Subgroups — #27180109

Order 12928.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265714	C₂	1	1	yes	maximal, characteristic
2	#20265308	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265254	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265247	C₂	1	2	yes	maximal, complemented
5	C₈×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₄₀₄	C₂	1	2	yes	maximal, complemented
7	#20265237	C₂	1	2	yes	maximal, complemented
8	#20265233	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265131	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108057	C₂²	1	2	yes	—
13	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₁	C₂²	1	2	yes	—
15	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
16	#15692250	C₄	1	2	yes	—
17	#15692246	C₄	1	1	yes	characteristic
18	#15692244	C₄	1	1	yes	characteristic
19	#15692240	C₄	1	2	yes	—
20	#15692231	C₄	1	1	yes	characteristic
21	#15692230	C₄	1	1	yes	characteristic
22	C₄×D₄₀₄	C₂²	1	2	yes	—
23	C₄×D₄₀₄	C₂²	1	1	yes	characteristic
24	#15692212	C₂²	1	2	yes	—
25	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
26	C₄⋊#11717579	C₂²	1	1	yes	characteristic
27	C₄⋊#11717579	C₂²	1	2	yes	—
28	#15692167	C₂²	1	2	yes	—
29	#15692167	C₂²	1	2	yes	—
30	#15692167	C₂²	1	2	yes	—
31	#15692166	C₂²	1	2	yes	—
32	#15692165	C₂²	1	2	yes	—
33	C₄×#11717579	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
41	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
42	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
47	#13108057	—	2	4	no	—
48	C₂×#11717579	C₂³	1	1	yes	characteristic
49	C₂×#11717579	C₂³	1	2	yes	—
50	C₂×#11717579	C₂³	1	1	yes	characteristic
51	C₈×D₁₀₁	—	2	2	no	—
52	C₈×D₁₀₁	—	2	4	no	—
53	C₈×D₁₀₁	—	2	2	no	—
54	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
56	#13108053	C₂×C₄	1	2	yes	—
57	#13108053	C₂×C₄	1	2	yes	—
58	#13108051	C₂×C₄	1	2	yes	—
59	#13108051	C₂×C₄	1	2	yes	—
60	#13108051	C₂×C₄	1	2	yes	—
61	#13108051	C₂×C₄	1	2	yes	—
62	#13108051	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
69	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
72	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₄²×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
86	D₄₀₄	—	2	4	no	—
87	D₄₀₄	—	2	2	no	—
88	C₄×D₁₀₁	—	2	2	no	—
89	C₄×D₁₀₁	—	2	2	no	—
90	C₄×D₁₀₁	—	2	1	no	—
91	C₄×D₁₀₁	—	2	1	no	—
92	C₄×D₁₀₁	—	2	2	no	—
93	C₄×D₁₀₁	—	2	2	no	—
94	C₈×C₁₀₁	—	2	1	no	—
95	C₈×C₁₀₁	—	2	2	no	—
96	C₈×C₁₀₁	#41	1	2	yes	—
97	#11717579	—	2	1	no	—
98	#11717579	—	2	2	no	—
99	#11717579	#41	1	2	yes	—
100	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₀₁	—	2	1	no	—
102	C₂×D₁₀₁	—	2	1	no	—
103	C₂×D₁₀₁	—	2	2	no	—
104	C₂×D₁₀₁	—	2	2	no	—
105	C₂×D₁₀₁	—	2	2	no	—
106	C₂×D₁₀₁	#131	1	2	yes	—
107	C₄×C₁₀₁	—	2	1	no	—
108	C₄×C₁₀₁	—	2	2	no	—
109	C₄×C₁₀₁	—	2	2	no	—
110	C₄×C₁₀₁	#131	1	2	yes	—
111	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
112	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
113	Dic₁₀₁	—	2	1	no	—
114	Dic₁₀₁	—	2	2	no	—
115	Dic₁₀₁	—	2	1	no	—
116	Dic₁₀₁	—	2	1	no	—
117	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
118	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
120	D₁₀₁	—	2	2	no	—
121	D₁₀₁	—	4	2	no	—
122	#2966	—	101	1	no	maximal, Sylow
123	C₁₀₁	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	—	101	1	no	—
125	#446	—	101	1	no	—
126	C₈×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
127	#436	—	101	1	no	—
128	#435	—	101	2	no	—
129	C₈×D₄	—	101	1	no	—
130	C₈×D₄	—	101	2	no	—
131	#433	—	101	2	no	—
132	#432	—	101	1	no	—
