Subgroups — #23767997

Order 10112.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098492	C₂	1	2	yes	maximal, complemented
2	#19098489	C₂	1	2	yes	maximal, complemented
3	#19098465	C₂	1	1	yes	maximal, characteristic
4	#19098465	C₂	1	1	yes	maximal, characteristic
5	#19098465	C₂	1	1	yes	maximal, characteristic
6	#19098461	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098453	C₂	1	2	yes	maximal, complemented
8	#19098452	C₂	1	2	yes	maximal, complemented
9	#19098429	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
18	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
24	C₇₉⋊#95	C₂²	1	2	yes	—
25	C₇₉⋊#95	C₂²	1	2	yes	—
26	C₇₉⋊#95	C₂²	1	2	yes	—
27	C₇₉⋊#95	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
33	#95×C₇₉	C₂²	1	1	yes	characteristic
34	#95×C₇₉	C₂²	1	1	yes	characteristic
35	C₂³×D₇₉	C₂³	1	2	yes	—
36	C₂×D₃₁₆	—	2	2	no	—
37	C₂×D₃₁₆	—	2	2	no	—
38	C₂×D₃₁₆	—	2	2	no	—
39	C₂²×Dic₇₉	C₂³	1	2	yes	—
40	C₂²×Dic₇₉	C₂³	1	2	yes	—
41	C₂²×Dic₇₉	C₂³	1	2	yes	—
42	C₄⋊Dic₇₉	—	2	2	no	—
43	C₄⋊Dic₇₉	—	2	2	no	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	Dic₇₉⋊C₄	—	4	2	no	—
46	(C₂×D₇₉)⋊C₄	—	2	4	no	—
47	(C₂×D₇₉)⋊C₄	—	2	4	no	—
48	(C₂×D₇₉)⋊C₄	—	4	2	no	—
49	(C₂×D₇₉)⋊C₄	—	4	2	no	—
50	(C₂×D₇₉)⋊C₄	—	4	2	no	—
51	(C₂×D₇₉)⋊C₄	—	4	2	no	—
52	C₄×Dic₇₉	—	4	2	no	—
53	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₇₉	—	2	2	no	—
61	C₄⋊C₄×C₇₉	—	2	2	no	—
62	C₄²×C₇₉	—	2	2	no	—
63	C₄²×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	—	2	2	no	—
65	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
66	C₂²×D₇₉	—	2	2	no	—
67	C₂²×D₇₉	—	2	2	no	—
68	C₂²×D₇₉	—	2	2	no	—
69	C₂²×D₇₉	—	2	2	no	—
70	C₂²×D₇₉	—	2	2	no	—
71	C₂²×D₇₉	—	2	2	no	—
72	C₂²×D₇₉	—	2	2	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	2	1	no	—
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	—	2	1	no	—
77	C₂×C₄×C₇₉	—	2	1	no	—
78	C₂×C₄×C₇₉	—	2	1	no	—
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	#41	1	2	yes	—
90	C₂×C₄×C₇₉	#41	1	2	yes	—
91	C₂×C₄×C₇₉	#41	1	2	yes	—
92	C₂×C₄×C₇₉	#41	1	2	yes	—
93	C₂×C₄×C₇₉	#41	1	2	yes	—
94	C₂×Dic₇₉	—	2	2	no	—
95	C₂×Dic₇₉	—	2	2	no	—
96	C₂×Dic₇₉	—	2	2	no	—
97	C₂×Dic₇₉	—	2	2	no	—
98	C₂×Dic₇₉	—	2	2	no	—
99	C₂×Dic₇₉	—	2	2	no	—
100	C₂×Dic₇₉	—	2	2	no	—
101	C₂×Dic₇₉	—	2	2	no	—
102	C₂×Dic₇₉	—	2	2	no	—
103	D₃₁₆	—	4	4	no	—
104	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
105	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
106	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
108	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×D₇₉	—	4	2	no	—
112	C₂×D₇₉	—	4	2	no	—
113	C₂×D₇₉	—	4	2	no	—
114	C₂×D₇₉	—	4	2	no	—
115	C₂×D₇₉	—	4	2	no	—
116	C₂×D₇₉	—	4	2	no	—
117	C₂×D₇₉	—	4	2	no	—
118	C₄×C₇₉	—	2	2	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	4	1	no	—
121	C₄×C₇₉	—	4	1	no	—
122	C₄×C₇₉	—	4	1	no	—
123	C₄×C₇₉	—	4	1	no	—
124	C₄×C₇₉	#126	1	4	yes	—
125	Dic₇₉	—	4	2	no	—
126	Dic₇₉	—	4	2	no	—
127	Dic₇₉	—	4	2	no	—
128	C₂×C₇₉	C₂×#126	1	1	yes	characteristic
129	C₂×C₇₉	#529	1	1	yes	characteristic
130	C₂×C₇₉	#540	1	1	yes	characteristic
131	C₂×C₇₉	#553	1	1	yes	characteristic
132	C₂×C₇₉	#555	1	1	yes	characteristic
133	C₂×C₇₉	#559	1	1	yes	characteristic
134	C₂×C₇₉	#559	1	1	yes	characteristic
135	D₇₉	—	8	2	no	—
136	#2513	—	79	1	no	maximal, Sylow
137	C₇₉	#2513	1	1	yes	characteristic, complemented, Sylow
138	#397	—	79	2	no	—
