Subgroups — #27173862

Order 12928.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692234	C₂	1	1	yes	maximal, characteristic
2	#20265081	C₂	1	2	yes	maximal, complemented
3	#20265072	C₂	1	2	yes	maximal, complemented
4	#20265062	C₂	1	2	yes	maximal, complemented
5	#20265058	C₂	1	2	yes	maximal, complemented
6	#20265049	C₂	1	2	yes	maximal
7	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal
8	C₂×#15692162	C₂	1	1	yes	maximal, characteristic
9	#380×C₁₀₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	#15692234	C₄	1	8	yes	complemented
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×#13108053	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
21	#15692162	C₂²	1	2	yes	—
22	#15692162	C₂²	1	2	yes	—
23	#15692162	C₂²	1	2	yes	—
24	#15692162	C₂²	1	2	yes	—
25	#15692162	C₂²	1	2	yes	—
26	#15692162	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
31	#95×C₁₀₁	C₂²	1	2	yes	—
32	C₂×Dic₂₀₂	—	4	1	no	—
33	C₂×Dic₂₀₂	—	4	1	no	—
34	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
35	C₂²×Dic₁₀₁	—	2	2	no	—
36	C₂²×Dic₁₀₁	—	4	1	no	—
37	C₂²×Dic₁₀₁	—	4	1	no	—
38	C₂²×Dic₁₀₁	—	4	1	no	—
39	C₂²×Dic₁₀₁	—	4	1	no	—
40	C₂²×Dic₁₀₁	Q₈	1	4	yes	complemented
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
46	#13108053	—	4	2	no	—
47	#13108053	—	4	2	no	—
48	#13108053	C₂×C₄	1	8	yes	complemented
49	C₂²⋊Dic₁₀₁	C₂×C₄	1	8	yes	—
50	C₄×Dic₁₀₁	—	4	2	no	—
51	C₄×Dic₁₀₁	—	4	2	no	—
52	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	—	4	1	no	—
54	C₂²×C₄×C₁₀₁	—	4	1	no	—
55	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₁	—	4	1	no	—
61	C₄⋊C₄×C₁₀₁	—	4	1	no	—
62	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
63	C₂³×C₁₀₁	—	2	1	no	—
64	C₂³×C₁₀₁	—	2	1	no	—
65	C₂³×C₁₀₁	—	2	1	no	—
66	C₂³×C₁₀₁	—	2	1	no	—
67	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
68	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
69	C₂³×C₁₀₁	#41	1	2	yes	—
70	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
71	C₂×C₄×C₁₀₁	—	2	1	no	—
72	C₂×C₄×C₁₀₁	—	2	1	no	—
73	C₂×C₄×C₁₀₁	—	2	1	no	—
74	C₂×C₄×C₁₀₁	—	2	1	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	—	2	2	no	—
81	C₂×C₄×C₁₀₁	—	4	4	no	—
82	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₀₁	—	2	2	no	—
85	C₂×Dic₁₀₁	—	2	2	no	—
86	C₂×Dic₁₀₁	—	2	2	no	—
87	C₂×Dic₁₀₁	—	2	2	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	—	4	4	no	—
94	C₂×Dic₁₀₁	—	4	4	no	—
95	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
96	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
97	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
98	Dic₂₀₂	—	4	4	no	—
99	C₂²×C₁₀₁	—	2	2	no	—
100	C₂²×C₁₀₁	—	2	2	no	—
101	C₂²×C₁₀₁	—	2	2	no	—
102	C₂²×C₁₀₁	—	2	2	no	—
103	C₂²×C₁₀₁	—	2	2	no	—
104	C₂²×C₁₀₁	—	2	2	no	—
105	C₂²×C₁₀₁	C₄×Q₈	1	4	yes	—
106	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
107	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₁₀₁	—	2	2	no	—
114	C₄×C₁₀₁	—	2	2	no	—
115	C₄×C₁₀₁	—	4	2	no	—
116	C₄×C₁₀₁	—	4	2	no	—
117	Dic₁₀₁	—	2	4	no	—
118	Dic₁₀₁	—	2	4	no	—
119	Dic₁₀₁	—	2	4	no	—
120	Dic₁₀₁	—	4	2	no	—
121	C₂×C₁₀₁	—	2	4	no	—
122	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₀₁	#518	1	1	yes	characteristic
124	C₂×C₁₀₁	#520	1	1	yes	characteristic
125	C₂×C₁₀₁	#538	1	1	yes	characteristic
126	C₂×C₁₀₁	#539	1	1	yes	characteristic
127	C₂×C₁₀₁	#543	1	1	yes	characteristic
128	C₂×C₁₀₁	#544	1	1	yes	characteristic
129	#2329	—	101	1	no	maximal, Sylow
130	C₁₀₁	#2329	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₂²⋊Q₈	—	101	1	no	—
132	#391	—	101	2	no	—
133	#389	—	101	2	no	—
134	#387	—	101	2	no	—
135	#387	—	101	2	no	—
136	#385	—	101	2	no	—
