Subgroups — #27187613

Order 12928.

600 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692334	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#87938	C₂	1	1	yes	maximal, characteristic
3	#20266128	C₂	1	4	yes	maximal, complemented
4	#20266124	C₂	1	4	yes	maximal
5	#20266123	C₂	1	2	yes	maximal, complemented
6	#20266121	C₂	1	2	yes	maximal, complemented
7	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×#87938	C₂²	1	2	yes	—
9	#15692334	—	2	4	no	—
10	C₂×C₄⋊#87938	—	2	4	no	—
11	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#87938	C₂²	1	2	yes	—
14	C₂×C₄⋊#87938	C₂²	1	2	yes	—
15	C₂×C₄⋊#87938	C₂²	1	8	yes	—
16	C₂×C₂²⋊#87938	C₂²	1	2	yes	—
17	C₂×C₄×#87938	C₂²	1	2	yes	—
18	(C₂×#87938)⋊C₄	C₂²	1	8	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	#15692214	C₄	1	1	yes	characteristic, complemented
21	#15692213	C₄	1	2	yes	complemented
22	#15692212	C₄	1	2	yes	complemented
23	#15692211	C₄	1	2	yes	complemented
24	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	—
25	C₂²⋊C₄×D₁₀₁	C₂²	1	2	yes	—
26	C₄²×D₁₀₁	C₂²	1	2	yes	—
27	#117×C₁₀₁	C₄	1	1	yes	characteristic, complemented
28	C₂²×#87938	—	2	4	no	—
29	C₂²×#87938	—	4	4	no	—
30	C₂²×#87938	C₂³	1	2	yes	—
31	C₂²×#87938	C₂³	1	4	yes	—
32	C₂²×#87938	C₂³	1	2	yes	—
33	C₄⋊#87938	—	2	4	no	—
34	C₄⋊#87938	—	2	4	no	—
35	C₄⋊#87938	—	2	2	no	—
36	C₄⋊#87938	—	2	2	no	—
37	C₄⋊#87938	—	4	8	no	—
38	C₂²⋊#87938	—	2	4	no	—
39	C₄×#87938	—	2	4	no	—
40	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
45	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
46	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
47	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	complemented
50	#13108053	C₂×C₄	1	4	yes	complemented
51	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
52	#13108051	C₂×C₄	1	4	yes	—
53	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
54	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	complemented
55	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	complemented
57	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
58	C₄²×C₁₀₁	C₂×C₄	1	2	yes	complemented
59	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₁₀₁	—	2	2	no	—
61	C₂²×D₁₀₁	—	2	2	no	—
62	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
64	C₂²×D₁₀₁	C₂×Q₈	1	2	yes	—
65	C₂²×D₁₀₁	C₂⁴	1	1	yes	characteristic
66	C₂×#87938	—	2	2	no	—
67	C₂×#87938	—	2	2	no	—
68	C₂×#87938	—	2	4	no	—
69	C₂×#87938	—	2	4	no	—
70	C₂×#87938	—	2	8	no	—
71	C₂×#87938	—	2	4	no	—
72	C₂×#87938	—	4	8	no	—
73	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	complemented
74	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₀₁	C₄⋊C₄	1	4	yes	complemented
