Subgroups — #34203609

Order 20224.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098384	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098340	C₂	1	2	yes	maximal
3	#23763086	C₂	1	1	yes	maximal, characteristic
4	#23761483	C₂	1	2	yes	maximal
5	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579072	C₄	1	1	yes	characteristic
8	C₂×#14579071	C₂²	1	1	yes	characteristic
9	C₂×#14579071	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
12	#19098409	C₄	1	1	yes	characteristic
13	#19098384	C₄	1	4	yes	complemented
14	#19098340	—	2	4	no	—
15	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#364×C₇₉	C₄	1	4	yes	complemented
18	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
22	#14579101	—	2	2	no	—
23	#14579101	C₂×C₄	1	4	yes	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	#14579072	—	2	2	no	—
26	#14579071	—	2	1	no	—
27	#14579071	—	2	1	no	—
28	#14579071	D₄	1	2	yes	—
29	#14579071	Q₈	1	2	yes	—
30	C₈×Dic₇₉	—	2	2	no	—
31	C₈×Dic₇₉	—	2	2	no	—
32	C₇₉⋊#95	—	2	2	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₉	—	2	2	no	—
37	M₃₂×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₇₉	—	2	2	no	—
39	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂²×Dic₇₉	—	2	1	no	—
42	C₂²×Dic₇₉	M₁₆	1	2	yes	—
43	C₂×C₇₉⋊C₈	—	2	1	no	—
44	C₂×C₇₉⋊C₈	—	2	1	no	—
45	C₂×C₇₉⋊C₈	—	2	1	no	—
46	C₂×C₇₉⋊C₈	—	2	1	no	—
47	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
49	C₄×Dic₇₉	—	2	1	no	—
50	C₄×Dic₇₉	—	2	1	no	—
51	C₄×Dic₇₉	—	2	1	no	—
52	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
53	C₇₉⋊C₁₆	—	2	4	no	—
54	C₂²×C₄×C₇₉	—	2	1	no	—
55	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₄²	1	2	yes	—
57	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
58	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
64	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
65	C₄²×C₇₉	—	2	1	no	—
66	C₄²×C₇₉	C₄²	1	2	yes	—
67	C₁₆×C₇₉	—	2	4	no	—
68	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₇₉	—	2	2	no	—
78	C₂×Dic₇₉	—	2	2	no	—
79	C₂×Dic₇₉	—	4	2	no	—
80	C₂×Dic₇₉	—	4	2	no	—
81	C₈×C₇₉	#97	1	4	yes	—
82	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
83	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
84	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
86	C₇₉⋊C₈	—	2	2	no	—
87	C₇₉⋊C₈	—	2	2	no	—
88	C₂²×C₇₉	—	2	2	no	—
89	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
90	C₂²×C₇₉	#423	1	1	yes	characteristic
91	C₂²×C₇₉	#432	1	1	yes	characteristic
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
95	C₄×C₇₉	#423	1	1	yes	characteristic
96	C₄×C₇₉	#432	1	1	yes	characteristic
97	Dic₇₉	—	4	2	no	—
98	Dic₇₉	—	4	2	no	—
99	#12660	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	—	2	1	no	—
101	C₂×C₇₉	—	2	1	no	—
102	C₂×C₇₉	#1767	1	1	yes	characteristic
103	C₂×C₇₉	#2166	1	2	yes	—
104	C₂×#364	—	79	1	no	—
105	C₂×#364	D₇₉	1	1	yes	characteristic
106	C₂×#347	—	79	2	no	—
107	#1726	—	79	1	no	—
108	#1377	—	79	2	no	—
109	C₇₉	#12660	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₃₂	—	79	2	no	—
111	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
112	C₂×#97	—	79	1	no	—
113	C₂×#97	—	79	1	no	—
114	C₂×#97	—	79	1	no	—
115	C₂×C₄×C₈	—	79	1	no	—
116	C₂×C₄×C₈	—	79	1	no	—
117	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	#376	—	79	1	no	—
119	#364	—	79	4	no	—
120	#364	Dic₇₉	1	4	yes	complemented
121	#347	—	158	4	no	—
122	C₂²×C₈	—	79	1	no	—
123	C₂²×C₈	—	79	1	no	—
124	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₂×C₄²	—	79	1	no	—
127	C₂×C₄²	—	79	1	no	—
128	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	79	4	no	—
131	M₃₂	—	158	2	no	—
132	M₃₂	C₂×Dic₇₉	1	4	yes	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	158	2	no	—
135	#97	—	79	2	no	—
136	#97	—	79	2	no	—
137	#97	—	158	2	no	—
138	#97	—	158	1	no	—
139	#97	—	158	1	no	—
140	C₄×C₈	—	158	2	no	—
141	C₄×C₈	—	158	2	no	—
142	C₄×C₈	C₄×D₇₉	1	2	yes	—
143	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
144	#95	—	158	2	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	79	2	no	—
147	C₂²×C₄	—	158	1	no	—
148	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
149	C₂×C₈	—	79	2	no	—
150	C₂×C₈	—	79	2	no	—
151	C₂×C₈	—	158	1	no	—
152	C₂×C₈	—	158	1	no	—
153	C₂×C₈	—	158	1	no	—
154	C₂×C₈	—	158	1	no	—
155	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
156	C₂×C₈	#11929607	1	2	yes	—
157	C₂×C₈	#11929607	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
159	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
160	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
162	C₂×C₈	#11929623	1	1	yes	characteristic
163	C₂×C₈	#11929625	1	1	yes	characteristic
164	C₄²	—	2	1	no	—
165	C₄²	—	79	2	no	—
166	C₄²	—	158	1	no	—
167	C₄²	—	158	1	no	—
168	C₄²	—	158	1	no	—
169	C₄²	C₄×Dic₇₉	1	2	yes	—
170	C₁₆	—	2	4	no	—
171	C₁₆	—	158	4	no	—
172	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	316	2	no	—
176	C₂×C₄	—	316	2	no	—
177	C₂×C₄	—	158	2	no	—
178	C₂×C₄	—	158	2	no	—
179	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
180	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
181	C₂×C₄	#14579119	1	1	yes	characteristic
182	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
183	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
184	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
185	C₈	—	158	2	no	—
186	C₈	—	158	2	no	—
187	C₈	#14579071	1	4	yes	—
188	C₈	#14579123	1	1	yes	characteristic
189	C₈	D₄×Dic₇₉	1	1	yes	characteristic
190	C₈	#14579126	1	1	yes	characteristic
191	C₈	Q₈×Dic₇₉	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	M₁₆×Dic₇₉	1	1	yes	characteristic
194	C₂²	#19098585	1	1	yes	characteristic
195	C₂²	#19098619	1	1	yes	characteristic
196	C₄	—	4	2	no	—
197	C₄	—	316	2	no	—
198	C₄	—	316	2	no	—
199	C₄	C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
200	C₄	C₂×#14579071	1	1	yes	characteristic
201	C₄	#19098585	1	1	yes	characteristic
202	C₄	#19098619	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#23763255	1	1	yes	characteristic
206	C₂	#23765215	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579119
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#432
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#12660
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #19098384	C₄	semidirect
17. #364×C₇₉	C₄	semidirect
109. C₇₉	#12660	semidirect
120. #364	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098384	2
C₂×#19098340	2
#23763086	2
#23761483	2
C₂×#364×C₇₉	2
#12660	79