137	#380	D₁₀₁	1	1	yes	characteristic
138	C₄×C₂²⋊C₄	—	101	2	no	—
139	C₂×#95	—	101	1	no	—
140	C₂²×Q₈	—	101	1	no	—
141	C₂³×C₄	—	101	1	no	—
142	C₂³×C₄	—	101	1	no	—
143	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
144	C₂²⋊Q₈	—	101	8	no	—
145	C₂×C₄⋊C₄	—	101	1	no	—
146	C₂×C₄⋊C₄	—	101	2	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	2	no	—
149	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	—	101	2	no	—
152	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
153	C₂×C₄²	—	101	2	no	—
154	C₂×C₄²	—	101	2	no	—
155	#95	—	101	2	no	—
156	#95	—	101	2	no	—
157	#95	—	101	2	no	—
158	#95	—	101	2	no	—
159	#95	—	101	2	no	—
160	#95	—	101	2	no	—
161	#95	C₂×D₁₀₁	1	2	yes	—
162	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
163	C₂×Q₈	—	101	4	no	—
164	C₂×Q₈	—	404	1	no	—
165	C₂×Q₈	—	404	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	101	4	no	—
169	C₂²×C₄	—	101	2	no	—
170	C₂²×C₄	—	101	2	no	—
171	C₂²×C₄	—	101	2	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	404	1	no	—
174	C₂²×C₄	—	404	1	no	—
175	C₂²×C₄	—	404	1	no	—
176	C₂²×C₄	—	404	1	no	—
177	C₂²×C₄	—	202	2	no	—
178	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
179	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
180	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
181	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	101	8	no	—
186	C₄⋊C₄	—	101	4	no	—
187	C₄⋊C₄	—	404	2	no	—
188	C₄⋊C₄	—	404	2	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	101	8	no	—
191	C₄²	—	404	2	no	—
192	C₄²	—	404	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
198	C₂³	Q₈×D₁₀₁	1	2	yes	—
199	C₂³	#13108073	1	2	yes	—
200	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
201	Q₈	—	404	4	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	101	4	no	—
214	C₂×C₄	—	101	4	no	—
215	C₂×C₄	—	101	4	no	—
216	C₂×C₄	—	404	4	no	—
217	C₂×C₄	—	404	4	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	—	202	2	no	—
225	C₂×C₄	—	202	2	no	—
226	C₂×C₄	—	202	2	no	—
227	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
228	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	#15692220	1	4	yes	—
236	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
237	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
238	C₂²	C₂×#13108073	1	1	yes	characteristic
239	C₂²	#15692300	1	1	yes	characteristic
240	C₂²	#15692300	1	1	yes	characteristic
241	C₂²	#15692301	1	1	yes	characteristic
242	C₂²	#15692302	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	404	2	no	—
248	C₄	—	202	4	no	—
249	C₄	—	202	4	no	—
250	C₄	—	202	4	no	—
251	C₂	—	2	4	no	—
252	C₂	C₂×#15692220	1	1	yes	characteristic
253	C₂	#20265650	1	1	yes	characteristic
254	C₂	#20265657	1	1	yes	characteristic
255	C₂	#20265764	1	1	yes	characteristic
256	C₂	#20265772	1	1	yes	characteristic
257	C₂	#20265802	1	1	yes	characteristic
258	C₂	#20265806	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#380×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	#380	D₁₀₁
hypocenter	C₁₀₁	#2329
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #380

Complements

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

N	H ≅ G/N	type
2. #20265081	C₂	semidirect
3. #20265072	C₂	semidirect
4. #20265062	C₂	semidirect
5. #20265058	C₂	semidirect
13. #15692234	C₄	semidirect
34. C₂×Dic₂₀₂	C₂×C₄	semidirect
40. C₂²×Dic₁₀₁	Q₈	semidirect
45. C₄⋊Dic₁₀₁	C₂×C₄	semidirect
48. #13108053	C₂×C₄	semidirect
130. C₁₀₁	#2329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692234	2
#20265081	2
#20265072	2
#20265062	2
#20265058	2
#20265049	2
C₂²⋊C₄×Dic₁₀₁	2
C₂×#15692162	2
#380×C₁₀₁	2
#2329	101