133	#405	—	101	2	no	—
134	C₂×C₄×C₈	—	101	1	no	—
135	C₂×M₁₆	—	101	2	no	—
136	C₂²×C₈	—	101	1	no	—
137	C₂²×C₈	—	101	1	no	—
138	C₂²×C₈	—	101	2	no	—
139	#126	—	101	2	no	—
140	#125	—	101	1	no	—
141	#124	—	101	1	no	—
142	#123	—	101	2	no	—
143	C₂²⋊Q₈	—	101	1	no	—
144	#121	—	101	1	no	—
145	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
146	C₄×D₄	—	101	2	no	—
147	C₄×D₄	—	101	1	no	—
148	#117	—	101	2	no	—
149	C₂×C₄²	—	101	1	no	—
150	C₄⋊C₈	—	101	1	no	—
151	C₄⋊C₈	—	101	2	no	—
152	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₁₀₁	1	2	yes	—
154	C₂²⋊C₈	—	101	2	no	—
155	C₂²⋊C₈	—	101	2	no	—
156	C₂²⋊C₈	—	101	2	no	—
157	#97	—	101	2	no	—
158	#97	—	101	2	no	—
159	C₄×C₈	—	101	1	no	—
160	C₄×C₈	—	101	1	no	—
161	C₄×C₈	—	101	1	no	—
162	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
163	C₄×C₈	C₂×D₁₀₁	1	2	yes	—
164	C₂×Q₈	C₄×D₁₀₁	1	1	yes	characteristic
165	C₂×D₄	—	101	2	no	—
166	C₂×D₄	—	101	1	no	—
167	C₂²×C₄	—	101	1	no	—
168	C₂²×C₄	—	101	1	no	—
169	C₂²×C₄	—	101	1	no	—
170	C₂²×C₄	—	101	2	no	—
171	M₁₆	—	202	4	no	—
172	C₂×C₈	—	101	1	no	—
173	C₂×C₈	—	101	1	no	—
174	C₂×C₈	—	101	2	no	—
175	C₂×C₈	—	202	2	no	—
176	C₂×C₈	—	202	2	no	—
177	C₂×C₈	—	202	4	no	—
178	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
180	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
181	C₄⋊C₄	—	101	2	no	—
182	C₄⋊C₄	—	101	2	no	—
183	C₄⋊C₄	—	101	1	no	—
184	C₄⋊C₄	—	101	2	no	—
185	C₄⋊C₄	C₄×D₁₀₁	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₁₀₁	1	2	yes	—
187	C₂²⋊C₄	—	101	2	no	—
188	C₂²⋊C₄	—	101	2	no	—
189	C₂²⋊C₄	—	101	2	no	—
190	C₂²⋊C₄	—	101	2	no	—
191	C₂²⋊C₄	—	101	2	no	—
192	C₄²	—	101	1	no	—
193	C₄²	—	101	1	no	—
194	C₄²	—	101	1	no	—
195	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
196	C₄²	C₂²×D₁₀₁	1	2	yes	—
197	C₂³	—	101	1	no	—
198	C₂³	—	101	2	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	202	4	no	—
201	D₄	—	202	2	no	—
202	C₂×C₄	—	101	1	no	—
203	C₂×C₄	—	101	1	no	—
204	C₂×C₄	—	101	1	no	—
205	C₂×C₄	—	101	2	no	—
206	C₂×C₄	—	202	2	no	—
207	C₂×C₄	—	202	1	no	—
208	C₂×C₄	—	202	2	no	—
209	C₂×C₄	—	202	2	no	—
210	C₂×C₄	—	202	1	no	—
211	C₂×C₄	—	202	2	no	—
212	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	101	2	no	—
220	C₈	—	202	1	no	—
221	C₈	—	202	2	no	—
222	C₈	#13108075	1	2	yes	—
223	C₂²	—	101	2	no	—
224	C₂²	—	202	2	no	—
225	C₂²	—	202	2	no	—
226	C₂²	—	202	2	no	—
227	C₂²	—	202	1	no	—
228	C₂²	—	202	1	no	—
229	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	202	1	no	—
234	C₄	—	202	2	no	—
235	C₄	—	202	1	no	—
236	C₄	—	202	1	no	—
237	C₄	#15692261	1	2	yes	—
238	C₄	#41×D₁₀₁	1	1	yes	characteristic
239	C₄	C₂×#13108075	1	1	yes	characteristic
240	C₂	—	404	2	no	—
241	C₂	—	202	2	no	—
242	C₂	C₄×#13108075	1	1	yes	characteristic
243	C₂	#131×D₁₀₁	1	1	yes	characteristic
244	C₂	C₂×#15692261	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₈×Q₈×C₁₀₁	C₂
prime radical	#15692230	C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₈×Q₈	D₁₀₁
hypocenter	C₁₀₁	#2966
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #20265308	C₂	semidirect
3. #20265254	C₂	semidirect
4. #20265247	C₂	semidirect
5. C₈×D₄₀₄	C₂	semidirect
6. C₈×D₄₀₄	C₂	semidirect
7. #20265237	C₂	semidirect
8. #20265233	C₂	semidirect
9. #20265131	C₂	semidirect
10. C₄×C₈×D₁₀₁	C₂	semidirect
11. C₈×Q₈×C₁₀₁	C₂	semidirect
123. C₁₀₁	#2966	semidirect
126. C₈×Q₈	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265714	2
#20265308	2
#20265254	2
#20265247	2
C₈×D₄₀₄	2
#20265237	2
#20265233	2
#20265131	2
C₄×C₈×D₁₀₁	2
C₈×Q₈×C₁₀₁	2
#2966	101