Subgroups — #34443779

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773940	C₂	1	2	yes	maximal
2	#23765685	C₂	1	4	yes	maximal, complemented
3	#23765677	C₂	1	2	yes	maximal, complemented
4	#23765674	C₂	1	2	yes	maximal, complemented
5	#23765189	C₂	1	2	yes	maximal
6	C₂×#19098389	C₂	1	1	yes	maximal, characteristic
7	#23764996	C₂	1	1	yes	maximal, characteristic
8	C₇₉×#2158	C₂	1	1	yes	maximal, characteristic
9	#19099191	—	2	2	no	—
10	C₂×#14579101	C₂²	1	1	yes	characteristic
11	#19098970	—	2	4	no	—
12	#19098967	C₂²	1	4	yes	—
13	#19098964	C₂²	1	4	yes	—
14	#19098606	—	2	2	no	—
15	C₂×#14579095	C₂²	1	2	yes	—
16	#19098518	C₂²	1	1	yes	characteristic
17	#19098390	C₂²	1	4	yes	—
18	#19098389	C₂²	1	2	yes	—
19	#19098389	C₂²	1	2	yes	—
20	#19098388	C₂²	1	4	yes	—
21	#19098338	C₂²	1	2	yes	—
22	C₁₆×Dic₇₉	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	#427×C₇₉	C₂²	1	2	yes	—
25	#367×C₇₉	C₂²	1	2	yes	—
26	#366×C₇₉	C₂²	1	1	yes	characteristic
27	#366×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#11929608	D₄	1	1	yes	characteristic
29	#14579161	—	4	2	no	—
30	#14579160	—	2	4	no	—
31	#14579153	—	4	2	no	—
32	#14579152	—	2	4	no	—
33	#14579138	—	4	4	no	—
34	#14579120	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇₉	—	2	2	no	—
36	#14579101	Q₈	1	4	yes	complemented
37	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
38	#14579095	—	2	2	no	—
39	#14579095	C₂³	1	4	yes	—
40	#14579094	—	2	4	no	—
41	#14579092	—	2	4	no	—
42	#14579071	D₄	1	2	yes	—
43	C₈×Dic₇₉	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	#117×C₇₉	—	2	2	no	—
46	C₂×C₁₆×C₇₉	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
49	#107×C₇₉	C₂³	1	2	yes	—
50	#107×C₇₉	C₂³	1	2	yes	—
51	#106×C₇₉	—	2	2	no	—
52	C₂×Dic₁₅₈	—	4	4	no	—
53	C₂²×Dic₇₉	—	4	2	no	—
54	#11929608	Q₁₆	1	4	yes	—
55	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
56	C₄⋊Dic₇₉	—	2	4	no	—
57	C₄⋊Dic₇₉	—	4	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
60	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
61	Dic₇₉⋊C₄	—	4	4	no	—
62	Dic₇₉⋊C₄	—	4	4	no	—
63	C₂²⋊Dic₇₉	D₈	1	2	yes	—
64	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
65	C₇₉⋊C₁₆	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₇₉	—	2	2	no	—
72	C₄⋊C₄×C₇₉	—	2	2	no	—
73	C₂²⋊C₄×C₇₉	—	4	2	no	—
74	C₄²×C₇₉	—	4	2	no	—
75	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
76	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
77	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₇₉	—	4	4	no	—
85	C₂×Dic₇₉	—	16	2	no	—
86	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
87	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
88	Dic₁₅₈	—	16	4	no	—
89	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
90	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
92	C₇₉⋊C₈	C₂×Q₁₆	1	4	yes	—
93	C₂²×C₇₉	—	2	1	no	—
94	C₂²×C₇₉	—	2	1	no	—
95	C₂²×C₇₉	#500	1	2	yes	—
96	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₉	—	8	2	no	—
98	C₄×C₇₉	—	8	2	no	—
99	C₄×C₇₉	#500	1	2	yes	—
100	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇₉	—	2	2	no	—
103	Dic₇₉	—	2	2	no	—
104	Dic₇₉	—	8	4	no	—
105	#33729	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	2	2	no	—
107	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
108	C₂×C₇₉	#3415	1	1	yes	characteristic
