Subgroups — C₂×#23765258 #34441543

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098657	C₂	1	2	yes	maximal
3	#23765258	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098366	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
8	C₂×#14579126	—	2	2	no	—
9	C₂×Dic₆₃₂	C₄	1	8	yes	complemented
10	C₂×Dic₆₃₂	C₂²	1	4	yes	—
11	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19098657	—	2	4	no	—
13	#19098657	C₂²	1	8	yes	—
14	C₂×#14579084	—	2	2	no	—
15	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
16	#19098366	C₂²	1	8	yes	—
17	C₁₆×Dic₇₉	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#107×C₇₉	C₂²	1	2	yes	—
20	#366×C₇₉	C₂²	1	4	yes	—
21	C₂²×Dic₁₅₈	—	2	2	no	—
22	C₂×Dic₃₁₆	—	2	2	no	—
23	C₂×Dic₃₁₆	—	2	4	no	—
24	C₂×Dic₃₁₆	C₂×C₄	1	8	yes	—
25	C₂×Dic₃₁₆	C₂³	1	4	yes	—
26	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
27	#14579126	—	2	8	no	—
28	#14579126	—	4	4	no	—
29	C₂×Dic₇₉⋊C₄	—	4	2	no	—
30	C₂×C₄×Dic₇₉	—	4	2	no	—
31	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
32	Dic₆₃₂	C₂×C₄	1	16	yes	complemented
33	C₂×C₇₉⋊C₁₆	—	2	2	no	—
34	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
35	#14579084	—	2	8	no	—
36	C₈×Dic₇₉	D₄	1	2	yes	—
37	C₈×Dic₇₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
40	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
42	#107×C₇₉	—	2	2	no	—
43	#107×C₇₉	C₂³	1	4	yes	—
44	C₂×Dic₁₅₈	—	2	8	no	—
45	C₂×Dic₁₅₈	—	2	4	no	—
46	C₂×Dic₁₅₈	—	4	2	no	—
47	C₂×Dic₁₅₈	—	4	4	no	—
48	C₂²×Dic₇₉	—	4	2	no	—
49	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
50	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
51	Dic₃₁₆	—	2	8	no	—
52	Dic₃₁₆	—	2	4	no	—
53	Dic₃₁₆	C₂²×C₄	1	8	yes	—
54	C₂×C₇₉⋊C₈	—	2	2	no	—
55	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
56	Dic₇₉⋊C₄	—	4	8	no	—
57	C₄×Dic₇₉	—	4	8	no	—
58	C₄×Dic₇₉	D₈	1	2	yes	—
59	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
60	C₇₉⋊C₁₆	—	2	2	no	—
61	C₂²×C₄×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	#41	1	2	yes	—
66	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₇₉	—	2	4	no	—
68	C₄⋊C₄×C₇₉	—	4	2	no	—
69	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
70	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	—	4	4	no	—
72	C₂×C₄×C₇₉	—	8	4	no	—
73	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#135	1	2	yes	—
76	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₇₉	—	2	2	no	—
78	C₂×Dic₇₉	—	4	4	no	—
79	C₂×Dic₇₉	—	4	8	no	—
80	C₂×Dic₇₉	Q₃₂	1	4	yes	complemented
81	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
82	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
83	Dic₁₅₈	—	2	8	no	—
84	Dic₁₅₈	—	4	4	no	—
85	Dic₁₅₈	—	4	8	no	—
86	C₈×C₇₉	C₄×D₄	1	2	yes	—
87	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
89	C₇₉⋊C₈	—	2	2	no	—
90	C₂²×C₇₉	C₄×D₈	1	2	yes	—
91	C₂²×C₇₉	C₂×Q₃₂	1	2	yes	—
92	C₂²×C₇₉	#508	1	2	yes	—
93	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₉	—	8	4	no	—
95	C₄×C₇₉	C₄×D₈	1	2	yes	—
96	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
98	Dic₇₉	—	2	2	no	—
99	Dic₇₉	—	4	8	no	—
100	Dic₇₉	C₂×Q₃₂	1	4	yes	complemented
101	C₂×C₄×Q₃₂	—	79	1	no	maximal, Sylow
102	C₂×C₇₉	C₄×Q₃₂	1	4	yes	complemented
103	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
104	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
106	C₂²×Q₃₂	—	79	1	no	—
107	C₂×C₄×Q₁₆	—	79	2	no	—
108	C₄×Q₃₂	—	79	8	no	—
109	C₂×#366	D₇₉	1	1	yes	characteristic
110	C₂×#358	—	79	2	no	—
111	C₂×C₄×C₁₆	—	79	1	no	—
112	C₇₉	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	79	2	no	—
