Subgroups — #23767973

Order 10112.

388 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19098505	C₂	1	2	yes	maximal
2	#19098504	C₂	1	2	yes	maximal
3	#19098470	C₂	1	1	yes	maximal, characteristic
4	#19098470	C₂	1	1	yes	maximal, characteristic
5	#19098461	C₂	1	1	yes	maximal, characteristic
6	#19098448	C₂	1	2	yes	maximal
7	#19098445	C₂	1	2	yes	maximal
8	#19098432	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
11	#391×C₇₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₇₉	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×Dic₇₉⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
22	C₇₉⋊#95	C₂²	1	1	yes	characteristic
23	C₇₉⋊#95	C₂²	1	1	yes	characteristic
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	C₇₉⋊#95	C₂²	1	2	yes	—
29	C₇₉⋊#95	C₂²	1	2	yes	—
30	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
33	#95×C₇₉	C₂²	1	2	yes	—
34	#95×C₇₉	C₂²	1	2	yes	—
35	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₇₉	C₂³	1	2	yes	—
37	C₂²×Dic₇₉	C₂³	1	2	yes	—
38	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
41	C₄⋊Dic₇₉	—	2	2	no	—
42	C₄⋊Dic₇₉	—	2	2	no	—
43	C₄⋊Dic₇₉	—	2	2	no	—
44	C₄⋊Dic₇₉	Q₈	1	4	yes	complemented
45	Dic₇₉⋊C₄	—	2	4	no	—
46	Dic₇₉⋊C₄	—	4	2	no	—
47	Dic₇₉⋊C₄	—	4	2	no	—
48	C₄×Dic₇₉	—	2	2	no	—
49	C₄×Dic₇₉	—	2	2	no	—
50	C₄×Dic₇₉	—	4	2	no	—
51	C₂×Q₈×C₇₉	—	2	2	no	—
52	C₂×Q₈×C₇₉	—	2	2	no	—
53	C₂×Q₈×C₇₉	—	2	2	no	—
54	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	—	2	2	no	—
61	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
62	Q₈×C₇₉	—	4	4	no	—
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₇₉	#41	1	2	yes	—
76	C₂×C₄×C₇₉	#41	1	2	yes	—
77	C₂×C₄×C₇₉	#41	1	2	yes	—
78	C₂×Dic₇₉	—	2	1	no	—
79	C₂×Dic₇₉	—	2	1	no	—
80	C₂×Dic₇₉	—	2	1	no	—
81	C₂×Dic₇₉	—	2	1	no	—
82	C₂×Dic₇₉	—	2	1	no	—
83	C₂×Dic₇₉	—	2	1	no	—
84	C₂×Dic₇₉	—	2	1	no	—
85	C₂×Dic₇₉	—	2	2	no	—
86	C₂×Dic₇₉	—	2	1	no	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	2	2	no	—
89	C₂×Dic₇₉	—	2	2	no	—
90	C₂×Dic₇₉	—	2	2	no	—
91	C₂×Dic₇₉	—	2	2	no	—
92	C₂×Dic₇₉	—	2	1	no	—
93	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
94	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
95	C₂×Dic₇₉	#41	1	2	yes	—
96	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
98	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
99	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
100	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₇₉	—	2	2	no	—
104	C₄×C₇₉	—	4	2	no	—
105	C₄×C₇₉	—	4	1	no	—
106	C₄×C₇₉	—	4	2	no	—
107	C₄×C₇₉	#126	1	4	yes	—
108	Dic₇₉	—	2	2	no	—
109	Dic₇₉	—	2	2	no	—
110	Dic₇₉	—	2	2	no	—
111	Dic₇₉	—	4	2	no	—
112	Dic₇₉	—	4	1	no	—
113	Dic₇₉	—	4	2	no	—
114	C₂×C₇₉	C₂×#126	1	1	yes	characteristic
115	C₂×C₇₉	#533	1	1	yes	characteristic
116	C₂×C₇₉	#542	1	1	yes	characteristic
117	C₂×C₇₉	#552	1	1	yes	characteristic
118	C₂×C₇₉	#554	1	1	yes	characteristic
119	C₂×C₇₉	#557	1	1	yes	characteristic
120	C₂×C₇₉	#559	1	1	yes	characteristic
121	#2511	—	79	1	no	maximal, Sylow
122	C₇₉	#2511	1	1	yes	characteristic, complemented, Sylow
123	#400	—	79	2	no	—