109	C₂×C₇₉	#3417	1	1	yes	characteristic
110	#3164	—	79	2	no	—
111	#2255	—	79	4	no	—
112	#2253	—	79	2	no	—
113	#2252	—	79	2	no	—
114	#2159	—	79	2	no	—
115	#2158	D₇₉	1	1	yes	characteristic
116	C₂×#366	—	79	1	no	—
117	#2108	—	79	1	no	—
118	C₇₉	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	158	2	no	—
120	C₂×M₃₂	—	79	1	no	—
121	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
122	#501	—	158	4	no	—
123	#500	—	79	4	no	—
124	#499	—	79	4	no	—
125	#428	—	158	2	no	—
126	#427	C₂×D₇₉	1	2	yes	—
127	C₂×#107	—	79	2	no	—
128	#405	—	79	1	no	—
129	#367	—	79	4	no	—
130	#367	C₂×D₇₉	1	2	yes	—
131	#366	—	79	2	no	—
132	#366	—	79	2	no	—
133	#366	—	79	4	no	—
134	#366	C₂×D₇₉	1	1	yes	characteristic
135	#366	C₂×D₇₉	1	1	yes	characteristic
136	#346	—	79	2	no	—
137	C₄×C₁₆	—	79	2	no	—
138	C₂×M₁₆	—	79	1	no	—
139	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
140	C₄⋊Q₈	—	316	2	no	—
141	C₄⋊Q₈	—	158	4	no	—
142	#125	—	316	2	no	—
143	#125	—	158	4	no	—
144	C₂²⋊Q₈	—	316	4	no	—
145	#117	—	2	2	no	—
146	#117	—	79	1	no	—
147	C₂×C₄⋊C₄	—	158	2	no	—
148	M₃₂	—	79	4	no	—
149	C₂×C₁₆	—	79	2	no	—
150	C₂×C₁₆	Dic₁₅₈	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
153	#107	—	79	4	no	—
154	#107	—	158	4	no	—
155	#107	—	158	2	no	—
156	#107	C₂²×D₇₉	1	2	yes	—
157	#107	C₂²×D₇₉	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	158	4	no	—
160	#97	—	79	2	no	—
161	C₄×C₈	—	79	2	no	—
162	C₂×Q₈	—	316	4	no	—
163	C₂²×C₄	—	316	2	no	—
164	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
165	M₁₆	—	79	4	no	—
166	C₂×C₈	—	79	2	no	—
167	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₇₉	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
170	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	316	4	no	—
174	C₄⋊C₄	—	316	4	no	—
175	C₄⋊C₄	—	316	4	no	—
176	C₄⋊C₄	—	316	2	no	—
177	C₄⋊C₄	—	79	1	no	—
178	C₄⋊C₄	—	79	1	no	—
179	C₄⋊C₄	—	158	4	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	79	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	—	79	2	no	—
184	C₁₆	—	79	4	no	—
185	C₁₆	C₂×Dic₁₅₈	1	2	yes	—
186	C₁₆	C₂×Dic₁₅₈	1	2	yes	—
187	C₂³	D₈×D₇₉	1	1	yes	characteristic
188	Q₈	—	1264	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	316	4	no	—
192	C₂×C₄	—	79	2	no	—
193	C₂×C₄	—	79	2	no	—
194	C₂×C₄	—	1264	2	no	—
195	C₂×C₄	#14579160	1	2	yes	—
196	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
199	C₈	—	79	4	no	—
200	C₈	#14579160	1	2	yes	—
201	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₅₈	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#19098967	1	2	yes	—
206	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	—	632	4	no	—
210	C₄	—	158	2	no	—
211	C₄	—	158	2	no	—
212	C₄	#19098967	1	2	yes	—
213	C₄	C₂×#14579160	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#19098967	1	1	yes	characteristic
217	C₂	#23776538	1	1	yes	characteristic
218	C₂	#23776550	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#33729
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #23765685	C₂	semidirect
3. #23765677	C₂	semidirect
4. #23765674	C₂	semidirect
36. #14579101	Q₈	semidirect
46. C₂×C₁₆×C₇₉	Q₈	semidirect
118. C₇₉	#33729	semidirect
150. C₂×C₁₆	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23773940	2
#23765685	2
#23765677	2
#23765674	2
#23765189	2
C₂×#19098389	2
#23764996	2
C₇₉×#2158	2
#33729	79