Subgroups — #27173751

Order 12928.

600 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692211	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692211	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13108053	C₂	1	1	yes	maximal, characteristic
4	#20265074	C₂	1	2	yes	maximal, complemented
5	#20265072	C₂	1	2	yes	maximal, complemented
6	#20265052	C₂	1	1	yes	maximal, characteristic
7	#20265050	C₂	1	1	yes	maximal, characteristic
8	#20265049	C₂	1	1	yes	maximal, characteristic
9	#20265046	C₂	1	2	yes	maximal, complemented
10	#20265041	C₂	1	2	yes	maximal, complemented
11	#385×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	#15692211	—	2	4	no	—
15	#15692211	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×#13108053	—	2	4	no	—
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	2	yes	—
20	C₂×#13108053	C₂²	1	1	yes	characteristic
21	C₂×#13108053	C₂²	1	2	yes	—
22	C₂×#13108053	C₂²	1	2	yes	—
23	C₂×#13108053	C₂²	1	2	yes	—
24	C₂×#13108053	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	#95×C₁₀₁	C₂²	1	2	yes	—
38	C₂²×Dic₁₀₁	—	2	2	no	—
39	C₂²×Dic₁₀₁	—	4	2	no	—
40	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
41	C₂²×Dic₁₀₁	D₄	1	4	yes	—
42	C₂²×Dic₁₀₁	Q₈	1	4	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₀₁	—	4	2	no	—
50	#13108053	—	2	4	no	—
51	#13108053	—	2	4	no	—
52	#13108053	—	2	2	no	—
53	#13108053	—	2	2	no	—
54	#13108053	—	4	2	no	—
55	#13108053	—	4	2	no	—
56	#13108053	C₂×C₄	1	8	yes	complemented
57	C₂²⋊Dic₁₀₁	—	2	2	no	—
58	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
59	C₄×Dic₁₀₁	—	2	2	no	—
60	C₄×Dic₁₀₁	—	2	2	no	—
61	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
62	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	—	4	2	no	—
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₄⋊C₄×C₁₀₁	—	4	2	no	—
70	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
71	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
72	C₂³×C₁₀₁	—	2	2	no	—
73	C₂³×C₁₀₁	—	2	2	no	—
74	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
76	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
77	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₁₀₁	—	2	2	no	—
79	C₂×C₄×C₁₀₁	—	2	2	no	—
80	C₂×C₄×C₁₀₁	—	2	1	no	—
81	C₂×C₄×C₁₀₁	—	2	1	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	4	4	no	—
88	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	1	no	—
92	C₂×Dic₁₀₁	—	2	1	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	4	4	no	—
99	C₂×Dic₁₀₁	C₄⋊C₄	1	16	yes	complemented
100	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
101	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
102	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
104	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
105	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
106	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
107	C₂²×C₁₀₁	—	2	4	no	—
108	C₂²×C₁₀₁	—	2	2	no	—
109	C₂²×C₁₀₁	—	2	2	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
112	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
113	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
115	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
117	C₄×C₁₀₁	—	2	4	no	—
118	C₄×C₁₀₁	—	4	1	no	—
119	C₄×C₁₀₁	—	4	2	no	—
120	C₄×C₁₀₁	—	4	1	no	—
121	Dic₁₀₁	—	2	4	no	—
122	Dic₁₀₁	—	4	1	no	—
123	Dic₁₀₁	—	4	2	no	—
124	Dic₁₀₁	—	4	1	no	—
125	Dic₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₁₀₁	—	2	4	no	—
128	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₁₀₁	#520	1	2	yes	—
130	C₂×C₁₀₁	#535	1	1	yes	characteristic
131	C₂×C₁₀₁	#536	1	1	yes	characteristic
132	C₂×C₁₀₁	#544	1	2	yes	—
133	#2317	—	101	1	no	maximal, Sylow
134	C₁₀₁	#2317	1	1	yes	characteristic, complemented, Sylow
135	C₂×#117	—	101	1	no	—
136	C₂×#117	—	101	1	no	—
