Subgroups — #27187886

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#87938	C₂	1	2	yes	maximal, complemented
2	#20266218	C₂	1	2	yes	maximal, complemented
3	#20266217	C₂	1	2	yes	maximal, complemented
4	#20266216	C₂	1	4	yes	maximal, complemented
5	#20266187	C₂	1	1	yes	maximal, characteristic, complemented
6	#20266167	C₂	1	2	yes	maximal, complemented
7	C₂×#15692309	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₀₁	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#87938	C₂²	1	4	yes	—
11	D₄×#87938	—	2	4	no	—
12	D₄×#87938	C₂²	1	4	yes	—
13	#15692334	—	2	2	no	—
14	#15692334	C₂²	1	2	yes	—
15	C₂×C₄×#87938	—	2	2	no	—
16	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
17	#15692324	C₂²	1	2	yes	—
18	#15692323	C₂²	1	2	yes	—
19	#15692318	C₂²	1	4	yes	—
20	#15692317	C₂²	1	4	yes	—
21	#15692309	C₂²	1	2	yes	—
22	#15692309	C₂²	1	2	yes	—
23	#41×D₁₀₁	C₂²	1	2	yes	—
24	#15692278	C₄	1	1	yes	characteristic, complemented
25	#15692277	C₄	1	1	yes	characteristic, complemented
26	#15692276	C₄	1	2	yes	complemented
27	#15692275	C₄	1	2	yes	complemented
28	#15692274	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
30	SD₁₆×D₁₀₁	C₂²	1	2	yes	—
31	D₈×D₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
33	#135×C₁₀₁	C₄	1	1	yes	characteristic, complemented
34	C₂²×#87938	—	2	4	no	—
35	C₂²×#87938	D₄	1	2	yes	—
36	D₁₀₁⋊C₈	—	2	1	no	—
37	D₁₀₁⋊C₈	C₂³	1	2	yes	—
38	C₂×#11717581	—	2	1	no	—
39	C₄⋊#87938	—	2	4	no	—
40	C₄⋊#87938	C₂³	1	4	yes	—
41	C₂²⋊#87938	—	2	2	no	—
42	C₂²⋊#87938	—	2	4	no	—
43	C₄×#87938	—	2	4	no	—
44	C₄×#87938	—	2	2	no	—
45	C₄×#87938	D₄	1	2	yes	—
46	C₄×#87938	C₂³	1	2	yes	—
47	#13108075	C₂×C₄	1	2	yes	complemented
48	#13108074	—	2	2	no	—
49	#13108074	C₂×C₄	1	2	yes	complemented
50	#13108073	—	2	2	no	—
51	#13108073	C₂×C₄	1	2	yes	complemented
52	Q₈×D₁₀₁	C₂³	1	2	yes	—
53	D₄×D₁₀₁	—	2	2	no	—
54	D₄×D₁₀₁	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₁	—	2	2	no	—
56	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
57	Dic₄₀₄	C₂×C₄	1	1	yes	characteristic, complemented
58	#13108064	C₂×C₄	1	2	yes	complemented
59	#13108063	C₂×C₄	1	2	yes	complemented
60	#13108062	C₂×C₄	1	2	yes	complemented
61	#13108061	C₂×C₄	1	2	yes	complemented
62	D₈₀₈	C₂×C₄	1	1	yes	characteristic, complemented
63	#13108059	C₂×C₄	1	2	yes	complemented
64	C₂×#11717579	C₂×C₄	1	1	yes	characteristic, complemented
65	C₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
68	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
69	#41×C₁₀₁	C₂×C₄	1	2	yes	complemented
70	Q₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₁₀₁	C₂×C₄	1	2	yes	complemented
72	D₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂²×D₁₀₁	—	2	2	no	—
75	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
76	C₂×#87938	—	2	4	no	—
77	C₂×#87938	—	4	2	no	—
78	C₂×#87938	—	4	8	no	—
79	C₂×#87938	C₂×D₄	1	4	yes	—
80	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
81	D₄×C₁₀₁	—	2	2	no	—
82	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₁₀₁	—	2	2	no	—
86	C₂²⋊D₁₀₁	—	2	2	no	—
87	C₂²⋊D₁₀₁	—	2	2	no	—
88	C₂×Dic₁₀₁	—	2	2	no	—
