Subgroups — #34199570

Order 20224.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	1	yes	maximal, characteristic
2	#23765156	C₂	1	1	yes	maximal, characteristic
3	#23761459	C₂	1	2	yes	maximal, complemented
4	#23761456	C₂	1	2	yes	maximal, complemented
5	C₂×#335×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19099120	—	2	1	no	—
7	C₂×#14579101	—	2	1	no	—
8	#19098968	C₂²	1	1	yes	characteristic
9	#19098968	C₂²	1	1	yes	characteristic
10	#19098965	—	2	1	no	—
11	#19098965	C₄	1	2	yes	—
12	#19098961	—	2	1	no	—
13	#19098601	—	2	1	no	—
14	#19098601	C₄	1	2	yes	—
15	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
16	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#335×C₇₉	—	2	1	no	—
20	#335×C₇₉	C₂²	1	2	yes	—
21	#14579161	—	2	1	no	—
22	#14579161	—	2	1	no	—
23	#14579153	—	2	2	no	—
24	#14579140	—	4	1	no	—
25	#14579140	—	4	1	no	—
26	C₄×Dic₁₅₈	—	4	1	no	—
27	C₄×Dic₁₅₈	—	4	1	no	—
28	#14579120	—	2	2	no	—
29	#14579101	—	4	2	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	#14579095	—	2	1	no	—
32	#14579095	—	2	1	no	—
33	#14579095	C₂×C₄	1	2	yes	—
34	#14579095	C₂×C₄	1	2	yes	—
35	#14579093	—	2	2	no	—
36	#14579093	—	2	2	no	—
37	C₂²×C₈×C₇₉	—	4	1	no	—
38	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₉	—	2	2	no	—
42	C₄⋊C₈×C₇₉	—	4	1	no	—
43	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₅₈	—	4	1	no	—
48	C₂×Dic₁₅₈	—	4	1	no	—
49	C₄⋊Dic₇₉	—	2	2	no	—
50	C₄⋊Dic₇₉	—	2	2	no	—
51	C₄⋊Dic₇₉	—	4	1	no	—
52	C₄⋊Dic₇₉	—	4	1	no	—
53	Dic₇₉⋊C₄	—	4	2	no	—
54	Dic₇₉⋊C₄	—	4	2	no	—
55	C₂²⋊Dic₇₉	—	4	2	no	—
56	C₄×Dic₇₉	—	4	2	no	—
57	C₇₉⋊C₁₆	—	4	2	no	—
58	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₂×C₈×C₇₉	—	4	2	no	—
63	C₂×C₈×C₇₉	—	8	2	no	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
67	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
68	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇₉	D₈	1	1	yes	characteristic
72	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
73	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₇₉	#102	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	—	2	1	no	—
77	C₂×C₄×C₇₉	—	4	1	no	—
78	C₂×C₄×C₇₉	#101	1	2	yes	—
79	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#103	1	2	yes	—
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₇₉	—	4	2	no	—
87	C₂×Dic₇₉	—	4	2	no	—
88	Dic₁₅₈	—	8	2	no	—
89	Dic₁₅₈	—	8	1	no	—
90	Dic₁₅₈	—	8	1	no	—
91	C₈×C₇₉	—	2	1	no	—
92	C₈×C₇₉	—	2	1	no	—
93	C₈×C₇₉	—	8	2	no	—
94	C₈×C₇₉	#103	1	2	yes	—
95	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
97	C₂²×C₇₉	—	2	1	no	—
98	C₂²×C₇₉	—	2	1	no	—
99	C₂²×C₇₉	#332	1	2	yes	—
100	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
101	C₄×C₇₉	—	2	1	no	—
102	C₄×C₇₉	—	2	1	no	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	#332	1	2	yes	—
105	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
106	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
107	C₄×C₇₉	#509	1	1	yes	characteristic
108	C₄×C₇₉	#510	1	1	yes	characteristic
109	Dic₇₉	—	8	2	no	—
110	Dic₇₉	—	8	2	no	—
111	#11942	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	—	2	2	no	—
113	C₂×C₇₉	C₂×#332	1	1	yes	characteristic
114	C₂×C₇₉	#2139	1	1	yes	characteristic
115	C₂×C₇₉	#2148	1	1	yes	characteristic
116	#3160	—	79	1	no	—
117	#2150	—	79	1	no	—
118	C₂×#335	D₇₉	1	1	yes	characteristic
119	#1365	—	79	2	no	—
120	#1364	—	79	2	no	—
121	C₇₉	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	158	1	no	—
