Subgroups — #27187863

Order 12928.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₀₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266212	C₂	1	2	yes	maximal, complemented
4	#20266209	C₂	1	2	yes	maximal, complemented
5	#20266208	C₂	1	2	yes	maximal, complemented
6	#20266204	C₂	1	2	yes	maximal, complemented
7	#20266184	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266151	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₀₁⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#20266142	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265687	C₂	1	1	yes	maximal, characteristic
12	#15692342	—	2	4	no	—
13	C₂×#13108084	C₂²	1	1	yes	characteristic
14	C₂×#13108084	C₂²	1	1	yes	characteristic
15	D₁₀₁⋊M₁₆	—	2	4	no	—
16	C₂×#13108083	C₂²	1	2	yes	—
17	C₂×#13108083	C₂²	1	2	yes	complemented
18	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
19	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×#11717581	C₂²	1	2	yes	complemented
22	#15692322	C₂²	1	2	yes	complemented
23	C₄⋊#11717581	C₂²	1	2	yes	complemented
24	#15692312	C₂²	1	1	yes	characteristic
25	#15692312	C₂²	1	1	yes	characteristic
26	C₂²⋊#11717581	C₂²	1	1	yes	characteristic
27	C₂²⋊#11717581	C₂²	1	1	yes	characteristic
28	C₂²⋊#11717581	C₂²	1	2	yes	complemented
29	C₂²⋊#11717581	C₂²	1	2	yes	complemented
30	#15692310	C₂²	1	2	yes	complemented
31	C₄×#11717581	C₂²	1	2	yes	complemented
32	C₂×#13108073	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
34	#15692237	C₄	1	2	yes	—
35	#15692235	C₄	1	1	yes	characteristic
36	#15692234	C₄	1	1	yes	characteristic
37	#15692226	C₄	1	1	yes	characteristic
38	#15692223	C₄	1	2	yes	—
39	D₄×Dic₁₀₁	C₂²	1	2	yes	—
40	#15692215	C₂²	1	2	yes	—
41	#15692213	C₂²	1	1	yes	characteristic
42	#121×C₁₀₁	C₄	1	1	yes	characteristic
43	#13108084	—	2	2	no	—
44	#13108084	D₄	1	4	yes	complemented
45	#13108083	—	2	4	no	—
46	#13108083	—	2	4	no	—
47	D₁₀₁⋊C₈	—	2	2	no	—
48	D₁₀₁⋊C₈	—	2	2	no	—
49	D₁₀₁⋊C₈	—	2	4	no	—
50	C₂×#11717581	—	2	4	no	—
51	C₂×#11717581	C₂³	1	1	yes	characteristic
52	C₂×#11717581	C₂³	1	1	yes	characteristic
53	C₂×#11717581	C₂³	1	1	yes	characteristic
54	C₂×#11717581	C₂³	1	1	yes	characteristic
55	C₂×#11717581	C₂³	1	2	yes	—
56	C₂×#11717581	C₂³	1	2	yes	—
57	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
58	#13108073	—	2	2	no	—
59	#13108073	—	2	2	no	—
60	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
63	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
65	C₂×C₄×D₁₀₁	—	2	2	no	—
66	C₂×C₄×D₁₀₁	—	2	2	no	—
67	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
73	#13108053	C₂×C₄	1	2	yes	—
74	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
75	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
76	#13108051	C₂×C₄	1	2	yes	—
77	#13108051	C₂×C₄	1	2	yes	—
78	C₄×Dic₁₀₁	C₂³	1	2	yes	—
79	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
84	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	4	1	no	—
89	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	D₄×C₁₀₁	—	2	2	no	—
92	D₄×C₁₀₁	—	4	2	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₁₀₁	—	2	4	no	—