89	C₂×Dic₁₀₁	#41	1	1	yes	characteristic
90	D₄₀₄	—	2	2	no	—
91	D₄₀₄	C₂²×C₄	1	2	yes	—
92	C₄×D₁₀₁	—	2	2	no	—
93	C₄×D₁₀₁	—	2	2	no	—
94	C₄×D₁₀₁	—	2	2	no	—
95	C₄×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₀₁	#41	1	1	yes	characteristic
98	C₄×D₁₀₁	C₂⁴	1	1	yes	characteristic
99	Dic₂₀₂	—	2	2	no	—
100	Dic₂₀₂	C₂²×C₄	1	2	yes	—
101	#11717581	—	2	2	no	—
102	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
104	#11717579	C₂²×C₄	1	1	yes	characteristic
105	#11717579	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
108	C₂×D₁₀₁	—	2	2	no	—
109	C₂×D₁₀₁	—	2	2	no	—
110	C₂×D₁₀₁	—	4	4	no	—
111	C₂×D₁₀₁	C₄×D₄	1	1	yes	characteristic
112	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
113	#87938	—	2	4	no	—
114	#87938	—	4	4	no	—
115	C₄×C₁₀₁	—	2	2	no	—
116	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
117	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
118	Dic₁₀₁	—	2	2	no	—
119	Dic₁₀₁	C₄×D₄	1	1	yes	characteristic
120	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₁₀₁	—	2	1	no	—
122	C₂×C₁₀₁	—	4	2	no	—
123	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
124	D₁₀₁	—	2	1	no	—
125	D₁₀₁	—	4	2	no	—
126	D₁₀₁	#575	1	2	yes	—
127	#2943	—	101	1	no	maximal, Sylow
128	C₁₀₁	#2943	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	101	1	no	—
130	C₄×#41	—	101	2	no	—
131	#442	—	101	2	no	—
132	#441	—	101	2	no	—
133	#440	—	101	4	no	—
134	#427	—	101	1	no	—
135	#416	—	101	2	no	—
136	C₂×#97	—	101	1	no	—
137	C₂×#41	—	101	2	no	—
138	#135	—	101	2	no	—
139	#135	—	101	1	no	—
140	#135	—	101	2	no	—
141	#135	—	101	1	no	—
142	#135	—	101	1	no	—
143	#135	#87938	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	101	1	no	—
145	C₂×SD₁₆	—	101	2	no	—
146	C₂×D₈	—	101	1	no	—
147	C₂²×C₈	—	101	1	no	—
148	C₂²×C₈	—	101	1	no	—
149	C₄×Q₈	—	101	4	no	—
150	C₄×D₄	—	101	4	no	—
151	C₄×D₄	—	202	4	no	—
152	#117	—	101	2	no	—
153	#117	—	202	2	no	—
154	C₂×C₄²	—	101	1	no	—
155	C₂×C₄²	—	202	2	no	—
156	#107	—	101	2	no	—
157	#106	—	101	2	no	—
158	#103	—	101	4	no	—
159	#102	—	101	4	no	—
160	#97	—	101	2	no	—
161	#97	—	101	2	no	—
162	#41	—	101	2	no	—
163	#41	—	101	2	no	—
164	#41	—	101	2	no	—
165	#41	—	202	2	no	—
166	#41	—	202	2	no	—
167	#41	C₂×#87938	1	2	yes	complemented
168	C₂×Q₈	—	101	2	no	—
169	C₂×D₄	—	101	2	no	—
170	C₂×D₄	—	202	2	no	—
171	C₂²×C₄	—	101	1	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	202	2	no	—
174	C₂²×C₄	—	202	4	no	—
175	Q₁₆	—	101	2	no	—
176	Q₁₆	—	101	1	no	—
177	Q₁₆	C₂×#87938	1	1	yes	characteristic, complemented
178	SD₁₆	—	101	2	no	—
179	SD₁₆	—	101	2	no	—
180	SD₁₆	—	101	2	no	—
181	SD₁₆	C₂×#87938	1	2	yes	complemented
182	D₈	—	101	2	no	—
183	D₈	—	101	1	no	—
184	D₈	C₂×#87938	1	1	yes	characteristic, complemented
185	C₂×C₈	—	101	1	no	—
186	C₂×C₈	—	101	1	no	—
187	C₂×C₈	—	101	1	no	—
188	C₂×C₈	—	101	1	no	—
189	C₂×C₈	—	101	1	no	—
190	C₂×C₈	—	101	2	no	—
191	C₂×C₈	—	202	1	no	—
192	C₂×C₈	—	202	1	no	—
193	C₂×C₈	C₂×#87938	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	101	4	no	—
195	C₄⋊C₄	—	202	4	no	—
196	C₂²⋊C₄	—	202	4	no	—
197	C₂²⋊C₄	—	202	2	no	—
198	C₄²	—	101	2	no	—
199	C₄²	—	101	2	no	—
