Subgroups — #27179657

Order 12928.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692300	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692263	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265657	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265180	C₂	1	2	yes	maximal, complemented
5	#20265179	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265175	C₂	1	2	yes	maximal, complemented
7	#20265171	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265170	C₂	1	2	yes	maximal, complemented
9	C₂×#15692176	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₀₁	C₂	1	2	yes	maximal, complemented
11	#418×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#15692300	—	2	4	no	—
13	#15692300	C₄	1	8	yes	complemented
14	C₂×#13108074	C₂²	1	1	yes	characteristic
15	C₂×#13108074	C₂²	1	1	yes	characteristic
16	C₂×#13108073	—	2	1	no	—
17	C₂×#13108073	C₂²	1	2	yes	—
18	C₂²×C₂²⋊D₁₀₁	—	2	2	no	—
19	C₂×D₄×D₁₀₁	—	2	1	no	—
20	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
21	#15692263	—	2	4	no	—
22	C₂×#13108057	C₂²	1	2	yes	complemented
23	C₂²×#11717579	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
25	C₄×Dic₂₀₂	—	2	1	no	—
26	C₄×D₄₀₄	—	2	1	no	—
27	#15692215	—	2	2	no	—
28	#15692213	C₂²	1	2	yes	complemented
29	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	complemented
31	#15692180	C₂²	1	2	yes	complemented
32	#15692178	C₂²	1	1	yes	characteristic, complemented
33	#15692178	C₂²	1	1	yes	characteristic, complemented
34	#15692177	C₂²	1	2	yes	complemented
35	#15692176	C₂²	1	2	yes	complemented
36	#15692176	C₂²	1	2	yes	complemented
37	#15692175	C₂²	1	1	yes	characteristic, complemented
38	#15692175	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
41	#117×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₀₁	C₂²	1	2	yes	complemented
43	#102×C₁₀₁	C₂²	1	2	yes	complemented
44	C₂³×D₁₀₁	—	2	2	no	—
45	#13108074	—	2	2	no	—
46	#13108074	C₂×C₄	1	4	yes	—
47	#13108074	C₂×C₄	1	4	yes	complemented
48	#13108074	D₄	1	4	yes	—
49	#13108073	—	2	4	no	—
50	#13108073	—	2	8	no	—
51	#13108073	C₂×C₄	1	8	yes	complemented
52	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
53	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
54	C₂×D₄₀₄	D₄	1	1	yes	characteristic
55	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
57	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
58	C₂×C₂²⋊D₁₀₁	—	2	16	no	—
59	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
60	C₂×C₂²⋊D₁₀₁	D₄	1	2	yes	—
61	D₄×D₁₀₁	—	2	8	no	—
62	D₄×D₁₀₁	—	2	4	no	—
63	D₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
64	C₂²×Dic₁₀₁	—	2	2	no	—
65	C₂²×Dic₁₀₁	—	2	1	no	—
66	C₂×C₄×D₁₀₁	—	2	1	no	—
67	C₂×C₄×D₁₀₁	—	2	2	no	—
68	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
69	#13108057	—	2	4	no	—
70	C₂×#11717579	—	2	2	no	—
71	C₂×#11717579	C₂³	1	1	yes	characteristic
72	C₂×#11717579	C₂³	1	1	yes	characteristic
73	C₈×D₁₀₁	—	2	4	no	—
74	C₄⋊Dic₁₀₁	—	2	1	no	—
75	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
76	#13108053	—	2	2	no	—
77	#13108051	—	2	2	no	—
78	C₄×Dic₁₀₁	—	2	2	no	—
79	C₂⁴×C₁₀₁	—	2	1	no	—
80	C₂×D₄×C₁₀₁	—	2	1	no	—
81	C₂×D₄×C₁₀₁	—	2	2	no	—
82	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
83	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	M₁₆×C₁₀₁	—	2	2	no	—
86	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₀₁	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
90	C₄²×C₁₀₁	—	2	1	no	—
91	C₂³×C₁₀₁	—	2	2	no	—
92	C₂³×C₁₀₁	—	2	4	no	—
93	C₂³×C₁₀₁	—	8	1	no	—
94	C₂³×C₁₀₁	—	8	1	no	—
95	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×D₁₀₁	—	2	1	no	—
97	C₂²×D₁₀₁	—	2	8	no	—
98	C₂²×D₁₀₁	—	8	2	no	—
99	C₂²×D₁₀₁	—	8	2	no	—
100	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
101	D₄×C₁₀₁	—	2	4	no	—
