Subgroups — C₂×#23765762 #34446489

Order 20224.

4694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃₁₆	C₂	1	3	yes	maximal
2	#23765762	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₇₉	C₂	1	3	yes	maximal, complemented
4	C₂×#19098410	C₂	1	6	yes	maximal, complemented
5	C₂×#95×D₇₉	C₂	1	2	yes	maximal, complemented
6	C₂×#376×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₇₉	C₂²	1	6	yes	—
9	C₂×C₄×D₃₁₆	C₄	1	48	yes	complemented
10	C₂×C₄×D₃₁₆	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₇₉	C₂²	1	3	yes	—
12	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₇₉	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₇₉	C₂²	1	24	yes	complemented
15	#19098410	C₂²	1	48	yes	complemented
16	C₂×C₇₉⋊#95	C₂²	1	6	yes	—
17	#95×D₇₉	C₂²	1	16	yes	complemented
18	C₂²×C₄²×C₇₉	C₂²	1	3	yes	—
19	#376×C₇₉	C₂²	1	8	yes	complemented
20	C₂×#95×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#95×C₇₉	C₂²	1	3	yes	—
22	C₂⁴×D₇₉	C₂³	1	2	yes	—
23	C₂²×D₃₁₆	C₂×C₄	1	24	yes	—
24	C₂²×D₃₁₆	C₂³	1	4	yes	—
25	C₂³×Dic₇₉	C₂³	1	6	yes	—
26	C₂²×C₄×D₇₉	—	4	24	no	—
27	C₂²×C₄×D₇₉	C₂×C₄	1	48	yes	complemented
28	C₂²×C₄×D₇₉	C₂³	1	24	yes	—
29	C₄×D₃₁₆	C₂×C₄	1	192	yes	complemented
30	C₂×C₄⋊Dic₇₉	C₂×C₄	1	24	yes	complemented
31	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
32	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	48	yes	complemented
33	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	24	yes	—
34	C₂×C₄×Dic₇₉	—	4	12	no	—
35	C₂×C₄×Dic₇₉	C₂³	1	24	yes	—
36	C₇₉⋊#95	C₂³	1	24	yes	—
37	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	C₂×C₄	1	24	yes	complemented
41	C₂×C₄²×C₇₉	C₂³	1	12	yes	—
42	#95×C₇₉	C₂³	1	4	yes	—
43	#95×C₇₉	C₂³	1	12	yes	—
44	C₂³×D₇₉	—	2	12	no	—
45	C₂³×D₇₉	—	2	4	no	—
46	C₂³×D₇₉	C₂²×C₄	1	24	yes	—
47	C₂³×D₇₉	C₂⁴	1	4	yes	—
48	C₂×D₃₁₆	C₄²	1	64	yes	complemented
49	C₂×D₃₁₆	C₂²×C₄	1	48	yes	—
50	C₂²×Dic₇₉	—	2	24	no	—
51	C₂²×Dic₇₉	C₂²×C₄	1	24	yes	—
52	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
53	C₂×C₄×D₇₉	—	4	96	no	—
54	C₂×C₄×D₇₉	—	4	48	no	—
55	C₂×C₄×D₇₉	C₂²×C₄	1	96	yes	complemented
56	C₄⋊Dic₇₉	C₂²×C₄	1	48	yes	complemented
57	(C₂×D₇₉)⋊C₄	C₂²×C₄	1	96	yes	complemented
58	C₄×Dic₇₉	—	4	24	no	—
59	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	—	2	6	no	—
61	C₂²×C₄×C₇₉	—	2	12	no	—
62	C₂²×C₄×C₇₉	—	2	6	no	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
66	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	48	yes	complemented
70	C₂³×C₇₉	C₂³×C₄	1	6	yes	—
71	C₂³×C₇₉	H(5,𝔽₂)	1	6	yes	—
72	C₂³×C₇₉	2₋¹⁺⁴	1	2	yes	—
73	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
74	C₂²×D₇₉	—	2	48	no	—
75	C₂²×D₇₉	—	2	48	no	—
76	C₂²×D₇₉	—	2	12	no	—
77	C₂²×D₇₉	—	2	4	no	—
78	C₂²×D₇₉	C₂×C₄²	1	32	yes	—
79	C₂²×D₇₉	C₂³×C₄	1	24	yes	—
80	C₂×C₄×C₇₉	—	2	24	no	—
81	C₂×C₄×C₇₉	—	2	24	no	—
82	C₂×C₄×C₇₉	—	2	6	no	—
83	C₂×C₄×C₇₉	—	2	12	no	—
84	C₂×C₄×C₇₉	—	2	6	no	—
85	C₂×C₄×C₇₉	C₂×C₄²	1	16	yes	—
