Subgroups — #27169360

Order 12928.

656 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108053	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692162	C₂	1	1	yes	maximal, characteristic
4	C₂×#15692162	C₂	1	2	yes	maximal
5	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
7	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×#13108053	—	2	4	no	—
10	C₂×#13108053	—	2	4	no	—
11	C₂×#13108053	C₄	1	8	yes	complemented
12	C₂×C₄×Dic₁₀₁	—	2	4	no	—
13	C₂×C₄×Dic₁₀₁	C₄	1	4	yes	complemented
14	#15692162	—	2	4	no	—
15	#15692162	—	2	2	no	—
16	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
18	#95×C₁₀₁	—	2	2	no	—
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₂²×Dic₁₀₁	—	2	2	no	—
24	C₂²×Dic₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	C₂²×Dic₁₀₁	—	2	2	no	—
27	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
28	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
29	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₀₁	D₄	1	2	yes	—
32	C₂²×Dic₁₀₁	D₄	1	4	yes	—
33	C₂²×Dic₁₀₁	D₄	1	2	yes	—
34	C₂²×Dic₁₀₁	Q₈	1	2	yes	—
35	C₂²×Dic₁₀₁	Q₈	1	2	yes	—
36	#13108053	—	2	16	no	—
37	C₄×Dic₁₀₁	—	2	8	no	—
38	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₁	—	2	2	no	—
40	C₂²×C₄×C₁₀₁	—	2	2	no	—
41	C₂²×C₄×C₁₀₁	—	2	2	no	—
42	C₂²×C₄×C₁₀₁	—	2	2	no	—
43	C₂²×C₄×C₁₀₁	—	2	2	no	—
44	C₂²×C₄×C₁₀₁	—	2	2	no	—
45	C₂²×C₄×C₁₀₁	—	2	2	no	—
46	C₂²×C₄×C₁₀₁	—	2	2	no	—
47	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
50	C₂²×C₄×C₁₀₁	Q₈	1	2	yes	—
51	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
55	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₁₀₁	#41	1	2	yes	—
61	C₂³×C₁₀₁	#41	1	2	yes	—
62	C₂×C₄×C₁₀₁	—	2	2	no	—
63	C₂×C₄×C₁₀₁	—	2	2	no	—
64	C₂×C₄×C₁₀₁	—	2	4	no	—
65	C₂×C₄×C₁₀₁	—	2	4	no	—
66	C₂×C₄×C₁₀₁	—	2	4	no	—
67	C₂×C₄×C₁₀₁	—	2	4	no	—
68	C₂×C₄×C₁₀₁	—	2	4	no	—
69	C₂×C₄×C₁₀₁	—	4	2	no	—
70	C₂×C₄×C₁₀₁	—	4	2	no	—
71	C₂×C₄×C₁₀₁	—	4	2	no	—
72	C₂×C₄×C₁₀₁	—	4	2	no	—
73	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	complemented
74	C₂×Dic₁₀₁	—	2	2	no	—
75	C₂×Dic₁₀₁	—	2	2	no	—
76	C₂×Dic₁₀₁	—	2	4	no	—
77	C₂×Dic₁₀₁	—	2	4	no	—
78	C₂×Dic₁₀₁	—	2	4	no	—
79	C₂×Dic₁₀₁	—	2	4	no	—
80	C₂×Dic₁₀₁	—	2	4	no	—
81	C₂×Dic₁₀₁	—	4	2	no	—
82	C₂×Dic₁₀₁	—	4	2	no	—
83	C₂×Dic₁₀₁	—	4	2	no	—
84	C₂×Dic₁₀₁	—	4	2	no	—
85	C₂×Dic₁₀₁	C₄²	1	4	yes	—
86	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
87	C₂×Dic₁₀₁	C₂²⋊C₄	1	8	yes	—
88	C₂×Dic₁₀₁	C₄⋊C₄	1	4	yes	—
89	C₂×Dic₁₀₁	C₄⋊C₄	1	8	yes	—
90	C₂×Dic₁₀₁	C₄⋊C₄	1	4	yes	complemented
91	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
92	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₀₁	#117	1	1	yes	characteristic
99	C₂²×C₁₀₁	#117	1	1	yes	characteristic
100	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
101	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
102	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
103	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
104	C₂²×C₁₀₁	#121	1	2	yes	—
105	C₂²×C₁₀₁	#121	1	2	yes	—
106	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₁₀₁	#124	1	2	yes	—
109	C₂²×C₁₀₁	#125	1	2	yes	—
110	C₂²×C₁₀₁	#127	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
114	C₄×C₁₀₁	—	2	4	no	—
115	C₄×C₁₀₁	—	4	4	no	—
116	Dic₁₀₁	—	2	4	no	—
117	Dic₁₀₁	—	4	4	no	—
118	Dic₁₀₁	#95	1	8	yes	complemented
119	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
120	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₁₀₁	#380	1	1	yes	characteristic
123	C₂×C₁₀₁	#380	1	1	yes	characteristic
124	C₂×C₁₀₁	#382	1	1	yes	characteristic
125	C₂×C₁₀₁	#384	1	1	yes	characteristic
126	C₂×C₁₀₁	#389	1	2	yes	—
127	C₂×C₁₀₁	#389	1	2	yes	—
128	C₂×C₁₀₁	#390	1	2	yes	—
129	C₂×C₁₀₁	#391	1	2	yes	—
130	C₄⋊#95	—	101	1	no	maximal, Sylow
131	C₁₀₁	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄⋊C₄	—	101	2	no	—
133	C₂²×C₄²	—	101	1	no	—
134	C₂×#95	—	101	2	no	—
135	C₂×#95	—	101	1	no	—
136	C₂×#95	D₁₀₁	1	1	yes	characteristic
137	C₂³×C₄	—	101	1	no	—
138	C₂³×C₄	—	101	1	no	—
139	C₂³×C₄	—	101	2	no	—