123	C₂×M₃₂	—	158	1	no	—
124	#500	—	79	1	no	—
125	#500	—	79	1	no	—
126	#499	—	79	2	no	—
127	#499	—	158	1	no	—
128	#498	—	158	1	no	—
129	#427	—	79	2	no	—
130	#427	—	158	1	no	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	79	2	no	—
134	C₄⋊C₁₆	—	158	1	no	—
135	#335	—	2	1	no	—
136	#335	C₂×D₇₉	1	2	yes	—
137	C₂²×C₈	—	4	1	no	—
138	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
139	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
140	C₄⋊Q₈	—	158	1	no	—
141	C₄⋊Q₈	—	158	1	no	—
142	#125	—	158	2	no	—
143	C₂²⋊Q₈	—	316	1	no	—
144	C₂²⋊Q₈	—	316	1	no	—
145	C₄×Q₈	—	316	1	no	—
146	C₄×Q₈	—	316	1	no	—
147	#117	—	158	2	no	—
148	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
149	M₃₂	—	316	2	no	—
150	C₂×C₁₆	—	158	2	no	—
151	#107	—	79	2	no	—
152	#107	—	79	2	no	—
153	#107	—	158	1	no	—
154	#107	—	158	1	no	—
155	#106	—	158	2	no	—
156	#106	—	158	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
160	C₄×C₈	D₃₁₆	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
163	C₂×Q₈	—	316	1	no	—
164	C₂×Q₈	—	316	1	no	—
165	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	#11929609	1	1	yes	characteristic
167	C₂²×C₄	#11929611	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11929613	1	2	yes	—
174	C₂×C₈	Dic₃₁₆	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
176	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
177	C₄⋊C₄	—	316	2	no	—
178	C₄⋊C₄	—	316	2	no	—
179	C₄⋊C₄	—	316	1	no	—
180	C₄⋊C₄	—	316	1	no	—
181	C₄⋊C₄	—	158	2	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₂²⋊C₄	—	316	2	no	—
184	C₄²	—	316	2	no	—
185	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
186	C₄²	#11929609	1	1	yes	characteristic
187	C₄²	#11929611	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	C₁₆	—	316	2	no	—
190	C₂³	#14579081	1	1	yes	characteristic
191	Q₈	—	632	2	no	—
192	Q₈	—	632	1	no	—
193	Q₈	—	632	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	316	2	no	—
198	C₂×C₄	—	316	2	no	—
199	C₂×C₄	#14579079	1	2	yes	—
200	C₂×C₄	#14579081	1	1	yes	characteristic
201	C₂×C₄	#14579081	1	1	yes	characteristic
202	C₂×C₄	#14579081	1	1	yes	characteristic
203	C₂×C₄	#14579086	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
206	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	8	2	no	—
210	C₈	#14579086	1	2	yes	—
211	C₈	C₂×#11929613	1	1	yes	characteristic
212	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#19098310	1	2	yes	—
216	C₂²	C₂×#14579081	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	632	2	no	—
221	C₄	—	632	2	no	—
222	C₄	#19098310	1	2	yes	—
223	C₄	C₂×#14579079	1	1	yes	characteristic
224	C₄	C₂×#14579086	1	1	yes	characteristic
225	C₄	#19098999	1	1	yes	characteristic
226	C₄	#19099004	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#19098310	1	1	yes	characteristic
229	C₂	#23765110	1	1	yes	characteristic
230	C₂	#23765147	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579081
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#335×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	#102
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#335	D₇₉
hypocenter	C₇₉	#11942
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #23761459	C₂	semidirect
4. #23761456	C₂	semidirect
121. C₇₉	#11942	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
#23765156	2
#23761459	2
#23761456	2
C₂×#335×C₇₉	2
#11942	79