Subgroups — C₂×#23765757 #34446479

Order 20224.

4110 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579120	C₂	1	2	yes	maximal
2	#23765757	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×Dic₇₉	C₂	1	8	yes	maximal, complemented
4	C₂×C₄×C₂²⋊Dic₇₉	C₂	1	4	yes	maximal
5	C₂³×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	C₄	1	32	yes	complemented
7	C₂×#14579120	C₂²	1	16	yes	—
8	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
9	C₂²×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₇₉	C₂²	1	8	yes	—
11	C₂²×C₄×Dic₇₉	C₂²	1	4	yes	—
12	C₄⋊C₄×Dic₇₉	C₂²	1	64	yes	complemented
13	C₄×C₂²⋊Dic₇₉	C₂²	1	32	yes	—
14	C₂×C₇₉⋊#95	C₂²	1	8	yes	—
15	C₂⁴×C₄×C₇₉	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₇₉	C₄	1	16	yes	complemented
18	C₂²×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₇₉	C₂²	1	8	yes	—
20	C₂³×Dic₇₉	C₂³	1	8	yes	—
21	#14579120	C₂×C₄	1	128	yes	complemented
22	C₂×C₄⋊Dic₇₉	C₂×C₄	1	32	yes	complemented
23	C₂×C₄⋊Dic₇₉	C₂³	1	16	yes	—
24	C₂×C₂²⋊Dic₇₉	C₂×C₄	1	32	yes	—
25	C₂×C₂²⋊Dic₇₉	C₂³	1	16	yes	—
26	C₂×C₄×Dic₇₉	—	4	16	no	—
27	C₂×C₄×Dic₇₉	C₂×C₄	1	32	yes	complemented
28	C₂×C₄×Dic₇₉	C₂³	1	32	yes	—
29	C₂×C₄×Dic₇₉	C₂³	1	16	yes	—
30	C₇₉⋊#95	C₂³	1	32	yes	—
31	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₇₉	—	4	8	no	—
33	C₂³×C₄×C₇₉	C₂×C₄	1	16	yes	—
34	C₂³×C₄×C₇₉	C₂×C₄	1	16	yes	—
35	C₂³×C₄×C₇₉	C₂×C₄	1	8	yes	—
36	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₇₉	C₂³	1	8	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	32	yes	—
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	64	yes	complemented
42	C₂×C₄⋊C₄×C₇₉	C₂³	1	16	yes	—
43	C₂²×Dic₇₉	—	2	16	no	—
44	C₂²×Dic₇₉	—	2	16	no	—
45	C₂²×Dic₇₉	C₂²×C₄	1	32	yes	—
46	C₂²×Dic₇₉	C₂⁴	1	16	yes	—
47	C₄⋊Dic₇₉	C₂²×C₄	1	64	yes	complemented
48	C₂²⋊Dic₇₉	C₂²×C₄	1	64	yes	—
49	C₄×Dic₇₉	—	4	32	no	—
50	C₄×Dic₇₉	C₂²×C₄	1	64	yes	complemented
51	C₂⁴×C₇₉	—	2	4	no	—
52	C₂⁴×C₇₉	—	2	4	no	—
53	C₂⁴×C₇₉	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₇₉	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₇₉	C₂⁴	1	2	yes	—
56	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	—	2	16	no	—
58	C₂²×C₄×C₇₉	—	4	32	no	—
59	C₂²×C₄×C₇₉	—	4	16	no	—
60	C₂²×C₄×C₇₉	C₄²	1	64	yes	complemented
61	C₂²×C₄×C₇₉	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₇₉	C₂²×C₄	1	32	yes	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₇₉	C₂²×C₄	1	32	yes	—
67	C₂²×C₄×C₇₉	C₂²×C₄	1	16	yes	—
68	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
70	C₄⋊C₄×C₇₉	C₂²×C₄	1	64	yes	—
71	C₂³×C₇₉	—	2	8	no	—
72	C₂³×C₇₉	—	2	8	no	—
73	C₂³×C₇₉	—	2	32	no	—
74	C₂³×C₇₉	—	2	4	no	—
75	C₂³×C₇₉	—	2	4	no	—
76	C₂³×C₇₉	C₂×C₄²	1	16	yes	—
77	C₂³×C₇₉	C₂³×C₄	1	8	yes	—
78	C₂³×C₇₉	C₂³×C₄	1	4	yes	—
79	C₂³×C₇₉	C₂³×C₄	1	4	yes	—
80	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
81	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
82	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
83	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	32	no	—
85	C₂×C₄×C₇₉	—	2	16	no	—
86	C₂×C₄×C₇₉	—	4	32	no	—