86	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
87	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
89	C₂×Dic₇₉	—	2	48	no	—
90	C₂×Dic₇₉	—	2	24	no	—
91	C₂×Dic₇₉	C₂³×C₄	1	24	yes	—
92	D₃₁₆	C₂×C₄²	1	64	yes	complemented
93	C₄×D₇₉	—	4	96	no	—
94	C₂²×C₇₉	C₂²×C₄²	1	4	yes	—
95	C₂²×C₇₉	#518	1	12	yes	—
96	C₂²×C₇₉	#520	1	12	yes	—
97	C₂²×C₇₉	C₂⁴×C₄	1	3	yes	—
98	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	3	yes	—
99	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×D₇₉	—	2	24	no	—
101	C₂×D₇₉	—	2	24	no	—
102	C₂×D₇₉	—	2	64	no	—
103	C₂×D₇₉	C₂²×C₄²	1	16	yes	—
104	C₄×C₇₉	—	2	12	no	—
105	C₄×C₇₉	—	2	12	no	—
106	C₄×C₇₉	C₂²×C₄²	1	8	yes	—
107	Dic₇₉	—	2	24	no	—
108	C₂×#2275	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	#2275	1	8	yes	complemented
110	C₂×C₇₉	C₂³×C₄²	1	1	yes	characteristic
111	C₂×C₇₉	C₂×#518	1	3	yes	—
112	C₂×C₇₉	C₂×#520	1	3	yes	—
113	D₇₉	—	2	16	no	—
114	C₂²×C₄×D₄	—	79	3	no	—
115	#2275	—	79	16	no	—
116	C₂×C₄×C₄⋊C₄	—	79	3	no	—
117	C₂×C₄×C₂²⋊C₄	—	79	6	no	—
118	C₂×#376	D₇₉	1	1	yes	characteristic, complemented
119	C₂²×#95	—	79	2	no	—
120	C₇₉	C₂×#2275	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	—	79	1	no	—
122	C₂⁴×C₄	—	79	6	no	—
123	C₂×C₄×D₄	—	79	24	no	—
124	C₂×C₄×D₄	—	79	48	no	—
125	C₂²×C₄⋊C₄	—	79	3	no	—
126	C₂²×C₂²⋊C₄	—	79	6	no	—
127	C₂²×C₄²	—	79	6	no	—
128	C₂²×C₄²	C₂×D₇₉	1	3	yes	—
129	C₄×C₄⋊C₄	—	79	24	no	—
130	C₄×C₂²⋊C₄	—	79	48	no	—
131	#376	C₂×D₇₉	1	8	yes	complemented
132	C₂×#95	—	79	6	no	—
133	C₂×#95	—	79	16	no	—
134	C₂×#95	C₂×D₇₉	1	1	yes	characteristic
135	C₂×#95	C₂×D₇₉	1	3	yes	—
136	C₂⁵	—	79	2	no	—
137	C₂²×D₄	—	79	24	no	—
138	C₂²×D₄	—	79	4	no	—
139	C₂³×C₄	—	316	24	no	—
140	C₂³×C₄	—	79	48	no	—
141	C₂³×C₄	—	79	24	no	—
142	C₂³×C₄	—	79	6	no	—
143	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
144	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
145	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
146	C₄×D₄	—	79	192	no	—
147	C₂×C₄⋊C₄	—	79	24	no	—
148	C₂×C₄⋊C₄	—	79	12	no	—
149	C₂×C₂²⋊C₄	—	79	48	no	—
150	C₂×C₂²⋊C₄	—	79	24	no	—
151	C₂×C₄²	—	316	12	no	—
152	C₂×C₄²	—	79	24	no	—
153	C₂×C₄²	C₄×D₇₉	1	24	yes	complemented
154	C₂×C₄²	C₂²×D₇₉	1	12	yes	—
155	#95	—	79	24	no	—
156	#95	C₂²×D₇₉	1	4	yes	—
157	#95	C₂²×D₇₉	1	12	yes	—
158	C₂⁴	—	79	4	no	—
159	C₂⁴	—	79	24	no	—
160	C₂⁴	—	158	12	no	—
161	C₂⁴	—	158	4	no	—
162	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
163	C₂×D₄	—	79	64	no	—
164	C₂×D₄	—	79	48	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	316	48	no	—
169	C₂²×C₄	—	316	96	no	—
170	C₂²×C₄	—	79	96	no	—
171	C₂²×C₄	—	79	12	no	—
172	C₂²×C₄	—	79	24	no	—
173	C₂²×C₄	—	158	24	no	—
174	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
175	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
176	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
177	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
