Subgroups — C₂×#23772506 #35153238

Order 20224.

1706 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772506	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19098694	C₂	1	8	yes	maximal, complemented
3	C₂×#19098624	C₂	1	4	yes	maximal
4	C₂×#19098620	C₂	1	2	yes	maximal
5	C₂×#533×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098694	C₂²	1	64	yes	complemented
7	#19098624	C₂²	1	32	yes	—
8	#19098620	C₂²	1	16	yes	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
12	C₂×#14579072	C₂²	1	4	yes	—
13	C₂×#14579072	C₂²	1	4	yes	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
15	#533×C₇₉	C₄	1	8	yes	—
16	#533×C₇₉	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₇₉	C₄	1	2	yes	—
18	C₂×#125×C₇₉	C₄	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₇₉	C₄	1	4	yes	—
20	C₂×C₄×Q₈×C₇₉	C₂²	1	4	yes	—
21	C₂×#117×C₇₉	C₂²	1	2	yes	—
22	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
24	C₄⋊(C₇₉⋊C₈)	C₂³	1	32	yes	—
25	C₄⋊(C₇₉⋊C₈)	C₂³	1	16	yes	—
26	C₂²⋊(C₇₉⋊C₈)	C₂³	1	16	yes	—
27	#14579072	C₂³	1	16	yes	—
28	#14579072	C₂³	1	16	yes	—
29	C₄×C₇₉⋊C₈	C₂³	1	16	yes	—
30	C₂²×Q₈×C₇₉	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	16	yes	—
34	#125×C₇₉	C₂×C₄	1	16	yes	—
35	C₂²⋊Q₈×C₇₉	C₂×C₄	1	32	yes	—
36	C₄×Q₈×C₇₉	C₂×C₄	1	16	yes	—
37	C₄×Q₈×C₇₉	C₂³	1	16	yes	—
38	#117×C₇₉	C₂×C₄	1	8	yes	—
39	#117×C₇₉	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
50	C₂×C₇₉⋊C₈	—	2	16	no	—
51	C₂×C₇₉⋊C₈	C₂⁴	1	16	yes	—
52	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×Q₈×C₇₉	—	4	8	no	—
54	C₂×Q₈×C₇₉	C₂²×C₄	1	16	yes	—
55	C₂²×C₄×C₇₉	—	2	4	no	—
56	C₂²×C₄×C₇₉	—	4	4	no	—
57	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₇₉	C₂²×C₄	1	16	yes	—
66	C₄⋊C₄×C₇₉	C₂²×C₄	1	32	yes	—
67	C₄⋊C₄×C₇₉	C₂²×C₄	1	16	yes	—
68	C₂²⋊C₄×C₇₉	C₂²×C₄	1	16	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
70	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
71	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
72	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
73	C₄²×C₇₉	C₂⁴	1	4	yes	—
74	C₄²×C₇₉	C₂⁴	1	8	yes	—
75	C₂³×C₇₉	—	2	2	no	—
76	C₂³×C₇₉	—	2	4	no	—
77	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
78	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₇₉	—	4	16	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	4	no	—
82	C₂×C₄×C₇₉	—	2	4	no	—
83	C₂×C₄×C₇₉	—	2	16	no	—
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	—	2	4	no	—
86	C₂×C₄×C₇₉	—	4	8	no	—
87	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₇₉	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₇₉	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₇₉	C₂³×C₄	1	16	yes	—
91	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₉	2₋¹⁺⁴	1	4	yes	—
93	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
94	C₇₉⋊C₈	—	2	16	no	—
95	C₂²×C₇₉	—	2	2	no	—
96	C₂²×C₇₉	—	2	4	no	—
97	C₂²×C₇₉	—	4	4	no	—
98	C₂²×C₇₉	#519	1	2	yes	—
99	C₂²×C₇₉	#568	1	4	yes	—
100	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₇₉	—	2	2	no	—
102	C₄×C₇₉	—	2	4	no	—
103	C₄×C₇₉	—	2	4	no	—
104	C₄×C₇₉	—	2	16	no	—
105	C₄×C₇₉	#519	1	2	yes	—
106	C₄×C₇₉	C₂×2₋¹⁺⁴	1	2	yes	—
107	C₂×#2987	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	—	4	2	no	—
109	C₂×C₇₉	#2987	1	4	yes	complemented
110	C₂×C₇₉	C₂×#519	1	1	yes	characteristic
111	C₂×C₇₉	C₂×#568	1	2	yes	—
112	C₂×#533	D₇₉	1	1	yes	characteristic
113	#2987	—	79	16	no	—
114	C₂×#446	—	79	8	no	—
115	C₂×#433	—	79	4	no	—
116	C₂×#432	—	79	2	no	—
117	C₇₉	C₂×#2987	1	1	yes	characteristic, complemented, Sylow
118	#533	Dic₇₉	1	8	yes	—
119	#533	C₂×D₇₉	1	8	yes	—
120	C₂×C₄⋊Q₈	Dic₇₉	1	2	yes	—