87	C₂×C₄×C₇₉	C₂×C₄²	1	32	yes	—
88	C₂×C₄×C₇₉	C₂×C₄²	1	64	yes	complemented
89	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
90	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
91	C₂×C₄×C₇₉	C₂³×C₄	1	16	yes	—
92	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
93	C₂×Dic₇₉	—	2	64	no	—
94	C₂×Dic₇₉	—	2	16	no	—
95	C₂×Dic₇₉	—	2	16	no	—
96	C₂×Dic₇₉	C₂³×C₄	1	32	yes	—
97	C₂²×C₇₉	—	2	4	no	—
98	C₂²×C₇₉	—	2	4	no	—
99	C₂²×C₇₉	—	2	16	no	—
100	C₂²×C₇₉	—	2	32	no	—
101	C₂²×C₇₉	C₂²×C₄²	1	8	yes	—
102	C₂²×C₇₉	C₂²×C₄²	1	4	yes	—
103	C₂²×C₇₉	#518	1	4	yes	—
104	C₂²×C₇₉	#519	1	4	yes	—
105	C₂²×C₇₉	#520	1	16	yes	—
106	C₂²×C₇₉	C₂⁴×C₄	1	2	yes	—
107	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
108	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	2	yes	—
109	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	2	yes	—
110	C₄×C₇₉	—	2	16	no	—
111	C₄×C₇₉	C₂²×C₄²	1	16	yes	—
112	Dic₇₉	—	2	32	no	—
113	C₂×#2274	—	79	1	no	maximal, Sylow
114	C₂×C₇₉	—	2	8	no	—
115	C₂×C₇₉	#2274	1	8	yes	complemented
116	C₂×C₇₉	C₂³×C₄²	1	1	yes	characteristic
117	C₂×C₇₉	C₂×#518	1	1	yes	characteristic
118	C₂×C₇₉	C₂×#519	1	1	yes	characteristic
119	C₂×C₇₉	C₂×#520	1	4	yes	—
120	C₂²×#117	—	79	2	no	—
121	C₂³×C₄⋊C₄	D₇₉	1	1	yes	characteristic
122	#2274	—	79	16	no	—
123	C₂×C₄×C₄⋊C₄	—	79	8	no	—
124	C₂×C₄×C₂²⋊C₄	—	79	4	no	—
125	C₇₉	C₂×#2274	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴×C₄	C₂×D₇₉	1	2	yes	—
127	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
128	C₂×#117	—	79	16	no	—
129	C₂×#117	—	79	32	no	—
130	C₂²×C₄⋊C₄	—	79	4	no	—
131	C₂²×C₄⋊C₄	Dic₇₉	1	16	yes	complemented
132	C₂²×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
133	C₂²×C₄⋊C₄	C₂×D₇₉	1	8	yes	—
134	C₂²×C₂²⋊C₄	—	79	4	no	—
135	C₂²×C₄²	—	79	8	no	—
136	C₂²×C₄²	—	79	4	no	—
137	C₄×C₄⋊C₄	—	79	64	no	—
138	C₄×C₂²⋊C₄	—	79	32	no	—
139	C₂×#95	—	79	8	no	—
140	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
141	C₂³×C₄	—	4	8	no	—
142	C₂³×C₄	—	79	8	no	—
143	C₂³×C₄	C₄×D₇₉	1	16	yes	—
144	C₂³×C₄	C₂×Dic₇₉	1	16	yes	—
145	C₂³×C₄	C₂×Dic₇₉	1	8	yes	—
146	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
147	C₂³×C₄	C₂²×D₇₉	1	8	yes	—
148	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
149	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
150	#117	—	79	128	no	—
151	C₂×C₄⋊C₄	—	79	32	no	—
152	C₂×C₄⋊C₄	—	79	16	no	—
153	C₂×C₄⋊C₄	C₂×Dic₇₉	1	32	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₇₉	1	64	yes	complemented
155	C₂×C₄⋊C₄	C₂²×D₇₉	1	16	yes	—
156	C₂×C₂²⋊C₄	—	79	32	no	—
157	C₂×C₂²⋊C₄	—	79	16	no	—
158	C₂×C₄²	—	316	16	no	—
159	C₂×C₄²	—	79	32	no	—
160	C₂×C₄²	—	79	32	no	—
161	C₂×C₄²	—	79	16	no	—
162	#95	—	79	32	no	—
163	C₂⁴	—	2	4	no	—
164	C₂⁴	—	2	4	no	—
165	C₂⁴	C₂×C₄×D₇₉	1	8	yes	—
166	C₂⁴	C₂²×Dic₇₉	1	4	yes	—
167	C₂⁴	C₂³×D₇₉	1	2	yes	—
168	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	4	16	no	—
171	C₂²×C₄	—	4	32	no	—
172	C₂²×C₄	—	79	16	no	—
173	C₂²×C₄	—	79	32	no	—
174	C₂²×C₄	—	158	16	no	—
175	C₂²×C₄	—	158	16	no	—
176	C₂²×C₄	C₄×Dic₇₉	1	64	yes	complemented
177	C₂²×C₄	C₂×C₄×D₇₉	1	8	yes	—