98	C₂²⋊D₁₀₁	—	2	4	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	1	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
104	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
107	C₄×D₁₀₁	—	2	2	no	—
108	C₄×D₁₀₁	—	2	2	no	—
109	C₄×D₁₀₁	—	2	1	no	—
110	C₄×D₁₀₁	—	2	1	no	—
111	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
112	C₄×D₁₀₁	#41	1	2	yes	—
113	Dic₂₀₂	—	2	2	no	—
114	#11717581	—	2	1	no	—
115	#11717581	—	2	1	no	—
116	#11717581	—	2	2	no	—
117	#11717581	—	2	1	no	—
118	#11717581	—	2	1	no	—
119	#11717581	C₂×D₄	1	4	yes	—
120	C₂²×C₁₀₁	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	1	no	—
122	C₂²×C₁₀₁	—	2	2	no	—
123	C₂²×C₁₀₁	—	2	1	no	—
124	C₂²×C₁₀₁	—	2	1	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₁₀₁	—	2	1	no	—
128	C₂×D₁₀₁	—	2	1	no	—
129	C₂×D₁₀₁	—	2	1	no	—
130	C₂×D₁₀₁	—	2	1	no	—
131	C₂×D₁₀₁	—	4	4	no	—
132	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
133	C₂×D₁₀₁	#131	1	2	yes	complemented
134	C₄×C₁₀₁	—	2	1	no	—
135	C₄×C₁₀₁	—	2	2	no	—
136	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	—	2	1	no	—
139	Dic₁₀₁	—	2	2	no	—
140	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₁₀₁	—	4	1	no	—
143	C₂×C₁₀₁	—	4	2	no	—
144	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
146	C₂×C₁₀₁	#568	1	1	yes	characteristic
147	D₁₀₁	—	2	2	no	—
148	D₁₀₁	—	2	2	no	—
149	#2933	—	101	1	no	maximal, Sylow
150	C₁₀₁	#2933	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	101	1	no	—
152	C₂²×M₁₆	—	101	1	no	—
153	#525	—	101	1	no	—
154	#436	—	101	2	no	—
155	#435	—	101	2	no	—
156	#435	—	101	2	no	—
157	C₈×D₄	—	101	2	no	—
158	#424	—	101	1	no	—
159	#408	—	101	1	no	—
160	C₂×C₂²⋊C₈	—	101	1	no	—
161	#405	—	101	1	no	—
162	C₂×#41	—	101	1	no	—
163	C₂³×C₄	—	101	1	no	—
164	#131	—	202	4	no	—
165	C₂×M₁₆	—	101	1	no	—
166	C₂×M₁₆	—	101	2	no	—
167	C₂×M₁₆	—	101	2	no	—
168	C₂×M₁₆	—	101	1	no	—
169	C₂×M₁₆	—	202	4	no	—
170	C₂²×C₈	—	101	2	no	—
171	C₂²×C₈	—	101	2	no	—
172	C₂²×C₈	—	101	1	no	—
173	C₂²×C₈	—	101	1	no	—
174	#123	—	101	2	no	—
175	C₂²⋊Q₈	—	101	1	no	—
176	C₂²⋊Q₈	—	101	1	no	—
177	#121	—	101	1	no	—
178	#121	#87938	1	1	yes	characteristic
179	#120	—	101	2	no	—
180	C₄×D₄	—	101	2	no	—
181	C₄×D₄	—	101	2	no	—
182	#117	—	101	1	no	—
183	C₄⋊C₈	—	101	2	no	—
184	C₄⋊C₈	—	101	2	no	—
185	C₂²⋊C₈	—	101	1	no	—
186	C₂²⋊C₈	—	101	1	no	—
187	C₂²⋊C₈	—	101	2	no	—
188	C₂²⋊C₈	—	101	2	no	—
189	C₂²⋊C₈	—	101	1	no	—
190	C₂²⋊C₈	—	101	1	no	—
191	#97	—	101	2	no	—
192	C₄×C₈	—	101	2	no	—
193	C₂⁴	—	101	1	no	—
194	#41	—	202	2	no	—
195	#41	—	202	2	no	—
196	C₂×Q₈	—	101	1	no	—
197	C₂×D₄	—	101	2	no	—
198	C₂×D₄	—	101	2	no	—
199	C₂×D₄	C₂×#87938	1	1	yes	characteristic
200	C₂×D₄	C₂×#87938	1	2	yes	—
201	C₂²×C₄	—	101	1	no	—
202	C₂²×C₄	—	101	2	no	—
203	C₂²×C₄	—	101	1	no	—
204	C₂²×C₄	—	101	1	no	—
205	C₂²×C₄	—	101	1	no	—
206	C₂²×C₄	—	101	1	no	—
207	C₂²×C₄	—	202	2	no	—
208	C₂²×C₄	—	202	2	no	—
209	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
210	M₁₆	—	101	4	no	—
211	M₁₆	—	202	4	no	—
212	M₁₆	—	202	4	no	—
213	M₁₆	—	202	2	no	—
214	C₂×C₈	—	101	2	no	—