137	C₂²×C₄⋊C₄	—	101	1	no	—
138	#389	—	101	2	no	—
139	#389	—	101	2	no	—
140	#385	—	101	1	no	—
141	#385	—	101	1	no	—
142	#385	—	101	1	no	—
143	#385	D₁₀₁	1	1	yes	characteristic
144	#384	—	101	2	no	—
145	#382	—	101	2	no	—
146	C₂³×C₄	—	101	1	no	—
147	C₂³×C₄	—	101	1	no	—
148	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
149	#117	—	101	8	no	—
150	#117	—	202	4	no	—
151	C₂×C₄⋊C₄	—	101	2	no	—
152	C₂×C₄⋊C₄	—	101	2	no	—
153	C₂×C₄⋊C₄	—	101	2	no	—
154	C₂×C₄⋊C₄	—	101	2	no	—
155	C₂×C₄⋊C₄	—	101	2	no	—
156	C₂×C₄⋊C₄	—	101	2	no	—
157	C₂×C₄⋊C₄	—	101	1	no	—
158	C₂×C₄⋊C₄	—	101	1	no	—
159	C₂×C₄⋊C₄	—	202	4	no	—
160	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
161	C₂×C₂²⋊C₄	—	101	1	no	—
162	C₂×C₂²⋊C₄	—	101	1	no	—
163	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂×C₄²	—	101	2	no	—
166	C₂×C₄²	—	101	1	no	—
167	C₂×C₄²	—	101	1	no	—
168	#95	—	101	2	no	—
169	#95	—	101	2	no	—
170	#95	—	101	2	no	—
171	#95	C₂×D₁₀₁	1	2	yes	—
172	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	101	4	no	—
176	C₂²×C₄	—	101	4	no	—
177	C₂²×C₄	—	101	4	no	—
178	C₂²×C₄	—	101	1	no	—
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	101	2	no	—
181	C₂²×C₄	—	101	1	no	—
182	C₂²×C₄	—	101	2	no	—
183	C₂²×C₄	—	101	1	no	—
184	C₂²×C₄	—	404	2	no	—
185	C₂²×C₄	—	202	2	no	—
186	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
188	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	101	8	no	—
192	C₄⋊C₄	—	404	2	no	—
193	C₄⋊C₄	—	404	2	no	—
194	C₄⋊C₄	—	404	2	no	—
195	C₄⋊C₄	—	202	4	no	—
196	C₄⋊C₄	—	202	4	no	—
197	C₄⋊C₄	—	202	2	no	—
198	C₄⋊C₄	—	202	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	101	4	no	—
201	C₂²⋊C₄	—	202	2	no	—
202	C₂²⋊C₄	C₄×D₁₀₁	1	4	yes	—
203	C₄²	—	101	8	no	—
204	C₄²	—	202	2	no	—
205	C₄²	—	202	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
209	C₂³	D₄×D₁₀₁	1	2	yes	—
210	C₂³	Q₈×D₁₀₁	1	2	yes	—
211	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	101	2	no	—
223	C₂×C₄	—	101	2	no	—
224	C₂×C₄	—	101	2	no	—
225	C₂×C₄	—	101	2	no	—
226	C₂×C₄	—	101	2	no	—
227	C₂×C₄	—	101	2	no	—
228	C₂×C₄	—	101	16	no	—
229	C₂×C₄	—	101	4	no	—
230	C₂×C₄	—	404	4	no	—
231	C₂×C₄	—	202	1	no	—
232	C₂×C₄	—	202	1	no	—
233	C₂×C₄	—	202	2	no	—
234	C₂×C₄	—	202	2	no	—
235	C₂×C₄	—	202	2	no	—
236	C₂×C₄	—	202	2	no	—
237	C₂×C₄	—	202	2	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	—	202	2	no	—
240	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	—
246	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
247	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
248	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
249	C₂²	#15692300	1	2	yes	—
250	C₂²	#15692302	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	101	4	no	—
256	C₄	—	101	4	no	—
257	C₄	—	404	2	no	—
258	C₄	—	404	1	no	—
259	C₄	—	404	1	no	—
260	C₄	—	202	4	no	—
261	C₂	—	2	4	no	—
262	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
263	C₂	#20265656	1	2	yes	—
264	C₂	#20265742	1	1	yes	characteristic
265	C₂	#20265749	1	1	yes	characteristic
266	C₂	#20265806	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#385×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	#385	D₁₀₁
hypocenter	C₁₀₁	#2317
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #385

Complements

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

N	H ≅ G/N	type
4. #20265074	C₂	semidirect
5. #20265072	C₂	semidirect
9. #20265046	C₂	semidirect
10. #20265041	C₂	semidirect
15. #15692211	C₄	semidirect
56. #13108053	C₂×C₄	semidirect
61. C₄×Dic₁₀₁	C₂×C₄	semidirect
99. C₂×Dic₁₀₁	C₄⋊C₄	semidirect
134. C₁₀₁	#2317	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692211	2
C₂²×#13108053	2
#20265074	2
#20265072	2
#20265052	2
#20265050	2
#20265049	2
#20265046	2
#20265041	2
#385×C₁₀₁	2
#2317	101