178	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
179	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
180	C₄⋊C₄	—	79	48	no	—
181	C₂²⋊C₄	—	79	96	no	—
182	C₄²	—	316	24	no	—
183	C₄²	C₂×C₄×D₇₉	1	48	yes	complemented
184	C₂³	—	79	32	no	—
185	C₂³	—	79	24	no	—
186	C₂³	—	158	12	no	—
187	C₂³	—	158	4	no	—
188	C₂³	—	158	48	no	—
189	C₂³	—	158	48	no	—
190	C₂³	C₂²×C₄×D₇₉	1	6	yes	—
191	C₂³	#14579207	1	6	yes	—
192	C₂³	#14579208	1	2	yes	—
193	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
194	D₄	—	79	64	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	24	no	—
199	C₂×C₄	—	2	24	no	—
200	C₂×C₄	—	316	96	no	—
201	C₂×C₄	—	79	24	no	—
202	C₂×C₄	—	158	24	no	—
203	C₂×C₄	—	158	48	no	—
204	C₂×C₄	C₄²×D₇₉	1	16	yes	—
205	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
206	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
207	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
208	C₂²	—	79	16	no	—
209	C₂²	—	158	64	no	—
210	C₂²	—	158	24	no	—
211	C₂²	—	158	24	no	—
212	C₂²	C₂×C₄²×D₇₉	1	4	yes	—
213	C₂²	#19099037	1	12	yes	—
214	C₂²	#19099043	1	12	yes	—
215	C₂²	C₂³×C₄×D₇₉	1	3	yes	—
216	C₂²	C₂×#14579207	1	3	yes	—
217	C₂²	C₂×#14579208	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	2	12	no	—
220	C₄	—	158	24	no	—
221	C₄	C₂×C₄²×D₇₉	1	8	yes	—
222	C₂	—	158	16	no	—
223	C₂	#23765762	1	8	yes	direct product factor
224	C₂	C₂²×C₄²×D₇₉	1	1	yes	characteristic
225	C₂	C₂×#19099037	1	3	yes	—
226	C₂	C₂×#19099043	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator	C₂×C₇₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂×#376×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂³×D₃₁₆	C₂²
cosocle	C₂³×C₇₉	C₂⁵
hypercenter	C₂×#376	D₇₉
hypocenter	C₇₉	C₂×#2275
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #23765762	C₂	direct (N × H)
3. C₂×C₄×C₄⋊Dic₇₉	C₂	semidirect
4. C₂×#19098410	C₂	semidirect
5. C₂×#95×D₇₉	C₂	semidirect
6. C₂×#376×C₇₉	C₂	semidirect
9. C₂×C₄×D₃₁₆	C₄	semidirect
14. C₄×C₄⋊Dic₇₉	C₂²	semidirect
15. #19098410	C₂²	semidirect
17. #95×D₇₉	C₂²	semidirect
19. #376×C₇₉	C₂²	semidirect
27. C₂²×C₄×D₇₉	C₂×C₄	semidirect
29. C₄×D₃₁₆	C₂×C₄	semidirect
30. C₂×C₄⋊Dic₇₉	C₂×C₄	semidirect
32. C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	semidirect
40. C₂×C₄²×C₇₉	C₂×C₄	semidirect
48. C₂×D₃₁₆	C₄²	semidirect
55. C₂×C₄×D₇₉	C₂²×C₄	semidirect
56. C₄⋊Dic₇₉	C₂²×C₄	semidirect
57. (C₂×D₇₉)⋊C₄	C₂²×C₄	semidirect
69. C₄²×C₇₉	C₂²×C₄	semidirect
92. D₃₁₆	C₂×C₄²	semidirect
109. C₂×C₇₉	#2275	semidirect
118. C₂×#376	D₇₉	semidirect
120. C₇₉	C₂×#2275	semidirect
131. #376	C₂×D₇₉	semidirect
153. C₂×C₄²	C₄×D₇₉	semidirect
183. C₄²	C₂×C₄×D₇₉	semidirect
223. C₂	#23765762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃₁₆	2
#23765762	2
C₂×C₄×C₄⋊Dic₇₉	2
C₂×#19098410	2
C₂×#95×D₇₉	2
C₂×#376×C₇₉	2
C₂×#2275	79

Subgroups — C2×#23765762 #34446489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765762 #34446489