Subgroups — C₂×#23763528 #34288453

Order 20224.

1182 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098433	C₂	1	2	yes	maximal
2	#23763528	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098326	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	—	2	2	no	—
6	C₂²×C₄⋊Dic₇₉	—	2	2	no	—
7	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
8	#19098433	—	2	4	no	—
9	#19098433	C₂²	1	8	yes	—
10	#19098326	C₂²	1	16	yes	—
11	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₉	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₇₉	C₂²	1	4	yes	—
15	C₂³×Dic₇₉	—	2	4	no	—
16	C₂×C₄⋊Dic₇₉	—	2	16	no	—
17	C₂×C₄⋊Dic₇₉	—	2	4	no	—
18	C₂×C₄⋊Dic₇₉	—	2	8	no	—
19	C₂×C₄⋊Dic₇₉	—	2	8	no	—
20	C₂×C₄⋊Dic₇₉	—	4	8	no	—
21	C₂×C₄⋊Dic₇₉	—	4	8	no	—
22	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
23	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	—	2	8	no	—
27	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
29	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇₉	D₄	1	2	yes	—
31	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
32	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
33	C₄×C₈×C₇₉	C₂×C₄	1	16	yes	complemented
34	C₂²×Dic₇₉	—	2	8	no	—
35	C₂²×Dic₇₉	—	4	8	no	—
36	C₂²×Dic₇₉	—	4	8	no	—
37	C₂²×Dic₇₉	—	4	8	no	—
38	C₄⋊Dic₇₉	—	2	16	no	—
39	C₄⋊Dic₇₉	—	4	8	no	—
40	C₄⋊Dic₇₉	—	4	16	no	—
41	C₄⋊Dic₇₉	—	4	16	no	—
42	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇₉	D₈	1	4	yes	—
46	C₂²×C₄×C₇₉	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₇₉	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₇₉	#41	1	8	yes	—
54	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	C₂×C₈×C₇₉	—	2	16	no	—
56	C₂×C₈×C₇₉	—	2	8	no	—
57	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
58	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
59	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
60	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
61	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₇₉	#124	1	4	yes	—
63	C₂³×C₇₉	C₂×D₈	1	2	yes	—
64	C₂³×C₇₉	C₂×SD₁₆	1	4	yes	—
65	C₂³×C₇₉	C₂×Q₁₆	1	2	yes	—
66	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#106	1	16	yes	—
68	C₂×C₄×C₇₉	#107	1	16	yes	—
69	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₇₉	#117	1	8	yes	—
73	C₂×C₄×C₇₉	#124	1	4	yes	—
74	C₂×C₄×C₇₉	#125	1	8	yes	—
75	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₇₉	C₂×SD₁₆	1	8	yes	—
77	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
78	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
81	C₂×Dic₇₉	—	4	8	no	—
82	C₂×Dic₇₉	—	4	8	no	—
83	C₂×Dic₇₉	—	4	8	no	—
84	C₂×Dic₇₉	—	8	16	no	—
85	C₈×C₇₉	—	2	8	no	—
86	C₂²×C₇₉	#382	1	4	yes	—
87	C₂²×C₇₉	C₂×#106	1	4	yes	—
88	C₂²×C₇₉	C₂×#107	1	4	yes	—
89	C₂²×C₇₉	#486	1	8	yes	—
90	C₂²×C₇₉	#487	1	8	yes	—
91	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₇₉	C₂×#124	1	2	yes	—
93	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
94	C₂²×C₇₉	C₂²×SD₁₆	1	2	yes	—
95	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
96	C₄×C₇₉	#382	1	4	yes	—
97	C₄×C₇₉	C₂×#106	1	8	yes	—
98	C₄×C₇₉	C₂×#107	1	8	yes	—
99	C₄×C₇₉	C₂×#117	1	2	yes	—
100	C₄×C₇₉	C₂×#125	1	2	yes	—
101	Dic₇₉	—	8	8	no	—
102	C₂×#1834	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	#1834	1	8	yes	complemented
104	C₂×C₇₉	C₂×#382	1	1	yes	characteristic
105	C₂×C₇₉	C₂²×#106	1	1	yes	characteristic
106	C₂×C₇₉	C₂²×#107	1	1	yes	characteristic
107	C₂×C₇₉	C₂×#486	1	2	yes	—
108	C₂×C₇₉	C₂×#487	1	2	yes	—
109	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
110	C₂×#384	—	79	2	no	—
111	#1834	—	79	8	no	—
112	C₂×#340	—	79	4	no	—
113	C₇₉	C₂×#1834	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₈	C₂×D₇₉	1	2	yes	—
115	C₂²×C₄⋊C₄	—	79	4	no	—
116	C₂²×C₄⋊C₄	—	158	2	no	—
117	C₂²×C₄⋊C₄	—	158	2	no	—
118	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
119	C₂×C₄×C₈	Dic₇₉	1	8	yes	complemented
120	C₂×C₄×C₈	C₂×D₇₉	1	4	yes	—
121	#384	—	79	8	no	—