114	C₂×C₄×Q₈	—	158	2	no	—
115	C₂×Q₃₂	—	79	4	no	—
116	C₂×Q₃₂	—	79	8	no	—
117	C₂²×C₁₆	—	79	1	no	—
118	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
119	C₄×Q₁₆	—	79	8	no	—
120	C₄×Q₁₆	—	158	4	no	—
121	C₂×#107	C₂×D₇₉	1	2	yes	—
122	C₂×#103	—	158	2	no	—
123	C₂×C₄×C₈	—	79	1	no	—
124	#366	C₂×D₇₉	1	4	yes	—
125	#358	—	79	8	no	—
126	C₄×C₁₆	—	79	4	no	—
127	C₂²×Q₈	—	158	2	no	—
128	C₂×Q₁₆	—	79	8	no	—
129	C₂×Q₁₆	—	79	4	no	—
130	C₂×Q₁₆	—	158	2	no	—
131	C₂×Q₁₆	—	158	4	no	—
132	C₂²×C₈	—	79	1	no	—
133	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
134	C₄×Q₈	—	316	4	no	—
135	C₄×Q₈	—	158	8	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	316	2	no	—
138	C₂×C₄²	—	316	2	no	—
139	C₂×C₄²	—	79	1	no	—
140	Q₃₂	—	79	16	no	—
141	C₂×C₁₆	—	79	2	no	—
142	C₂×C₁₆	—	158	2	no	—
143	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
144	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
145	#107	—	2	2	no	—
146	#107	C₂²×D₇₉	1	4	yes	—
147	#103	—	158	8	no	—
148	C₄×C₈	—	79	2	no	—
149	C₄×C₈	—	79	2	no	—
150	C₂×Q₈	—	316	2	no	—
151	C₂×Q₈	—	316	4	no	—
152	C₂×Q₈	—	158	8	no	—
153	C₂×Q₈	—	158	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	316	2	no	—
156	C₂²×C₄	—	79	1	no	—
157	C₂²×C₄	—	79	1	no	—
158	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
159	Q₁₆	—	79	8	no	—
160	Q₁₆	—	158	4	no	—
161	Q₁₆	—	158	8	no	—
162	C₂×C₈	—	79	2	no	—
163	C₂×C₈	—	158	2	no	—
164	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
165	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
166	C₂×C₈	#11929625	1	2	yes	—
167	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	316	8	no	—
171	C₄²	—	316	8	no	—
172	C₄²	—	79	2	no	—
173	C₄²	—	79	2	no	—
174	C₁₆	—	158	2	no	—
175	C₁₆	C₂×C₄×D₇₉	1	4	yes	—
176	C₂³	D₈×D₇₉	1	1	yes	characteristic
177	Q₈	—	316	8	no	—
178	Q₈	—	316	4	no	—
179	Q₈	—	158	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	316	8	no	—
183	C₂×C₄	—	316	4	no	—
184	C₂×C₄	—	79	4	no	—
185	C₂×C₄	—	79	2	no	—
186	C₂×C₄	—	79	2	no	—
187	C₂×C₄	—	158	2	no	—
188	C₂×C₄	#14579123	1	2	yes	—
189	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
190	C₂×C₄	#14579184	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
192	C₈	—	158	2	no	—
193	C₈	#14579123	1	2	yes	—
194	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
195	C₈	C₂×#11929625	1	1	yes	characteristic
196	C₂²	#19098645	1	2	yes	—
197	C₂²	Q₃₂×D₇₉	1	2	yes	—
198	C₂²	#19098992	1	2	yes	—
199	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	316	8	no	—
202	C₄	—	79	4	no	—
203	C₄	—	158	2	no	—
204	C₄	#19098645	1	2	yes	—
205	C₄	C₂×#14579123	1	1	yes	characteristic
206	C₄	C₂×#14579184	1	1	yes	characteristic
207	C₂	#23765258	1	4	yes	direct product factor
208	C₂	C₂×#19098645	1	1	yes	characteristic
209	C₂	C₂×Q₃₂×D₇₉	1	1	yes	characteristic
210	C₂	C₂×#19098992	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#366×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#366	D₇₉
hypocenter	C₇₉	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #23765258	C₂	direct (N × H)
9. C₂×Dic₆₃₂	C₄	semidirect
32. Dic₆₃₂	C₂×C₄	semidirect
80. C₂×Dic₇₉	Q₃₂	semidirect
100. Dic₇₉	C₂×Q₃₂	semidirect
102. C₂×C₇₉	C₄×Q₃₂	semidirect
112. C₇₉	C₂×C₄×Q₃₂	semidirect
207. C₂	#23765258	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×Dic₆₃₂	2
C₂×#19098657	2
#23765258	2
C₂×#19098366	2
C₂×C₁₆×Dic₇₉	2
C₂×#366×C₇₉	2
C₂×C₄×Q₃₂	79

Subgroups — C2×#23765258 #34441543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765258 #34441543