78	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₀₁	—	2	4	no	—
82	C₄×D₁₀₁	—	2	4	no	—
83	C₄×D₁₀₁	C₄⋊C₄	1	4	yes	complemented
84	C₄×D₁₀₁	C₄⋊C₄	1	4	yes	complemented
85	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
86	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
87	C₄×D₁₀₁	C₂×Q₈	1	2	yes	—
88	C₄×D₁₀₁	C₂×Q₈	1	2	yes	—
89	C₂²×C₁₀₁	—	2	2	no	—
90	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₁₀₁	—	2	2	no	—
93	C₂×D₁₀₁	—	2	2	no	—
94	C₂×D₁₀₁	—	2	2	no	—
95	C₂×D₁₀₁	—	2	4	no	—
96	C₂×D₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₁₀₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×D₁₀₁	H(5,𝔽₂)	1	2	yes	—
100	C₂×D₁₀₁	2₋¹⁺⁴	1	2	yes	—
101	#87938	—	4	2	no	—
102	#87938	—	4	4	no	—
103	#87938	—	4	2	no	—
104	C₄×C₁₀₁	—	2	4	no	—
105	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
106	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
107	Dic₁₀₁	—	2	4	no	—
108	Dic₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
109	Dic₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₁₀₁	—	2	2	no	—
111	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₁₀₁	#520	1	2	yes	—
113	D₁₀₁	—	2	2	no	—
114	D₁₀₁	#535	1	1	yes	characteristic
115	D₁₀₁	#536	1	1	yes	characteristic
116	D₁₀₁	#544	1	2	yes	—
117	#2317	—	101	1	no	maximal, Sylow
118	C₁₀₁	#2317	1	1	yes	characteristic, complemented, Sylow
119	C₂×#117	—	101	1	no	—
120	C₂×#117	—	101	1	no	—
121	C₂²×C₄⋊C₄	—	101	1	no	—
122	#389	—	101	4	no	—
123	#385	—	101	4	no	—
124	#384	—	101	2	no	—
125	#382	—	101	2	no	—
126	C₂³×C₄	—	101	1	no	—
127	C₂³×C₄	—	101	2	no	—
128	#117	—	101	1	no	—
129	#117	—	101	2	no	—
130	#117	—	101	2	no	—
131	#117	—	101	2	no	—
132	#117	—	202	4	no	—
133	#117	#87938	1	1	yes	characteristic, complemented
134	C₂×C₄⋊C₄	—	101	2	no	—
135	C₂×C₄⋊C₄	—	101	8	no	—
136	C₂×C₄⋊C₄	—	101	2	no	—
137	C₂×C₄⋊C₄	—	101	2	no	—
138	C₂×C₄⋊C₄	—	101	1	no	—
139	C₂×C₄⋊C₄	—	101	1	no	—
140	C₂×C₄⋊C₄	—	202	4	no	—
141	C₂×C₂²⋊C₄	—	101	2	no	—
142	C₂×C₂²⋊C₄	—	101	2	no	—
143	C₂×C₄²	—	101	2	no	—
144	C₂×C₄²	—	101	2	no	—
145	#95	—	101	8	no	—
146	C₂⁴	—	101	1	no	—
147	C₂²×C₄	—	101	1	no	—
148	C₂²×C₄	—	101	4	no	—
149	C₂²×C₄	—	101	4	no	—
150	C₂²×C₄	—	101	1	no	—
151	C₂²×C₄	—	101	1	no	—
152	C₂²×C₄	—	101	1	no	—
153	C₂²×C₄	—	101	1	no	—
154	C₂²×C₄	—	101	4	no	—
155	C₂²×C₄	—	101	2	no	—
156	C₂²×C₄	—	101	4	no	—
157	C₂²×C₄	—	101	2	no	—
158	C₂²×C₄	—	404	4	no	—
159	C₂²×C₄	—	202	4	no	—
160	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
161	C₄⋊C₄	—	101	4	no	—
162	C₄⋊C₄	—	101	2	no	—
163	C₄⋊C₄	—	404	8	no	—
164	C₄⋊C₄	—	202	4	no	—
165	C₄⋊C₄	—	202	4	no	—
166	C₄⋊C₄	—	202	2	no	—
167	C₄⋊C₄	—	202	2	no	—
168	C₄⋊C₄	C₂×#87938	1	2	yes	complemented
169	C₂²⋊C₄	—	101	2	no	—
170	C₂²⋊C₄	—	101	4	no	—
