Subgroups — #27174391

Order 12928.

560 conjugacy classes of subgroups.

The subgroup lattice has 276 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	1	yes	maximal, characteristic, complemented
2	#20265116	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265113	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265107	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265105	C₂	1	2	yes	maximal, complemented
6	#20265102	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265091	C₂	1	2	yes	maximal
8	C₄×#13108053	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊Dic₁₀₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	#15692214	—	2	4	no	—
13	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×#13108053	C₂²	1	2	yes	—
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	#15692162	C₂²	1	1	yes	characteristic
24	#15692162	C₂²	1	2	yes	—
25	#15692162	C₂²	1	2	yes	—
26	#15692162	C₂²	1	1	yes	characteristic
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₀₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₄⋊Dic₁₀₁	—	2	2	no	—
42	C₄⋊Dic₁₀₁	—	2	2	no	—
43	#13108053	—	2	4	no	—
44	#13108053	—	2	4	no	—
45	C₂²⋊Dic₁₀₁	—	2	4	no	—
46	C₂²⋊Dic₁₀₁	—	2	4	no	—
47	C₂²⋊Dic₁₀₁	—	2	4	no	—
48	C₂²⋊Dic₁₀₁	—	2	4	no	—
49	C₄×Dic₁₀₁	—	2	4	no	—
50	C₄×Dic₁₀₁	—	2	4	no	—
51	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
52	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₀₁	—	2	2	no	—
54	C₂×D₄×C₁₀₁	—	2	2	no	—
55	C₂×D₄×C₁₀₁	—	2	2	no	—
56	C₂²×C₄×C₁₀₁	—	2	4	no	—
57	C₂²×C₄×C₁₀₁	—	2	2	no	—
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₁	—	2	2	no	—
66	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
68	C₄²×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	—	2	1	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	—	2	1	no	—
73	C₂³×C₁₀₁	—	2	1	no	—
74	C₂³×C₁₀₁	—	2	1	no	—
75	C₂³×C₁₀₁	—	2	2	no	—
76	C₂³×C₁₀₁	—	2	2	no	—
77	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
78	C₂³×C₁₀₁	#41	1	2	yes	—
79	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
80	D₄×C₁₀₁	—	2	8	no	—
81	C₂×C₄×C₁₀₁	—	2	1	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	2	1	no	—
84	C₂×C₄×C₁₀₁	—	2	8	no	—
85	C₂×C₄×C₁₀₁	—	2	1	no	—
86	C₂×C₄×C₁₀₁	—	2	1	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	4	4	no	—
90	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	#41	1	4	yes	—
93	C₂×C₄×C₁₀₁	#41	1	2	yes	—
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	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	2	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	—	2	2	no	—
105	C₂×Dic₁₀₁	#41	1	4	yes	—
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	—	2	2	no	—
108	C₂²×C₁₀₁	—	2	2	no	—
109	C₂²×C₁₀₁	—	2	4	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	4	1	no	—
112	C₂²×C₁₀₁	—	4	2	no	—
113	C₂²×C₁₀₁	—	4	2	no	—
114	C₂²×C₁₀₁	—	4	2	no	—
115	C₂²×C₁₀₁	—	4	1	no	—
116	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
118	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
119	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
121	C₄×C₁₀₁	—	2	2	no	—
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	—	4	1	no	—
124	C₄×C₁₀₁	—	4	1	no	—
125	C₄×C₁₀₁	#123	1	4	yes	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	—	4	2	no	—
128	Dic₁₀₁	—	4	2	no	—
129	Dic₁₀₁	—	4	2	no	—
130	C₂×C₁₀₁	—	4	2	no	—
131	C₂×C₁₀₁	—	4	2	no	—
132	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