121	C₂×#125	Dic₇₉	1	2	yes	—
122	C₂×C₂²⋊Q₈	Dic₇₉	1	4	yes	—
123	C₂×C₄×Q₈	C₂×D₇₉	1	4	yes	—
124	C₂×#117	C₂×D₇₉	1	2	yes	—
125	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
126	#446	—	79	64	no	—
127	#433	—	79	32	no	—
128	#432	—	79	16	no	—
129	C₂×C₄⋊C₈	—	79	4	no	—
130	C₂×C₄⋊C₈	—	79	8	no	—
131	C₂×C₂²⋊C₈	—	79	4	no	—
132	C₂×#97	—	79	4	no	—
133	C₂×#97	—	79	4	no	—
134	C₂×C₄×C₈	—	79	4	no	—
135	C₂²×Q₈	C₂×Dic₇₉	1	4	yes	—
136	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
137	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
138	C₂²×C₈	—	79	8	no	—
139	C₄⋊Q₈	C₂×Dic₇₉	1	16	yes	—
140	#125	C₂×Dic₇₉	1	16	yes	—
141	C₂²⋊Q₈	C₂×Dic₇₉	1	32	yes	—
142	C₄×Q₈	C₂×Dic₇₉	1	16	yes	—
143	C₄×Q₈	C₂²×D₇₉	1	16	yes	—
144	#117	C₂×Dic₇₉	1	8	yes	—
145	#117	C₂²×D₇₉	1	8	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
147	C₂×C₄⋊C₄	C₂×Dic₇₉	1	8	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
150	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
153	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
154	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
155	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
156	C₄⋊C₈	—	79	16	no	—
157	C₄⋊C₈	—	79	32	no	—
158	C₂²⋊C₈	—	79	16	no	—
159	#97	—	79	16	no	—
160	#97	—	79	16	no	—
161	C₄×C₈	—	79	16	no	—
162	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
163	C₂×Q₈	—	4	8	no	—
164	C₂×Q₈	C₂²×Dic₇₉	1	16	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
168	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
170	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
173	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
174	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
175	C₂×C₈	—	79	16	no	—
176	C₂×C₈	—	158	16	no	—
177	C₄⋊C₄	C₂²×Dic₇₉	1	16	yes	—
178	C₄⋊C₄	C₂²×Dic₇₉	1	32	yes	—
179	C₄⋊C₄	C₂²×Dic₇₉	1	16	yes	—
180	C₂²⋊C₄	C₂²×Dic₇₉	1	16	yes	—
181	C₄²	C₂²×Dic₇₉	1	4	yes	—
182	C₄²	C₂²×Dic₇₉	1	4	yes	—
183	C₄²	C₂²×Dic₇₉	1	4	yes	—
184	C₄²	C₂²×Dic₇₉	1	8	yes	—
185	C₄²	C₂³×D₇₉	1	8	yes	—
186	C₄²	C₂³×D₇₉	1	4	yes	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₂³×Dic₇₉	1	2	yes	—
190	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
191	Q₈	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂³×Dic₇₉	1	2	yes	—
200	C₂×C₄	C₂³×Dic₇₉	1	4	yes	—
201	C₂×C₄	C₂³×Dic₇₉	1	4	yes	—
202	C₂×C₄	C₂³×Dic₇₉	1	16	yes	—
203	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
204	C₂×C₄	#14579209	1	4	yes	—
205	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
206	C₈	—	158	16	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	#19099041	1	2	yes	—
211	C₂²	#19099330	1	4	yes	—
212	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	16	no	—
216	C₄	—	2	2	no	—
217	C₄	#19099041	1	2	yes	—
218	C₄	C₂×#14579209	1	2	yes	—
219	C₂	—	4	2	no	—
220	C₂	#23772506	1	4	yes	direct product factor
221	C₂	C₂×#19099041	1	1	yes	characteristic
222	C₂	C₂×#19099330	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×#533×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	C₂×#533	D₇₉
hypocenter	C₇₉	C₂×#2987
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #23772506	C₂	direct (N × H)
2. C₂×#19098694	C₂	semidirect
6. #19098694	C₂²	semidirect
109. C₂×C₇₉	#2987	semidirect
117. C₇₉	C₂×#2987	semidirect
220. C₂	#23772506	direct (N × H)

Maximal subgroups

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

subgroup	index
#23772506	2
C₂×#19098694	2
C₂×#19098624	2
C₂×#19098620	2
C₂×#533×C₇₉	2
C₂×#2987	79

Subgroups — C2×#23772506 #35153238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23772506 #35153238