124	#400	—	79	2	no	—
125	#391	—	79	1	no	—
126	#391	—	79	1	no	—
127	#391	D₇₉	1	1	yes	characteristic
128	#389	—	79	1	no	—
129	#387	—	79	2	no	—
130	#387	—	79	2	no	—
131	#384	—	79	1	no	—
132	C₄×C₄⋊C₄	—	79	1	no	—
133	C₄×C₄⋊C₄	—	79	1	no	—
134	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	79	2	no	—
136	C₂×C₄⋊C₄	—	79	2	no	—
137	C₂×C₄⋊C₄	—	79	1	no	—
138	C₂×C₄⋊C₄	—	79	1	no	—
139	C₂×C₄⋊C₄	—	79	1	no	—
140	C₂×C₄⋊C₄	—	79	2	no	—
141	C₂×C₄⋊C₄	—	79	1	no	—
142	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
143	C₂×C₄²	—	79	2	no	—
144	C₂×C₄²	—	79	1	no	—
145	C₂×C₄²	—	79	1	no	—
146	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
147	#95	—	79	1	no	—
148	#95	—	79	1	no	—
149	#95	—	79	2	no	—
150	#95	—	79	2	no	—
151	#95	—	79	2	no	—
152	#95	—	79	2	no	—
153	#95	—	79	2	no	—
154	#95	—	79	2	no	—
155	#95	C₂×D₇₉	1	2	yes	—
156	#95	C₂×D₇₉	1	2	yes	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂²×C₄	—	79	1	no	—
161	C₂²×C₄	—	79	1	no	—
162	C₂²×C₄	—	79	1	no	—
163	C₂²×C₄	—	79	1	no	—
164	C₂²×C₄	—	79	2	no	—
165	C₂²×C₄	—	79	2	no	—
166	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
167	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
169	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	316	2	no	—
173	C₄⋊C₄	—	316	2	no	—
174	C₄⋊C₄	—	79	4	no	—
175	C₄⋊C₄	—	158	4	no	—
176	C₄⋊C₄	—	158	2	no	—
177	C₄⋊C₄	—	158	2	no	—
178	C₄⋊C₄	—	158	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	316	2	no	—
181	C₄²	—	158	2	no	—
182	C₄²	—	158	2	no	—
183	C₂³	C₂³×D₇₉	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	79	2	no	—
197	C₂×C₄	—	79	2	no	—
198	C₂×C₄	—	79	2	no	—
199	C₂×C₄	—	158	1	no	—
200	C₂×C₄	—	158	2	no	—
201	C₂×C₄	—	158	2	no	—
202	C₂×C₄	—	158	2	no	—
203	C₂×C₄	—	158	1	no	—
204	C₂×C₄	—	158	2	no	—
205	C₂×C₄	—	158	2	no	—
206	C₂×C₄	—	158	2	no	—
207	C₂×C₄	—	158	1	no	—
208	C₂×C₄	—	158	1	no	—
209	C₂×C₄	—	158	1	no	—
210	C₂×C₄	—	158	1	no	—
211	C₂×C₄	—	158	1	no	—
212	C₂×C₄	—	158	1	no	—
213	C₂×C₄	—	158	1	no	—
214	C₂×C₄	Q₈×D₇₉	1	2	yes	—
215	C₂×C₄	#11929623	1	2	yes	—
216	C₂×C₄	#11929623	1	2	yes	—
217	C₂×C₄	#11929624	1	2	yes	—
218	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
219	C₂²	#41×D₇₉	1	1	yes	characteristic
220	C₂²	C₂×#11929623	1	1	yes	characteristic
221	C₂²	C₂×#11929623	1	1	yes	characteristic
222	C₂²	C₂×#11929624	1	1	yes	characteristic
223	C₂²	#14579207	1	1	yes	characteristic
224	C₂²	#14579209	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	316	2	no	—
230	C₄	—	316	1	no	—
231	C₄	—	316	2	no	—
232	C₄	—	158	2	no	—
233	C₄	—	158	2	no	—
234	C₄	—	158	2	no	—
235	C₄	#14579154	1	4	yes	—
236	C₂	C₂×#14579154	1	1	yes	characteristic
237	C₂	#19099115	1	1	yes	characteristic
238	C₂	#19099186	1	1	yes	characteristic
239	C₂	#19099254	1	1	yes	characteristic
240	C₂	#19099273	1	1	yes	characteristic
241	C₂	#19099288	1	1	yes	characteristic
242	C₂	#19099295	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#391×C₇₉	C₂
hypercenter	#391	D₇₉
hypocenter	C₇₉	#2511
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
44. C₄⋊Dic₇₉	Q₈	semidirect
122. C₇₉	#2511	semidirect