200	C₄²	—	202	2	no	—
201	C₄²	—	202	4	no	—
202	C₂³	—	101	1	no	—
203	C₂³	—	202	2	no	—
204	Q₈	—	101	2	no	—
205	Q₈	—	202	2	no	—
206	Q₈	C₂²×#87938	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	101	2	no	—
209	D₄	—	202	2	no	—
210	D₄	—	202	2	no	—
211	D₄	—	202	2	no	—
212	D₄	—	202	2	no	—
213	D₄	C₂²×#87938	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	101	1	no	—
216	C₂×C₄	—	101	1	no	—
217	C₂×C₄	—	101	1	no	—
218	C₂×C₄	—	101	1	no	—
219	C₂×C₄	—	101	1	no	—
220	C₂×C₄	—	101	4	no	—
221	C₂×C₄	—	404	8	no	—
222	C₂×C₄	—	404	2	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	—	202	2	no	—
225	C₂×C₄	—	202	2	no	—
226	C₂×C₄	—	202	4	no	—
227	C₂×C₄	—	202	2	no	—
228	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
229	C₈	—	101	1	no	—
230	C₈	—	101	1	no	—
231	C₈	—	202	2	no	—
232	C₈	C₂²×#87938	1	1	yes	characteristic
233	C₈	C₂²×#87938	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	101	1	no	—
236	C₂²	—	101	1	no	—
237	C₂²	—	404	4	no	—
238	C₂²	—	202	2	no	—
239	C₂²	—	202	2	no	—
240	C₂²	D₄×#87938	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	101	1	no	—
243	C₄	—	101	1	no	—
244	C₄	—	404	4	no	—
245	C₄	—	202	4	no	—
246	C₄	—	202	2	no	—
247	C₄	D₄×#87938	1	1	yes	characteristic
248	C₄	C₂³×#87938	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	101	2	no	—
252	C₂	—	404	2	no	—
253	C₂	—	202	1	no	—
254	C₂	C₂×D₄×#87938	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#87938
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₄	C₂³×#87938
Fitting	#135×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×C₄×D₄
prime radical	#135×D₁₀₁	C₂
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	#135	#87938
hypocenter	C₁₀₁	#2943
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#87938	C₂	semidirect
2. #20266218	C₂	semidirect
3. #20266217	C₂	semidirect
4. #20266216	C₂	semidirect
5. #20266187	C₂	semidirect
6. #20266167	C₂	semidirect
7. C₂×#15692309	C₂	semidirect
24. #15692278	C₄	semidirect
25. #15692277	C₄	semidirect
26. #15692276	C₄	semidirect
27. #15692275	C₄	semidirect
28. #15692274	C₄	semidirect
33. #135×C₁₀₁	C₄	semidirect
47. #13108075	C₂×C₄	semidirect
49. #13108074	C₂×C₄	semidirect
51. #13108073	C₂×C₄	semidirect
57. Dic₄₀₄	C₂×C₄	semidirect
58. #13108064	C₂×C₄	semidirect
59. #13108063	C₂×C₄	semidirect
60. #13108062	C₂×C₄	semidirect
61. #13108061	C₂×C₄	semidirect
62. D₈₀₈	C₂×C₄	semidirect
63. #13108059	C₂×C₄	semidirect
64. C₂×#11717579	C₂×C₄	semidirect
65. C₈×D₁₀₁	C₂×C₄	semidirect
66. C₈×D₁₀₁	C₂×C₄	semidirect
69. #41×C₁₀₁	C₂×C₄	semidirect
70. Q₁₆×C₁₀₁	C₂×C₄	semidirect
71. SD₁₆×C₁₀₁	C₂×C₄	semidirect
72. D₈×C₁₀₁	C₂×C₄	semidirect
73. C₂×C₈×C₁₀₁	C₂×C₄	semidirect
128. C₁₀₁	#2943	semidirect
143. #135	#87938	semidirect
167. #41	C₂×#87938	semidirect
177. Q₁₆	C₂×#87938	semidirect
181. SD₁₆	C₂×#87938	semidirect
184. D₈	C₂×#87938	semidirect
193. C₂×C₈	C₂×#87938	semidirect

Maximal subgroups

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

subgroup	index
#41×#87938	2
#20266218	2
#20266217	2
#20266216	2
#20266187	2
#20266167	2
C₂×#15692309	2
#135×D₁₀₁	2
#2943	101