139	#397	—	79	2	no	—
140	#390	—	79	1	no	—
141	#390	—	79	1	no	—
142	#390	—	79	1	no	—
143	#389	—	79	1	no	—
144	#388	—	79	2	no	—
145	#388	—	79	2	no	—
146	#382	—	79	1	no	—
147	C₄×C₄⋊C₄	—	79	1	no	—
148	C₄×C₄⋊C₄	D₇₉	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	79	1	no	—
150	C₂×C₄⋊C₄	—	79	2	no	—
151	C₂×C₄⋊C₄	—	79	1	no	—
152	C₂×C₄⋊C₄	—	79	1	no	—
153	C₂×C₄⋊C₄	—	79	1	no	—
154	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	79	2	no	—
157	C₂×C₂²⋊C₄	—	79	2	no	—
158	C₂×C₂²⋊C₄	—	79	2	no	—
159	C₂×C₂²⋊C₄	—	79	2	no	—
160	C₂×C₂²⋊C₄	—	79	2	no	—
161	C₂×C₂²⋊C₄	—	79	2	no	—
162	C₂×C₄²	—	79	2	no	—
163	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
164	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
166	#95	—	79	2	no	—
167	#95	—	79	2	no	—
168	#95	—	79	2	no	—
169	#95	—	79	2	no	—
170	#95	C₂×D₇₉	1	1	yes	characteristic
171	#95	C₂×D₇₉	1	1	yes	characteristic
172	C₂⁴	—	79	2	no	—
173	C₂×D₄	—	158	2	no	—
174	C₂×D₄	—	158	2	no	—
175	C₂×D₄	—	158	2	no	—
176	C₂²×C₄	—	79	2	no	—
177	C₂²×C₄	—	79	2	no	—
178	C₂²×C₄	—	79	2	no	—
179	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	316	2	no	—
189	C₄⋊C₄	—	158	2	no	—
190	C₄⋊C₄	—	158	2	no	—
191	C₄⋊C₄	—	158	2	no	—
192	C₂²⋊C₄	—	316	2	no	—
193	C₂²⋊C₄	—	316	2	no	—
194	C₂²⋊C₄	—	316	2	no	—
195	C₂²⋊C₄	—	316	2	no	—
196	C₂²⋊C₄	—	158	4	no	—
197	C₂²⋊C₄	—	158	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	316	2	no	—
202	C₂³	—	158	2	no	—
203	C₂³	—	158	2	no	—
204	C₂³	—	158	2	no	—
205	C₂³	—	158	2	no	—
206	C₂³	—	158	2	no	—
207	C₂³	—	158	2	no	—
208	C₂³	—	158	2	no	—
209	C₂³	C₂³×D₇₉	1	1	yes	characteristic
210	D₄	—	316	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	158	2	no	—
228	C₂×C₄	—	158	2	no	—
229	C₂×C₄	—	158	2	no	—
230	C₂×C₄	—	158	2	no	—
231	C₂×C₄	—	158	2	no	—
232	C₂×C₄	—	158	2	no	—
233	C₂×C₄	—	158	2	no	—
234	C₂×C₄	—	158	2	no	—
235	C₂×C₄	—	158	2	no	—
236	C₂×C₄	#11929623	1	2	yes	—
237	C₂×C₄	#11929624	1	2	yes	—
238	C₂×C₄	#11929624	1	2	yes	—
239	C₂×C₄	#11929624	1	2	yes	—
240	C₂×C₄	#11929625	1	2	yes	—
241	C₂²	—	316	2	no	—
242	C₂²	—	316	2	no	—
243	C₂²	—	316	2	no	—
244	C₂²	—	316	2	no	—
245	C₂²	—	316	2	no	—
246	C₂²	—	316	2	no	—
247	C₂²	—	316	2	no	—
248	C₂²	C₂×#11929623	1	1	yes	characteristic
249	C₂²	C₂×#11929624	1	1	yes	characteristic
250	C₂²	C₂×#11929624	1	1	yes	characteristic
251	C₂²	C₂×#11929624	1	1	yes	characteristic
252	C₂²	C₂×#11929625	1	1	yes	characteristic
253	C₂²	#14579207	1	1	yes	characteristic
254	C₂²	#14579207	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	316	2	no	—
262	C₄	—	316	2	no	—
263	C₄	—	316	2	no	—
264	C₄	#14579156	1	4	yes	—
265	C₂	—	632	2	no	—
266	C₂	C₂×#14579156	1	1	yes	characteristic
267	C₂	#19099097	1	1	yes	characteristic
268	C₂	#19099168	1	1	yes	characteristic
269	C₂	#19099259	1	1	yes	characteristic
270	C₂	#19099278	1	1	yes	characteristic
271	C₂	#19099298	1	1	yes	characteristic
272	C₂	#19099298	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₇₉	C₂
prime radical	C₂²×D₃₁₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#2513
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #19098492	C₂	semidirect
2. #19098489	C₂	semidirect
6. #19098461	C₂	semidirect
7. #19098453	C₂	semidirect
8. #19098452	C₂	semidirect
9. #19098429	C₂	semidirect
10. C₄×C₄⋊Dic₇₉	C₂	semidirect
11. C₄×C₄⋊C₄×C₇₉	C₂	semidirect
137. C₇₉	#2513	semidirect
148. C₄×C₄⋊C₄	D₇₉	semidirect