Subgroups — #27187892

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266245	C₂	1	1	yes	maximal, characteristic, complemented
2	#20266244	C₂	1	1	yes	maximal, characteristic
3	#20266216	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#87938	C₂	1	4	yes	maximal, complemented
5	#20266187	C₂	1	1	yes	maximal, characteristic, complemented
6	#20266170	C₂	1	1	yes	maximal, characteristic, complemented
7	#20266169	C₂	1	1	yes	maximal, characteristic
8	#20266140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic
10	D₁₀₁⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#87938	—	2	2	no	—
12	Q₈×#87938	C₂²	1	4	yes	complemented
13	D₄×#87938	—	2	2	no	—
14	D₄×#87938	C₂²	1	4	yes	—
15	#15692334	—	2	2	no	—
16	#15692334	C₂²	1	1	yes	characteristic
17	#15692334	C₂²	1	1	yes	characteristic
18	#15692334	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#87938	—	2	2	no	—
20	#15692324	C₂²	1	2	yes	complemented
21	#15692323	C₂²	1	2	yes	complemented
22	#15692318	C₂²	1	4	yes	complemented
23	#15692317	C₂²	1	4	yes	—
24	#15692309	C₂²	1	2	yes	complemented
25	C₈×#87938	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×#13108063	C₄	1	1	yes	characteristic, complemented
29	C₂×#13108062	C₄	1	1	yes	characteristic, complemented
30	C₂×#13108061	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₁₀₁	C₄	1	2	yes	complemented
32	SD₁₆×D₁₀₁	C₄	1	2	yes	complemented
33	SD₁₆×D₁₀₁	C₂²	1	2	yes	—
34	SD₁₆×D₁₀₁	C₂²	1	2	yes	—
35	C₂×C₈×D₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₀₁	C₄	1	1	yes	characteristic, complemented
37	C₂²×#87938	—	2	4	no	—
38	#13108084	—	2	1	no	—
39	#13108083	—	2	1	no	—
40	D₁₀₁⋊C₈	C₂³	1	2	yes	—
41	C₄⋊#87938	—	2	4	no	—
42	C₄⋊#87938	D₄	1	2	yes	—
43	C₄⋊#87938	C₂³	1	2	yes	—
44	C₄⋊#87938	C₂³	1	2	yes	—
45	C₂²⋊#87938	—	2	1	no	—
46	C₂²⋊#87938	—	2	1	no	—
47	C₂²⋊#87938	—	2	4	no	—
48	C₂²⋊#87938	D₄	1	2	yes	—
49	C₄×#87938	—	2	4	no	—
50	C₄×#87938	—	2	1	no	—
51	C₄×#87938	—	2	1	no	—
52	C₄×#87938	C₂³	1	2	yes	—
53	C₂³×D₁₀₁	—	2	1	no	—
54	C₂×Dic₂₀₂	—	2	1	no	—
55	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₁₀₁	—	2	1	no	—
57	Q₈×D₁₀₁	—	2	1	no	—
58	Q₈×D₁₀₁	C₂×C₄	1	2	yes	complemented
59	Q₈×D₁₀₁	C₂³	1	2	yes	—
60	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
62	D₄×D₁₀₁	—	2	1	no	—
63	D₄×D₁₀₁	—	2	1	no	—
64	D₄×D₁₀₁	C₂×C₄	1	2	yes	—
65	D₄×D₁₀₁	C₂³	1	2	yes	—
66	C₂×C₄×D₁₀₁	—	2	1	no	—
67	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
68	#13108063	C₂×C₄	1	2	yes	complemented
69	#13108063	C₂×C₄	1	2	yes	complemented
70	#13108062	C₂×C₄	1	2	yes	complemented
71	#13108062	C₂×C₄	1	2	yes	complemented
72	#13108061	C₂×C₄	1	2	yes	complemented
73	#13108061	C₂×C₄	1	2	yes	complemented
74	C₂×#11717579	C₂×C₄	1	1	yes	characteristic, complemented
75	C₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
76	C₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
77	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
78	C₈×D₁₀₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₀₁	C₂×C₄	1	2	yes	complemented
82	SD₁₆×C₁₀₁	C₂×C₄	1	2	yes	complemented
83	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂³×C₁₀₁	—	2	1	no	—
85	C₂²×D₁₀₁	—	2	1	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	8	2	no	—
89	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
90	C₂×#87938	—	2	2	no	—
91	C₂×#87938	—	2	2	no	—
92	C₂×#87938	—	8	4	no	—
93	C₂×#87938	C₂×D₄	1	4	yes	—
94	Q₈×C₁₀₁	—	2	1	no	—