133	C₂×C₁₀₁	#525	1	1	yes	characteristic
134	C₂×C₁₀₁	#529	1	2	yes	—
135	C₂×C₁₀₁	#529	1	2	yes	—
136	C₂×C₁₀₁	#532	1	1	yes	characteristic
137	#2395	—	101	1	no	maximal, Sylow
138	C₁₀₁	#2395	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic
140	C₂×#117	—	101	1	no	—
141	#400	—	101	1	no	—
142	#399	—	101	1	no	—
143	#398	—	101	1	no	—
144	#397	—	101	1	no	—
145	#397	—	101	2	no	—
146	#394	—	101	2	no	—
147	C₄×C₄⋊C₄	—	101	2	no	—
148	C₄×C₂²⋊C₄	—	101	2	no	—
149	C₄×C₂²⋊C₄	—	101	1	no	—
150	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
153	C₄×D₄	—	2	4	no	—
154	#117	—	202	4	no	—
155	C₂×C₄⋊C₄	—	101	1	no	—
156	C₂×C₄⋊C₄	—	101	1	no	—
157	C₂×C₄⋊C₄	—	101	2	no	—
158	C₂×C₄⋊C₄	—	101	2	no	—
159	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	101	2	no	—
161	C₂×C₂²⋊C₄	—	101	2	no	—
162	C₂×C₂²⋊C₄	—	101	2	no	—
163	C₂×C₂²⋊C₄	—	101	2	no	—
164	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂×C₄²	—	101	2	no	—
167	C₂×C₄²	—	101	2	no	—
168	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
169	#95	—	101	2	no	—
170	#95	—	101	1	no	—
171	#95	—	101	2	no	—
172	#95	—	101	1	no	—
173	#95	—	101	2	no	—
174	#95	—	101	2	no	—
175	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
176	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	101	2	no	—
184	C₂²×C₄	—	101	2	no	—
185	C₂²×C₄	—	101	2	no	—
186	C₂²×C₄	—	101	2	no	—
187	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
188	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	202	4	no	—
195	C₄⋊C₄	—	202	2	no	—
196	C₄⋊C₄	—	202	2	no	—
197	C₄⋊C₄	—	202	4	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	202	4	no	—
201	C₂²⋊C₄	—	202	4	no	—
202	C₂²⋊C₄	—	202	4	no	—
203	C₂²⋊C₄	—	202	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	202	4	no	—
206	C₄²	—	202	4	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
216	C₂³	#13108074	1	2	yes	—
217	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
218	D₄	—	2	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	101	4	no	—
229	C₂×C₄	—	202	2	no	—
230	C₂×C₄	—	202	2	no	—
231	C₂×C₄	—	202	2	no	—
232	C₂×C₄	—	202	2	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×C₂²⋊D₁₀₁	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
242	C₂×C₄	#13108073	1	4	yes	—
243	C₂×C₄	#13108074	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
255	C₂²	#41×D₁₀₁	1	2	yes	—
256	C₂²	C₂×#13108073	1	2	yes	—
257	C₂²	C₂×#13108074	1	1	yes	characteristic
258	C₂²	C₂×#13108074	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	404	2	no	—
264	C₄	—	404	2	no	—
265	C₄	—	404	2	no	—
266	C₄	—	202	4	no	—
267	C₄	#15692236	1	4	yes	—
268	C₂	—	4	2	no	—
269	C₂	—	4	2	no	—
270	C₂	C₂×#15692236	1	1	yes	characteristic
271	C₂	#20265688	1	1	yes	characteristic
272	C₂	#20265707	1	2	yes	—
273	C₂	#20265709	1	2	yes	—
274	C₂	#20265723	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	#2395
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15692214	C₂	semidirect
2. #20265116	C₂	semidirect
3. #20265113	C₂	semidirect
4. #20265107	C₂	semidirect
5. #20265105	C₂	semidirect
6. #20265102	C₂	semidirect
8. C₄×#13108053	C₂	semidirect
9. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
10. C₄×C₂²⋊Dic₁₀₁	C₂	semidirect
138. C₁₀₁	#2395	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
#20265116	2
#20265113	2
#20265107	2
#20265105	2
#20265102	2
#20265091	2
C₄×#13108053	2
C₄×C₂²⋊Dic₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
#2395	101