102	D₄×C₁₀₁	—	2	2	no	—
103	D₄×C₁₀₁	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	1	no	—
106	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₀₁	—	2	4	no	—
111	C₂²⋊D₁₀₁	—	2	8	no	—
112	C₂²⋊D₁₀₁	—	2	8	no	—
113	C₂²⋊D₁₀₁	C₂²⋊C₄	1	8	yes	—
114	C₂×Dic₁₀₁	—	2	1	no	—
115	C₂×Dic₁₀₁	—	2	8	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	8	1	no	—
118	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
119	D₄₀₄	—	2	1	no	—
120	D₄₀₄	C₂²⋊C₄	1	4	yes	—
121	D₄₀₄	C₂²×C₄	1	2	yes	—
122	C₄×D₁₀₁	—	2	4	no	—
123	C₄×D₁₀₁	—	8	2	no	—
124	C₄×D₁₀₁	C₂²×C₄	1	4	yes	—
125	C₄×D₁₀₁	C₂×D₄	1	4	yes	—
126	Dic₂₀₂	—	2	1	no	—
127	Dic₂₀₂	C₂²⋊C₄	1	4	yes	—
128	Dic₂₀₂	C₂²×C₄	1	2	yes	—
129	C₈×C₁₀₁	—	2	2	no	—
130	#11717579	—	2	1	no	—
131	#11717579	—	2	1	no	—
132	C₂²×C₁₀₁	—	2	1	no	—
133	C₂²×C₁₀₁	—	2	1	no	—
134	C₂²×C₁₀₁	—	2	4	no	—
135	C₂²×C₁₀₁	—	4	2	no	—
136	C₂²×C₁₀₁	—	4	2	no	—
137	C₂²×C₁₀₁	—	8	4	no	—
138	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₀₁	—	2	2	no	—
141	C₂×D₁₀₁	—	2	2	no	—
142	C₂×D₁₀₁	—	2	2	no	—
143	C₂×D₁₀₁	—	4	1	no	—
144	C₂×D₁₀₁	—	4	1	no	—
145	C₂×D₁₀₁	—	8	8	no	—
146	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₁₀₁	—	4	1	no	—
148	C₄×C₁₀₁	—	4	1	no	—
149	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
152	Dic₁₀₁	—	2	2	no	—
153	Dic₁₀₁	—	4	2	no	—
154	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₁₀₁	—	2	2	no	—
156	C₂×C₁₀₁	—	4	4	no	—
157	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₁₀₁	#576	1	1	yes	characteristic
159	C₂×C₁₀₁	#577	1	1	yes	characteristic
160	D₁₀₁	—	2	4	no	—
161	D₁₀₁	—	4	2	no	—
162	#2897	—	101	1	no	maximal, Sylow
163	C₁₀₁	#2897	1	1	yes	characteristic, complemented, Sylow
164	C₂×H(5,𝔽₂)	—	101	1	no	—
165	C₂×#131	—	101	1	no	—
166	#520	—	101	1	no	—
167	#418	—	101	2	no	—
168	#418	D₁₀₁	1	1	yes	characteristic, complemented
169	#417	—	101	1	no	—
170	#417	—	101	2	no	—
171	#416	—	101	1	no	—
172	#416	—	101	2	no	—
173	C₂×#102	—	101	1	no	—
174	C₂×#102	—	101	2	no	—
175	H(5,𝔽₂)	—	101	8	no	—
176	H(5,𝔽₂)	—	202	4	no	—
177	C₂×#41	—	101	1	no	—
178	C₂×#41	—	101	2	no	—
179	C₂×#41	—	101	1	no	—
180	C₂×#41	—	202	1	no	—
181	C₂²×D₄	—	101	2	no	—
182	C₂²×D₄	—	202	1	no	—
183	C₂²×D₄	—	202	2	no	—
184	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
185	#131	—	202	4	no	—
186	C₂×M₁₆	—	101	2	no	—
187	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
188	C₂²×C₈	—	101	1	no	—
189	C₂²×C₈	—	101	2	no	—
190	C₄×Q₈	—	202	1	no	—
191	C₄×D₄	—	202	1	no	—
192	C₄×D₄	—	202	2	no	—
193	#117	—	101	2	no	—
194	#117	C₂×D₁₀₁	1	1	yes	characteristic, complemented
195	C₂×C₄⋊C₄	—	101	1	no	—
196	C₂×C₄⋊C₄	—	101	2	no	—
197	#103	—	101	2	no	—
198	#103	—	101	1	no	—
199	#103	—	101	1	no	—
200	#102	—	101	2	no	—
201	#102	—	101	2	no	—
202	#102	—	101	2	no	—
203	#102	—	101	1	no	—
204	#102	—	101	1	no	—
205	#102	C₂×D₁₀₁	1	2	yes	complemented
206	#102	C₂×D₁₀₁	1	2	yes	complemented
207	C₂⁴	—	2	1	no	—
208	C₂⁴	—	202	2	no	—
209	#41	—	101	4	no	—
210	#41	—	101	4	no	—
211	#41	—	101	8	no	—
212	#41	—	101	4	no	—
213	#41	—	202	4	no	—
214	#41	—	202	8	no	—
215	#41	—	202	2	no	—
216	C₂×Q₈	—	101	1	no	—
217	C₂×Q₈	—	101	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	101	2	no	—
221	C₂×D₄	—	101	8	no	—
222	C₂×D₄	—	101	1	no	—
223	C₂×D₄	—	101	1	no	—
224	C₂×D₄	—	202	4	no	—
225	C₂×D₄	—	202	2	no	—
226	C₂×D₄	—	202	8	no	—
227	C₂×D₄	—	202	16	no	—
228	C₂×D₄	—	202	2	no	—
229	C₂×D₄	—	202	1	no	—
230	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
231	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
232	C₂²×C₄	—	101	2	no	—
233	C₂²×C₄	—	202	2	no	—
234	C₂²×C₄	—	202	1	no	—
235	C₂²×C₄	—	202	2	no	—
236	C₂²×C₄	—	202	1	no	—
237	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	202	4	no	—
240	C₂×C₈	—	101	1	no	—
241	C₂×C₈	—	101	1	no	—
242	C₂×C₈	—	202	4	no	—
