—
81	C₂×Dic₁₀₁	—	2	2	no	—
82	C₂×Dic₁₀₁	—	2	4	no	—
83	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
84	C₂²×C₁₀₁	—	2	4	no	—
85	C₂²×C₁₀₁	—	2	4	no	—
86	C₂²×C₁₀₁	—	2	4	no	—
87	C₂²×C₁₀₁	—	4	2	no	—
88	C₂²×C₁₀₁	—	4	2	no	—
89	C₂²×C₁₀₁	—	4	2	no	—
90	C₂²×C₁₀₁	—	4	2	no	—
91	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
94	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
95	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
98	C₄×C₁₀₁	—	2	4	no	—
99	C₄×C₁₀₁	—	2	2	no	—
100	C₄×C₁₀₁	#117	1	4	yes	complemented
101	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
102	Dic₁₀₁	—	2	4	no	—
103	Dic₁₀₁	—	4	4	no	—
104	Dic₁₀₁	—	4	2	no	—
105	C₂×C₁₀₁	—	4	4	no	—
106	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
107	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₁₀₁	#520	1	1	yes	characteristic
109	C₂×C₁₀₁	#547	1	1	yes	characteristic
110	C₂×C₁₀₁	#550	1	1	yes	characteristic
111	C₂×C₁₀₁	#555	1	1	yes	characteristic
112	C₂×C₁₀₁	#556	1	1	yes	characteristic
113	#2354	—	101	1	no	maximal, Sylow
114	C₁₀₁	#2354	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic
116	C₂×#117	—	101	2	no	—
117	#390	—	101	2	no	—
118	#388	—	101	4	no	—
119	#384	—	101	1	no	—
120	#380	—	101	2	no	—
121	C₄×C₄⋊C₄	—	101	2	no	—
122	C₄×C₄⋊C₄	—	101	1	no	—
123	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
125	C₄×D₄	Dic₁₀₁	1	8	yes	complemented
126	#117	—	202	8	no	—
127	C₂×C₄⋊C₄	—	101	1	no	—
128	C₂×C₄⋊C₄	—	101	2	no	—
129	C₂×C₄⋊C₄	—	101	2	no	—
130	C₂×C₄⋊C₄	—	101	1	no	—
131	C₂×C₄⋊C₄	—	101	2	no	—
132	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	101	4	no	—
134	C₂×C₂²⋊C₄	—	101	4	no	—
135	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
136	C₂×C₄²	—	101	4	no	—
137	C₂×C₄²	—	101	2	no	—
138	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
139	#95	—	101	4	no	—
140	#95	—	101	2	no	—
141	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂×Dic₁₀₁	1	4	yes	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	101	2	no	—
148	C₂²×C₄	—	101	2	no	—
149	C₂²×C₄	—	101	4	no	—
150	C₂²×C₄	C₂×Dic₁₀₁	1	8	yes	—
151	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
155	C₄⋊C₄	—	101	4	no	—
156	C₄⋊C₄	—	404	2	no	—
157	C₄⋊C₄	—	202	4	no	—
158	C₄⋊C₄	—	202	2	no	—
159	C₄⋊C₄	—	202	4	no	—
160	C₄⋊C₄	C₂×Dic₁₀₁	1	4	yes	—
161	C₂²⋊C₄	—	404	4	no	—
162	C₂²⋊C₄	—	202	8	no	—
163	C₂²⋊C₄	C₂×Dic₁₀₁	1	8	yes	—
164	C₄²	—	202	8	no	—
165	C₄²	—	202	4	no	—
166	C₄²	C₂×Dic₁₀₁	1	4	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₂²×Dic₁₀₁	1	4	yes	—
174	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	101	4	no	—
182	C₂×C₄	—	202	4	no	—
183	C₂×C₄	—	202	2	no	—
184	C₂×C₄	—	202	2	no	—
185	C₂×C₄	—	202	4	no	—
186	C₂×C₄	—	202	2	no	—
187	C₂×C₄	—	202	4	no	—
188	C₂×C₄	—	202	2	no	—
189	C₂×C₄	—	202	2	no	—
190	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
191	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
192	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
193	C₂×C₄	C₂²×Dic₁₀₁	1	2	yes	—
194	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
195	C₂×C₄	#13108073	1	2	yes	—
196	C₂×C₄	#13108074	1	2	yes	—
197	C₂×C₄	#13108074	1	2	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
206	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
207	C₂²	C₂×#13108073	1	1	yes	characteristic
208	C₂²	C₂×#13108074	1	1	yes	characteristic
209	C₂²	C₂×#13108074	1	1	yes	characteristic
210	C₂²	#15692301	1	1	yes	characteristic
211	C₂²	#15692302	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	404	4	no	—
215	C₄	—	404	2	no	—
216	C₄	—	202	4	no	—
217	C₄	#15692214	1	4	yes	complemented
218	C₄	D₄×Dic₁₀₁	1	4	yes	—
219	C₂	—	4	4	no	—
220	C₂	C₂×#15692214	1	1	yes	characteristic
221	C₂	C₂×D₄×Dic₁₀₁	1	1	yes	characteristic
222	C₂	#20265660	1	1	yes	characteristic
223	C₂	#20265832	1	1	yes	characteristic
224	C₂	#20265848	1	1	yes	characteristic
225	C₂	#20265892	1	1	yes	characteristic
226	C₂	#20265897	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2354
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15692214	C₂	semidirect
3. #20265070	C₂	semidirect
4. #20265047	C₂	semidirect
5. #20265037	C₂	semidirect
6. C₄⋊C₄×Dic₁₀₁	C₂	semidirect
7. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
23. C₄×D₄×C₁₀₁	C₄	semidirect
34. C₄⋊Dic₁₀₁	D₄	semidirect
100. C₄×C₁₀₁	#117	semidirect
114. C₁₀₁	#2354	semidirect
125. C₄×D₄	Dic₁₀₁	semidirect
217. C₄	#15692214	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
#20265080	2
#20265070	2
#20265047	2
#20265037	2
C₄⋊C₄×Dic₁₀₁	2
C₄×C₄⋊Dic₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
#2354	101