140	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
141	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	101	8	no	—
143	C₂×C₄⋊C₄	—	202	4	no	—
144	C₂×C₄⋊C₄	—	202	4	no	—
145	C₂×C₄²	—	101	4	no	—
146	C₂×C₄²	—	202	4	no	—
147	#95	—	2	2	no	—
148	#95	—	202	4	no	—
149	#95	—	202	2	no	—
150	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	101	2	no	—
160	C₂²×C₄	—	101	4	no	—
161	C₂²×C₄	—	101	4	no	—
162	C₂²×C₄	—	101	2	no	—
163	C₂²×C₄	—	101	2	no	—
164	C₂²×C₄	—	101	2	no	—
165	C₂²×C₄	—	101	2	no	—
166	C₂²×C₄	—	101	2	no	—
167	C₂²×C₄	—	101	4	no	—
168	C₂²×C₄	—	202	2	no	—
169	C₂²×C₄	—	202	2	no	—
170	C₂²×C₄	—	202	2	no	—
171	C₂²×C₄	—	202	2	no	—
172	C₂²×C₄	—	202	2	no	—
173	C₂²×C₄	—	202	2	no	—
174	C₂²×C₄	—	202	2	no	—
175	C₂²×C₄	—	202	2	no	—
176	C₂²×C₄	Dic₂₀₂	1	2	yes	—
177	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
178	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
179	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
180	C₄⋊C₄	—	202	16	no	—
181	C₄²	—	202	8	no	—
182	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
183	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
184	C₂³	D₄×D₁₀₁	1	2	yes	—
185	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
186	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
187	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
188	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
189	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
190	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
191	C₂³	#13108073	1	2	yes	—
192	C₂³	#13108074	1	2	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	101	4	no	—
205	C₂×C₄	—	101	4	no	—
206	C₂×C₄	—	101	8	no	—
207	C₂×C₄	—	101	4	no	—
208	C₂×C₄	—	101	8	no	—
209	C₂×C₄	—	101	4	no	—
210	C₂×C₄	—	404	2	no	—
211	C₂×C₄	—	404	2	no	—
212	C₂×C₄	—	404	2	no	—
213	C₂×C₄	—	404	2	no	—
214	C₂×C₄	—	202	4	no	—
215	C₂×C₄	—	202	4	no	—
216	C₂×C₄	—	202	4	no	—
217	C₂×C₄	—	202	2	no	—
218	C₂×C₄	—	202	2	no	—
219	C₂×C₄	—	202	4	no	—
220	C₂×C₄	—	202	4	no	—
221	C₂×C₄	#13108053	1	4	yes	complemented
222	C₂²	C₄²×D₁₀₁	1	1	yes	characteristic
223	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
224	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
225	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
226	C₂²	C₄⋊C₄×D₁₀₁	1	2	yes	—
227	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
228	C₂²	C₂×#13108053	1	1	yes	characteristic
229	C₂²	#15692211	1	1	yes	characteristic
230	C₂²	#15692212	1	1	yes	characteristic
231	C₂²	#15692215	1	2	yes	—
232	C₂²	C₄×C₂²⋊D₁₀₁	1	2	yes	—
233	C₂²	#15692220	1	2	yes	—
234	C₂²	C₄×Dic₂₀₂	1	2	yes	—
235	C₂²	#15692227	1	2	yes	—
236	C₂²	#15692228	1	2	yes	—
237	C₂²	#15692232	1	2	yes	—
238	C₂²	#15692233	1	2	yes	—
239	C₂²	#15692241	1	2	yes	—
240	C₂²	#15692245	1	2	yes	—
241	C₂²	#15692251	1	1	yes	characteristic
242	C₂²	#15692253	1	1	yes	characteristic
243	C₂²	#15692254	1	1	yes	characteristic
244	C₂²	#15692254	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	4	4	no	—
247	C₄	—	101	8	no	—
248	C₄	—	404	4	no	—
249	C₄	—	202	4	no	—
250	C₂	#95×D₁₀₁	1	1	yes	characteristic
251	C₂	#20265028	1	1	yes	characteristic
252	C₂	C₄×#13108053	1	1	yes	characteristic
253	C₂	#20265035	1	1	yes	characteristic
254	C₂	#20265036	1	1	yes	characteristic
255	C₂	#20265041	1	1	yes	characteristic
256	C₂	#20265046	1	1	yes	characteristic
257	C₂	#20265072	1	2	yes	—
258	C₂	#20265074	1	2	yes	—
259	C₂	#20265077	1	2	yes	—
260	C₂	#20265081	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂×C₄²
Fitting	C₂×#95×C₁₀₁	C₂
socle, prime radical, cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	C₄⋊#95
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
11. C₂×#13108053	C₄	semidirect
13. C₂×C₄×Dic₁₀₁	C₄	semidirect
73. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
90. C₂×Dic₁₀₁	C₄⋊C₄	semidirect
118. Dic₁₀₁	#95	semidirect
131. C₁₀₁	C₄⋊#95	semidirect
221. C₂×C₄	#13108053	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108053	2
C₂²×C₄×Dic₁₀₁	2
C₂×#15692162	2
C₂×#95×C₁₀₁	2
C₄⋊#95	101