Subgroups — #34203109

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	2	yes	maximal
2	#23761478	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	4	no	—
5	C₂×#14579101	—	2	4	no	—
6	C₂×#14579101	C₄	1	8	yes	complemented
7	C₂²×C₇₉⋊C₁₆	C₂²	1	4	yes	—
8	C₂³×C₈×C₇₉	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₇₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₉	—	2	4	no	—
12	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	#14579101	—	2	16	no	—
15	C₂×C₇₉⋊C₁₆	—	2	4	no	—
16	C₂×C₇₉⋊C₁₆	—	2	4	no	—
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	8	yes	complemented
18	C₂³×C₄×C₇₉	C₈	1	2	yes	—
19	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	—	2	2	no	—
21	C₂²×C₈×C₇₉	—	2	2	no	—
22	C₂²×C₈×C₇₉	C₈	1	4	yes	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	D₄	1	4	yes	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₈	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₉	—	2	4	no	—
35	C₄×C₈×C₇₉	—	2	4	no	—
36	C₇₉⋊C₁₆	—	2	8	no	—
37	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	—	2	4	no	—
39	C₂²×C₄×C₇₉	—	2	8	no	—
40	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
47	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
48	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	—	2	8	no	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	C₄²	1	2	yes	—
53	C₂×C₈×C₇₉	C₂²⋊C₄	1	8	yes	—
54	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
56	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₇₉	—	2	4	no	—
63	C₄²×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	—	2	2	no	—
65	C₂³×C₇₉	—	2	4	no	—
66	C₂³×C₇₉	C₄×C₈	1	2	yes	—
67	C₂³×C₇₉	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
69	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	16	no	—
71	C₂×C₄×C₇₉	—	2	4	no	—
72	C₂×C₄×C₇₉	—	2	4	no	—
73	C₂×C₄×C₇₉	#95	1	4	yes	—
74	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₇₉	#97	1	4	yes	—
76	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
78	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₇₉	M₃₂	1	8	yes	complemented
81	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₇₉	—	2	4	no	—
88	C₈×C₇₉	#95	1	4	yes	—
89	C₈×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
90	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₇₉	—	2	4	no	—
92	C₂²×C₇₉	—	2	4	no	—
93	C₂²×C₇₉	—	2	4	no	—
94	C₂²×C₇₉	#336	1	4	yes	—
95	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₇₉	C₂×M₃₂	1	4	yes	—
99	C₄×C₇₉	—	2	8	no	—
100	C₄×C₇₉	#336	1	4	yes	—
101	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
102	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
103	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
105	#12544	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	2	4	no	—
107	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
108	C₂×C₇₉	C₄×M₃₂	1	2	yes	—
109	C₂×C₇₉	#2112	1	2	yes	—
110	C₂×C₇₉	#2150	1	2	yes	—
111	C₂²×M₃₂	—	79	2	no	—
112	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
113	#1374	—	79	4	no	—
114	C₇₉	#12544	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₈	Dic₇₉	1	1	yes	characteristic
116	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
117	C₂²×C₄²	Dic₇₉	1	1	yes	characteristic
118	C₂×M₃₂	—	79	8	no	—
119	C₂×M₃₂	—	158	4	no	—
120	C₂×M₃₂	—	158	4	no	—
121	C₂²×C₁₆	—	79	4	no	—
122	C₂×C₄×C₈	—	2	4	no	—
123	C₂×C₄×C₈	Dic₇₉	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
125	C₂³×C₄	C₇₉⋊C₈	1	2	yes	—
126	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	C₇₉⋊C₈	1	4	yes	—
130	C₂²×C₈	Dic₁₅₈	1	2	yes	—
131	C₂²×C₈	C₄×D₇₉	1	4	yes	—
132	C₂²×C₈	D₃₁₆	1	2	yes	—
133	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
134	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₇₉	1	4	yes	—
136	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₇₉⋊C₈	1	2	yes	—
140	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
141	M₃₂	—	158	16	no	—
142	C₂×C₁₆	—	79	8	no	—
143	C₂×C₁₆	—	158	4	no	—
144	C₂×C₁₆	—	158	4	no	—
145	C₄×C₈	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	C₂⁴	C₂×C₇₉⋊C₈	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
151	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
152	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
153	C₂²×C₄	C₂×C₇₉⋊C₈	1	4	yes	—
154	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
156	C₂²×C₄	#11929608	1	2	yes	—
157	C₂²×C₄	#11929608	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
163	C₂×C₈	(C₂×D₇₉)⋊C₄	1	8	yes	—
164	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
165	C₂×C₈	Dic₇₉⋊C₄	1	8	yes	—
166	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
167	C₂×C₈	C₂×C₇₉⋊C₈	1	4	yes	—
168	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
169	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
170	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
172	C₄²	—	2	4	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₁₆	—	158	8	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄×C₇₉⋊C₈	1	2	yes	—
178	C₂³	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
179	C₂³	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
180	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	C₇₉⋊#95	1	4	yes	—
185	C₂×C₄	C₄×C₇₉⋊C₈	1	2	yes	—
186	C₂×C₄	#14579072	1	4	yes	—
187	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
188	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
189	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
190	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	4	yes	—
191	C₂×C₄	#14579101	1	8	yes	complemented
192	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
197	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₇₉⋊#95	1	4	yes	—
200	C₈	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
201	C₈	C₂×Dic₇₉⋊C₄	1	2	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#19098318	1	4	yes	—
206	C₂²	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
207	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
208	C₂²	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
209	C₂²	C₂×#14579101	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#19098318	1	4	yes	—
212	C₄	C₂×C₇₉⋊#95	1	1	yes	characteristic
213	C₄	C₂×#14579072	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
215	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#19098318	1	1	yes	characteristic
218	C₂	C₄×#14579101	1	2	yes	—
219	C₂	#23765011	1	2	yes	—
220	C₂	#23765156	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	#12544
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23761478	C₂	semidirect
6. C₂×#14579101	C₄	semidirect
17. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
80. C₂×C₄×C₇₉	M₃₂	semidirect
114. C₇₉	#12544	semidirect
191. C₂×C₄	#14579101	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
#23761478	2
C₂²×C₄×C₈×C₇₉	2
#12544	79