178	C₂²×C₄	C₂×C₄×D₇₉	1	8	yes	—
179	C₂²×C₄	C₂×C₄×D₇₉	1	32	yes	—
180	C₂²×C₄	C₂²×Dic₇₉	1	16	yes	—
181	C₂²×C₄	C₂²×Dic₇₉	1	32	yes	—
182	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
183	C₂²×C₄	C₂²×Dic₇₉	1	16	yes	—
184	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
185	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
186	C₄⋊C₄	—	79	64	no	—
187	C₄⋊C₄	C₂²×Dic₇₉	1	64	yes	—
188	C₂²⋊C₄	—	79	64	no	—
189	C₄²	—	316	32	no	—
190	C₄²	—	79	64	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	8	no	—
195	C₂³	—	2	32	no	—
196	C₂³	C₂×C₄×Dic₇₉	1	16	yes	—
197	C₂³	C₂²×C₄×D₇₉	1	8	yes	—
198	C₂³	C₂²×C₄×D₇₉	1	4	yes	—
199	C₂³	C₂³×Dic₇₉	1	4	yes	—
200	C₂³	C₂³×Dic₇₉	1	2	yes	—
201	C₂³	#14579206	1	4	yes	—
202	C₂³	#14579209	1	4	yes	—
203	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	32	no	—
206	C₂×C₄	—	4	32	no	—
207	C₂×C₄	—	79	32	no	—
208	C₂×C₄	—	158	16	no	—
209	C₂×C₄	—	158	16	no	—
210	C₂×C₄	—	158	64	no	—
211	C₂×C₄	C₂×C₄×Dic₇₉	1	32	yes	—
212	C₂×C₄	C₂×C₄×Dic₇₉	1	64	yes	complemented
213	C₂×C₄	C₂²×C₄×D₇₉	1	8	yes	—
214	C₂×C₄	C₂²×C₄×D₇₉	1	8	yes	—
215	C₂×C₄	C₂³×Dic₇₉	1	16	yes	—
216	C₂×C₄	C₂³×Dic₇₉	1	8	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	2	32	no	—
221	C₂²	C₂²×C₄×Dic₇₉	1	8	yes	—
222	C₂²	C₂²×C₄×Dic₇₉	1	4	yes	—
223	C₂²	#19099038	1	4	yes	—
224	C₂²	#19099041	1	4	yes	—
225	C₂²	#19099044	1	16	yes	—
226	C₂²	C₂³×C₄×D₇₉	1	2	yes	—
227	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
228	C₂²	C₂×#14579206	1	2	yes	—
229	C₂²	C₂×#14579209	1	2	yes	—
230	C₄	—	2	16	no	—
231	C₄	—	158	32	no	—
232	C₄	C₂²×C₄×Dic₇₉	1	16	yes	—
233	C₂	—	2	8	no	—
234	C₂	#23765757	1	8	yes	direct product factor
235	C₂	C₂³×C₄×Dic₇₉	1	1	yes	characteristic
236	C₂	C₂×#19099038	1	1	yes	characteristic
237	C₂	C₂×#19099041	1	1	yes	characteristic
238	C₂	C₂×#19099044	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator	C₂×C₇₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂³×C₄⋊C₄×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂⁵×C₇₉	C₂³
cosocle	C₂³×C₇₉	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#2274
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #23765757	C₂	direct (N × H)
3. C₂×C₄⋊C₄×Dic₇₉	C₂	semidirect
6. C₂×#14579120	C₄	semidirect
12. C₄⋊C₄×Dic₇₉	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₇₉	C₄	semidirect
21. #14579120	C₂×C₄	semidirect
22. C₂×C₄⋊Dic₇₉	C₂×C₄	semidirect
27. C₂×C₄×Dic₇₉	C₂×C₄	semidirect
41. C₂×C₄⋊C₄×C₇₉	C₂×C₄	semidirect
47. C₄⋊Dic₇₉	C₂²×C₄	semidirect
50. C₄×Dic₇₉	C₂²×C₄	semidirect
60. C₂²×C₄×C₇₉	C₄²	semidirect
88. C₂×C₄×C₇₉	C₂×C₄²	semidirect
115. C₂×C₇₉	#2274	semidirect
125. C₇₉	C₂×#2274	semidirect
131. C₂²×C₄⋊C₄	Dic₇₉	semidirect
154. C₂×C₄⋊C₄	C₂×Dic₇₉	semidirect
176. C₂²×C₄	C₄×Dic₇₉	semidirect
212. C₂×C₄	C₂×C₄×Dic₇₉	semidirect
234. C₂	#23765757	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×#14579120	2
#23765757	2
C₂×C₄⋊C₄×Dic₇₉	2
C₂×C₄×C₂²⋊Dic₇₉	2
C₂³×C₄⋊C₄×C₇₉	2
C₂×#2274	79

Subgroups — C2×#23765757 #34446479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765757 #34446479