215	C₂×C₈	—	101	1	no	—
216	C₂×C₈	—	101	1	no	—
217	C₂×C₈	—	101	1	no	—
218	C₂×C₈	—	101	1	no	—
219	C₂×C₈	—	101	2	no	—
220	C₂×C₈	—	202	4	no	—
221	C₂×C₈	—	202	4	no	—
222	C₂×C₈	—	202	2	no	—
223	C₂×C₈	—	202	2	no	—
224	C₄⋊C₄	—	101	2	no	—
225	C₄⋊C₄	—	101	1	no	—
226	C₄⋊C₄	—	101	2	no	—
227	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic
228	C₂²⋊C₄	—	101	2	no	—
229	C₂²⋊C₄	—	101	2	no	—
230	C₂²⋊C₄	—	101	2	no	—
231	C₂²⋊C₄	—	101	2	no	—
232	C₂²⋊C₄	C₂×#87938	1	2	yes	—
233	C₄²	—	101	2	no	—
234	C₂³	—	101	1	no	—
235	C₂³	—	101	1	no	—
236	C₂³	—	404	1	no	—
237	C₂³	—	202	2	no	—
238	C₂³	—	202	2	no	—
239	C₂³	C₂²×#87938	1	1	yes	characteristic
240	C₂³	C₂²×#87938	1	2	yes	—
241	Q₈	—	202	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	202	4	no	—
245	D₄	—	202	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	101	1	no	—
248	C₂×C₄	—	101	2	no	—
249	C₂×C₄	—	101	2	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	101	1	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	202	1	no	—
254	C₂×C₄	—	202	1	no	—
255	C₂×C₄	—	202	2	no	—
256	C₂×C₄	—	202	2	no	—
257	C₂×C₄	—	202	1	no	—
258	C₂×C₄	—	202	1	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	C₂²×#87938	1	2	yes	—
262	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
263	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
264	C₈	—	101	4	no	—
265	C₈	—	202	1	no	—
266	C₈	—	202	2	no	—
267	C₈	—	202	1	no	—
268	C₈	—	202	1	no	—
269	C₈	—	202	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	101	2	no	—
277	C₂²	—	101	2	no	—
278	C₂²	—	404	4	no	—
279	C₂²	—	202	1	no	—
280	C₂²	—	202	1	no	—
281	C₂²	—	202	1	no	—
282	C₂²	—	202	1	no	—
283	C₂²	C₂³×#87938	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	101	1	no	—
287	C₄	—	101	1	no	—
288	C₄	—	202	1	no	—
289	C₄	—	202	2	no	—
290	C₄	—	202	2	no	—
291	C₄	D₄×#87938	1	2	yes	—
292	C₂	—	4	1	no	—
293	C₂	—	4	2	no	—
294	C₂	—	202	2	no	—
295	C₂	—	202	2	no	—
296	C₂	C₂×D₄×#87938	1	1	yes	characteristic
297	C₂	D₁₀₁⋊#131	1	1	yes	characteristic
298	C₂	#20266255	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#121×C₁₀₁	C₄
prime radical	#20265687	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#121	#87938
hypocenter	C₁₀₁	#2933
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#15692342	C₂	semidirect
2. C₂×D₁₀₁⋊M₁₆	C₂	semidirect
3. #20266212	C₂	semidirect
4. #20266209	C₂	semidirect
5. #20266208	C₂	semidirect
6. #20266204	C₂	semidirect
7. #20266184	C₂	semidirect
8. #20266151	C₂	semidirect
9. D₁₀₁⋊(C₂²⋊C₈)	C₂	semidirect
10. #20266142	C₂	semidirect
17. C₂×#13108083	C₂²	semidirect
21. C₂²×#11717581	C₂²	semidirect
22. #15692322	C₂²	semidirect
23. C₄⋊#11717581	C₂²	semidirect
28. C₂²⋊#11717581	C₂²	semidirect
29. C₂²⋊#11717581	C₂²	semidirect
30. #15692310	C₂²	semidirect
31. C₄×#11717581	C₂²	semidirect
44. #13108084	D₄	semidirect
133. C₂×D₁₀₁	#131	semidirect
150. C₁₀₁	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692342	2
C₂×D₁₀₁⋊M₁₆	2
#20266212	2
#20266209	2
#20266208	2
#20266204	2
#20266184	2
#20266151	2
D₁₀₁⋊(C₂²⋊C₈)	2
#20266142	2
#20265687	2
#2933	101