171	C₂²⋊C₄	—	202	4	no	—
172	C₂²⋊C₄	C₂×#87938	1	2	yes	—
173	C₄²	—	101	4	no	—
174	C₄²	—	101	2	no	—
175	C₄²	—	202	4	no	—
176	C₄²	C₂×#87938	1	2	yes	complemented
177	C₂³	—	101	1	no	—
178	C₂³	—	101	2	no	—
179	C₂³	—	101	2	no	—
180	C₂³	—	101	1	no	—
181	C₂³	—	202	2	no	—
182	C₂³	—	202	2	no	—
183	C₂³	C₂²×#87938	1	1	yes	characteristic
184	C₂×C₄	—	101	4	no	—
185	C₂×C₄	—	101	1	no	—
186	C₂×C₄	—	101	1	no	—
187	C₂×C₄	—	101	2	no	—
188	C₂×C₄	—	101	2	no	—
189	C₂×C₄	—	101	4	no	—
190	C₂×C₄	—	101	4	no	—
191	C₂×C₄	—	101	2	no	—
192	C₂×C₄	—	101	2	no	—
193	C₂×C₄	—	101	4	no	—
194	C₂×C₄	—	404	8	no	—
195	C₂×C₄	—	202	4	no	—
196	C₂×C₄	—	202	4	no	—
197	C₂×C₄	—	202	2	no	—
198	C₂×C₄	—	202	2	no	—
199	C₂×C₄	—	202	4	no	—
200	C₂×C₄	—	202	8	no	—
201	C₂×C₄	—	202	4	no	—
202	C₂×C₄	—	202	4	no	—
203	C₂×C₄	C₄⋊#87938	1	4	yes	complemented
204	C₂×C₄	C₂²×#87938	1	4	yes	—
205	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
206	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	—	101	2	no	—
209	C₂²	—	101	2	no	—
210	C₂²	—	101	1	no	—
211	C₂²	—	101	1	no	—
212	C₂²	—	101	2	no	—
213	C₂²	—	202	2	no	—
214	C₂²	—	202	2	no	—
215	C₂²	—	202	4	no	—
216	C₂²	—	202	2	no	—
217	C₂²	C₂×C₄⋊#87938	1	2	yes	—
218	C₂²	C₂³×#87938	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	101	2	no	—
221	C₄	—	101	2	no	—
222	C₄	—	404	4	no	—
223	C₄	—	404	2	no	—
224	C₄	—	404	2	no	—
225	C₄	—	202	4	no	—
226	C₄	C₂×C₄⋊#87938	1	2	yes	—
227	C₄	C₂×C₄⋊#87938	1	2	yes	—
228	C₂	—	2	2	no	—
229	C₂	—	101	1	no	—
230	C₂	—	101	1	no	—
231	C₂	—	101	2	no	—
232	C₂	—	202	2	no	—
233	C₂	C₂²×C₄⋊#87938	1	1	yes	characteristic
234	C₂	#20266246	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#117×C₁₀₁	C₄
prime radical	C₂³×D₁₀₁	C₂³
cosocle	C₂²×D₁₀₁	C₂⁴
hypercenter	#117	#87938
hypocenter	C₁₀₁	#2317
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20266128	C₂	semidirect
5. #20266123	C₂	semidirect
6. #20266121	C₂	semidirect
20. #15692214	C₄	semidirect
21. #15692213	C₄	semidirect
22. #15692212	C₄	semidirect
23. #15692211	C₄	semidirect
27. #117×C₁₀₁	C₄	semidirect
49. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
50. #13108053	C₂×C₄	semidirect
53. C₄×Dic₁₀₁	C₂×C₄	semidirect
54. C₄×Dic₁₀₁	C₂×C₄	semidirect
56. C₄⋊C₄×C₁₀₁	C₂×C₄	semidirect
58. C₄²×C₁₀₁	C₂×C₄	semidirect
73. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
77. C₂×Dic₁₀₁	C₄⋊C₄	semidirect
83. C₄×D₁₀₁	C₄⋊C₄	semidirect
84. C₄×D₁₀₁	C₄⋊C₄	semidirect
118. C₁₀₁	#2317	semidirect
133. #117	#87938	semidirect
168. C₄⋊C₄	C₂×#87938	semidirect
176. C₄²	C₂×#87938	semidirect
203. C₂×C₄	C₄⋊#87938	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692334	2
C₂²×C₄⋊#87938	2
#20266128	2
#20266124	2
#20266123	2
#20266121	2
#117×D₁₀₁	2
#2317	101