122	#384	—	158	4	no	—
123	#340	—	79	16	no	—
124	C₂³×C₄	—	158	4	no	—
125	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
126	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
127	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
128	C₂²×C₈	—	2	8	no	—
129	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
130	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
131	C₂×C₄⋊C₄	—	316	8	no	—
132	C₂×C₄⋊C₄	—	316	8	no	—
133	C₂×C₄⋊C₄	—	79	8	no	—
134	C₂×C₄⋊C₄	—	158	4	no	—
135	C₂×C₄⋊C₄	—	158	8	no	—
136	C₂×C₄⋊C₄	—	158	8	no	—
137	C₂×C₄⋊C₄	—	158	16	no	—
138	C₂×C₄²	Dic₁₅₈	1	4	yes	—
139	C₂×C₄²	D₃₁₆	1	2	yes	—
140	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
141	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
142	C₄×C₈	C₂×Dic₇₉	1	16	yes	complemented
143	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
144	C₂²×C₄	—	316	8	no	—
145	C₂²×C₄	—	316	8	no	—
146	C₂²×C₄	—	316	8	no	—
147	C₂²×C₄	—	158	8	no	—
148	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
149	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
150	C₂²×C₄	D₆₃₂	1	4	yes	—
151	C₂²×C₄	#11929613	1	8	yes	—
152	C₂²×C₄	Dic₃₁₆	1	4	yes	—
153	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
154	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
156	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
157	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
158	C₂²×C₄	#11929624	1	8	yes	—
159	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	2	16	no	—
162	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
163	C₄⋊C₄	—	316	8	no	—
164	C₄⋊C₄	—	316	16	no	—
165	C₄⋊C₄	—	316	16	no	—
166	C₄⋊C₄	—	158	16	no	—
167	C₄²	C₄⋊Dic₇₉	1	8	yes	—
168	C₄²	C₂²×Dic₇₉	1	4	yes	—
169	C₄²	C₂×Dic₁₅₈	1	4	yes	—
170	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
171	C₂³	#14579149	1	4	yes	—
172	C₂³	C₂×D₆₃₂	1	2	yes	—
173	C₂³	C₂×#11929613	1	4	yes	—
174	C₂³	C₂×Dic₃₁₆	1	2	yes	—
175	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
176	C₂×C₄	—	316	8	no	—
177	C₂×C₄	—	316	8	no	—
178	C₂×C₄	—	316	8	no	—
179	C₂×C₄	—	632	16	no	—
180	C₂×C₄	#14579093	1	16	yes	—
181	C₂×C₄	#14579095	1	16	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
185	C₂×C₄	#14579120	1	8	yes	—
186	C₂×C₄	#14579149	1	4	yes	—
187	C₂×C₄	#14579153	1	8	yes	—
188	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
189	C₂×C₄	C₂×#11929613	1	8	yes	—
190	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
191	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#11929624	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₂²	#19098429	1	4	yes	—
196	C₂²	C₂×#14579093	1	4	yes	—
197	C₂²	C₂×#14579095	1	4	yes	—
198	C₂²	#19098910	1	8	yes	—
199	C₂²	#19098914	1	8	yes	—
200	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
201	C₂²	C₂×#14579149	1	2	yes	—
202	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
203	C₂²	C₂²×#11929613	1	2	yes	—
204	C₂²	C₂²×Dic₃₁₆	1	1	yes	characteristic
205	C₄	—	632	8	no	—
206	C₄	#19098429	1	4	yes	—
207	C₄	C₂×#14579093	1	8	yes	—
208	C₄	C₂×#14579095	1	8	yes	—
209	C₄	C₂×#14579120	1	2	yes	—
210	C₄	C₂×#14579153	1	2	yes	—
211	C₂	#23763528	1	8	yes	direct product factor
212	C₂	C₂×#19098429	1	1	yes	characteristic
213	C₂	C₂²×#14579093	1	1	yes	characteristic
214	C₂	C₂²×#14579095	1	1	yes	characteristic
215	C₂	C₂×#19098910	1	2	yes	—
216	C₂	C₂×#19098914	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×#1834
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23763528	C₂	direct (N × H)
13. C₂×C₄×C₈×C₇₉	C₄	semidirect
33. C₄×C₈×C₇₉	C₂×C₄	semidirect
103. C₂×C₇₉	#1834	semidirect
113. C₇₉	C₂×#1834	semidirect
119. C₂×C₄×C₈	Dic₇₉	semidirect
142. C₄×C₈	C₂×Dic₇₉	semidirect
211. C₂	#23763528	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098433	2
#23763528	2
C₂×#19098326	2
C₂²×C₄×C₈×C₇₉	2
C₂×#1834	79

Subgroups — C2×#23763528 #34288453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763528 #34288453