243	C₂×C₈	—	202	2	no	—
244	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
245	C₄⋊C₄	—	101	2	no	—
246	C₄⋊C₄	—	202	2	no	—
247	C₄⋊C₄	—	202	1	no	—
248	C₄⋊C₄	C₂²×D₁₀₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₀₁	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	202	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	202	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	8	1	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	808	2	no	—
259	C₂³	—	808	2	no	—
260	C₂³	—	101	2	no	—
261	C₂³	—	202	8	no	—
262	C₂³	—	202	1	no	—
263	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
264	Q₈	—	101	4	no	—
265	Q₈	—	101	2	no	—
266	Q₈	—	202	1	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	4	no	—
269	D₄	—	101	8	no	—
270	D₄	—	101	2	no	—
271	D₄	—	101	4	no	—
272	D₄	—	202	8	no	—
273	D₄	—	202	8	no	—
274	D₄	—	202	4	no	—
275	D₄	—	202	1	no	—
276	D₄	C₂×C₄×D₁₀₁	1	4	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	808	2	no	—
280	C₂×C₄	—	808	1	no	—
281	C₂×C₄	—	101	2	no	—
282	C₂×C₄	—	101	4	no	—
283	C₂×C₄	—	101	4	no	—
284	C₂×C₄	—	202	8	no	—
285	C₂×C₄	—	202	2	no	—
286	C₂×C₄	—	202	1	no	—
287	C₂×C₄	—	202	4	no	—
288	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
289	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
290	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	202	1	no	—
294	C₈	—	202	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	808	8	no	—
302	C₂²	—	101	4	no	—
303	C₂²	—	404	1	no	—
304	C₂²	—	404	1	no	—
305	C₂²	—	202	2	no	—
306	C₂²	—	202	2	no	—
307	C₂²	—	202	2	no	—
308	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
309	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	101	4	no	—
313	C₄	—	404	2	no	—
314	C₄	—	202	2	no	—
315	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
316	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
317	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	4	no	—
320	C₂	—	404	2	no	—
321	C₂	—	202	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
323	C₂	#20265981	1	1	yes	characteristic
324	C₂	#20265987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#418×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#15692300	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#418	D₁₀₁
hypocenter	C₁₀₁	#2897
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
2. C₂×#15692263	C₂	semidirect
3. #20265657	C₂	semidirect
4. #20265180	C₂	semidirect
5. #20265179	C₂	semidirect
6. #20265175	C₂	semidirect
7. #20265171	C₂	semidirect
8. #20265170	C₂	semidirect
9. C₂×#15692176	C₂	semidirect
10. #102×D₁₀₁	C₂	semidirect
11. #418×C₁₀₁	C₂	semidirect
13. #15692300	C₄	semidirect
22. C₂×#13108057	C₂²	semidirect
23. C₂²×#11717579	C₂²	semidirect
24. C₂×C₈×D₁₀₁	C₂²	semidirect
28. #15692213	C₂²	semidirect
29. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
30. C₄⋊C₄×D₁₀₁	C₂²	semidirect
31. #15692180	C₂²	semidirect
32. #15692178	C₂²	semidirect
33. #15692178	C₂²	semidirect
34. #15692177	C₂²	semidirect
35. #15692176	C₂²	semidirect
36. #15692176	C₂²	semidirect
37. #15692175	C₂²	semidirect
38. #15692175	C₂²	semidirect
40. C₂×M₁₆×C₁₀₁	C₂²	semidirect
41. #117×C₁₀₁	C₂²	semidirect
42. #102×C₁₀₁	C₂²	semidirect
43. #102×C₁₀₁	C₂²	semidirect
47. #13108074	C₂×C₄	semidirect
51. #13108073	C₂×C₄	semidirect
63. D₄×D₁₀₁	C₂×C₄	semidirect
82. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
163. C₁₀₁	#2897	semidirect
168. #418	D₁₀₁	semidirect
187. C₂×M₁₆	C₂×D₁₀₁	semidirect
194. #117	C₂×D₁₀₁	semidirect
205. #102	C₂×D₁₀₁	semidirect
206. #102	C₂×D₁₀₁	semidirect
230. C₂×D₄	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692300	2
C₂×#15692263	2
#20265657	2
#20265180	2
#20265179	2
#20265175	2
#20265171	2
#20265170	2
C₂×#15692176	2
#102×D₁₀₁	2
#418×C₁₀₁	2
#2897	101