95	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
96	D₄×C₁₀₁	—	2	1	no	—
97	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₁₀₁	—	2	2	no	—
101	C₂²⋊D₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	1	no	—
103	C₂×Dic₁₀₁	#41	1	1	yes	characteristic
104	D₄₀₄	—	2	1	no	—
105	D₄₀₄	C₂²×C₄	1	2	yes	—
106	C₄×D₁₀₁	—	2	2	no	—
107	C₄×D₁₀₁	—	2	2	no	—
108	C₄×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
110	C₄×D₁₀₁	#41	1	1	yes	characteristic
111	C₄×D₁₀₁	C₂⁴	1	1	yes	characteristic
112	Dic₂₀₂	—	2	1	no	—
113	Dic₂₀₂	—	2	2	no	—
114	Dic₂₀₂	—	2	2	no	—
115	Dic₂₀₂	C₂²×C₄	1	2	yes	—
116	#11717581	—	2	2	no	—
117	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
119	#11717579	C₂²×C₄	1	1	yes	characteristic
120	#11717579	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	—	8	1	no	—
123	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	—	4	4	no	—
127	C₂×D₁₀₁	—	8	2	no	—
128	C₂×D₁₀₁	—	8	1	no	—
129	C₂×D₁₀₁	C₄×D₄	1	1	yes	characteristic
130	C₂×D₁₀₁	C₂²×D₄	1	1	yes	characteristic
131	#87938	—	2	4	no	—
132	#87938	—	4	2	no	—
133	#87938	—	4	2	no	—
134	C₄×C₁₀₁	—	2	2	no	—
135	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	C₄×D₄	1	1	yes	characteristic
139	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₁₀₁	—	2	1	no	—
141	C₂×C₁₀₁	—	4	2	no	—
142	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₀₁	—	2	1	no	—
144	D₁₀₁	—	4	2	no	—
145	D₁₀₁	#577	1	2	yes	—
146	#2949	—	101	1	no	maximal, Sylow
147	C₁₀₁	#2949	1	1	yes	characteristic, complemented, Sylow
148	C₂²×SD₁₆	—	101	1	no	—
149	#519	—	101	1	no	—
150	#518	—	101	1	no	—
151	#440	—	101	4	no	—
152	C₄×SD₁₆	—	101	4	no	—
153	#427	—	101	1	no	—
154	#419	—	101	1	no	—
155	#418	—	101	1	no	—
156	#405	—	101	1	no	—
157	C₂²×Q₈	—	101	1	no	—
158	C₂²×D₄	—	101	1	no	—
159	C₂×SD₁₆	—	101	1	no	—
160	C₂×SD₁₆	—	101	1	no	—
161	C₂×SD₁₆	—	101	2	no	—
162	C₂×SD₁₆	—	101	2	no	—
163	C₂×SD₁₆	—	101	2	no	—
164	C₂×SD₁₆	—	101	2	no	—
165	C₂×SD₁₆	—	101	1	no	—
166	C₂×SD₁₆	#87938	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	101	1	no	—
168	C₂²×C₈	—	101	1	no	—
169	C₄×Q₈	—	101	4	no	—
170	C₄×Q₈	—	202	2	no	—
171	C₄×D₄	—	101	4	no	—
172	C₄×D₄	—	202	2	no	—
173	#117	—	101	1	no	—
174	#117	—	101	1	no	—
175	#117	—	101	1	no	—
176	#117	—	202	2	no	—
177	C₂×C₂²⋊C₄	—	202	2	no	—
178	#107	—	101	2	no	—
179	#106	—	101	2	no	—
180	#103	—	101	4	no	—
181	#102	—	101	4	no	—
182	#97	—	101	2	no	—
183	C₄×C₈	—	101	2	no	—
184	C₂⁴	—	202	1	no	—
185	C₂×Q₈	—	101	2	no	—
186	C₂×Q₈	—	101	2	no	—
187	C₂×Q₈	—	101	1	no	—
188	C₂×Q₈	—	202	1	no	—
189	C₂×Q₈	—	202	1	no	—
190	C₂×Q₈	—	202	1	no	—
191	C₂×Q₈	C₂×#87938	1	1	yes	characteristic, complemented
192	C₂×D₄	—	101	2	no	—
193	C₂×D₄	—	101	2	no	—
194	C₂×D₄	—	101	1	no	—
195	C₂×D₄	—	202	1	no	—
196	C₂×D₄	—	202	1	no	—
197	C₂×D₄	—	202	1	no	—
198	C₂×D₄	C₂×#87938	1	1	yes	characteristic
199	C₂²×C₄	—	101	1	no	—
200	C₂²×C₄	—	202	1	no	—
201	C₂²×C₄	—	202	4	no	—
202	SD₁₆	—	101	2	no	—
203	SD₁₆	—	101	2	no	—
204	SD₁₆	—	101	2	no	—
205	SD₁₆	—	101	2	no	—
206	SD₁₆	—	101	2	no	—
207	SD₁₆	—	101	2	no	—
208	SD₁₆	C₂×#87938	1	2	yes	complemented
209	SD₁₆	C₂×#87938	1	2	yes	complemented
210	M₁₆	—	202	1	no	—
211	M₁₆	—	202	1	no	—
212	C₂×C₈	—	101	1	no	—
213	C₂×C₈	—	101	1	no	—
214	C₂×C₈	—	101	1	no	—
215	C₂×C₈	—	101	1	no	—
216	C₂×C₈	—	101	1	no	—
217	C₂×C₈	—	101	2	no	—
218	C₂×C₈	C₂×#87938	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	101	2	no	—
220	C₄⋊C₄	—	101	2	no	—
221	C₄⋊C₄	—	101	2	no	—
222	C₄⋊C₄	—	202	4	no	—
223	C₂²⋊C₄	—	101	2	no	—
224	C₂²⋊C₄	—	202	4	no	—
225	C₂²⋊C₄	—	202	1	no	—
226	C₂²⋊C₄	—	202	1	no	—
227	C₄²	—	101	2	no	—
228	C₄²	—	202	1	no	—
229	C₄²	—	202	1	no	—
230	C₄²	—	202	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	808	2	no	—
233	C₂³	—	101	1	no	—
234	C₂³	—	202	2	no	—
235	C₂³	—	202	2	no	—
236	C₂³	—	202	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	101	2	no	—
239	Q₈	—	202	2	no	—
240	Q₈	—	202	2	no	—
241	Q₈	—	202	1	no	—
242	Q₈	C₂²×#87938	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	101	2	no	—
245	D₄	—	202	2	no	—
246	D₄	—	202	2	no	—
247	D₄	—	202	1	no	—
248	D₄	C₂²×#87938	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	808	4	no	—
251	C₂×C₄	—	101	1	no	—
252	C₂×C₄	—	101	1	no	—
253	C₂×C₄	—	101	1	no	—
254	C₂×C₄	—	101	1	no	—
255	C₂×C₄	—	101	1	no	—
256	C₂×C₄	—	101	4	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	1	no	—
262	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
263	C₈	—	101	1	no	—
264	C₈	—	101	1	no	—
265	C₈	—	202	2	no	—
266	C₈	C₂²×#87938	1	1	yes	characteristic
267	C₈	C₂²×#87938	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	808	1	no	—
271	C₂²	—	808	2	no	—
272	C₂²	—	101	1	no	—
273	C₂²	—	101	1	no	—
274	C₂²	—	404	4	no	—
275	C₂²	—	202	2	no	—
276	C₂²	—	202	2	no	—
277	C₂²	D₄×#87938	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	101	1	no	—
280	C₄	—	101	1	no	—
281	C₄	—	404	2	no	—
282	C₄	—	404	2	no	—
283	C₄	—	202	4	no	—
284	C₄	—	202	2	no	—
285	C₄	D₄×#87938	1	1	yes	characteristic
286	C₄	C₂³×#87938	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	101	2	no	—
290	C₂	—	404	2	no	—
291	C₂	—	202	1	no	—
292	C₂	C₂×D₄×#87938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#87938
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₄	C₂³×#87938
Fitting	C₂×SD₁₆×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×C₄×D₄
prime radical	C₂×D₄×D₁₀₁	C₂²
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	C₂×SD₁₆	#87938
hypocenter	C₁₀₁	#2949
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20266245	C₂	semidirect
3. #20266216	C₂	semidirect
4. SD₁₆×#87938	C₂	semidirect
5. #20266187	C₂	semidirect
6. #20266170	C₂	semidirect
8. #20266140	C₂	semidirect
12. Q₈×#87938	C₂²	semidirect
20. #15692324	C₂²	semidirect
21. #15692323	C₂²	semidirect
22. #15692318	C₂²	semidirect
24. #15692309	C₂²	semidirect
25. C₈×#87938	C₂²	semidirect
28. C₂×#13108063	C₄	semidirect
29. C₂×#13108062	C₄	semidirect
30. C₂×#13108061	C₄	semidirect
31. SD₁₆×D₁₀₁	C₄	semidirect
32. SD₁₆×D₁₀₁	C₄	semidirect
36. C₂×SD₁₆×C₁₀₁	C₄	semidirect
55. C₂×Dic₂₀₂	C₂×C₄	semidirect
58. Q₈×D₁₀₁	C₂×C₄	semidirect
68. #13108063	C₂×C₄	semidirect
69. #13108063	C₂×C₄	semidirect
70. #13108062	C₂×C₄	semidirect
71. #13108062	C₂×C₄	semidirect
72. #13108061	C₂×C₄	semidirect
73. #13108061	C₂×C₄	semidirect
74. C₂×#11717579	C₂×C₄	semidirect
75. C₈×D₁₀₁	C₂×C₄	semidirect
76. C₈×D₁₀₁	C₂×C₄	semidirect
79. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
81. SD₁₆×C₁₀₁	C₂×C₄	semidirect
82. SD₁₆×C₁₀₁	C₂×C₄	semidirect
83. C₂×C₈×C₁₀₁	C₂×C₄	semidirect
147. C₁₀₁	#2949	semidirect
166. C₂×SD₁₆	#87938	semidirect
191. C₂×Q₈	C₂×#87938	semidirect
208. SD₁₆	C₂×#87938	semidirect
209. SD₁₆	C₂×#87938	semidirect
218. 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
#20266245	2
#20266244	2
#20266216	2
SD₁₆×#87938	2
#20266187	2
#20266170	2
#20266169	2
#20266140	2
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