Subgroups — C₂×#23765680 #34443753

Order 20224.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098968	C₂	1	1	yes	maximal, characteristic
2	#23765680	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098390	C₂	1	4	yes	maximal
4	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	—	2	1	no	—
7	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
8	#19098968	—	2	2	no	—
9	#19098968	C₂²	1	4	yes	—
10	C₂×#14579095	—	2	1	no	—
11	C₂×#14579095	C₂²	1	2	yes	—
12	#19098390	C₂²	1	16	yes	—
13	C₄×C₇₉⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
15	#500×C₇₉	—	2	2	no	—
16	#500×C₇₉	C₂²	1	4	yes	—
17	C₂×#107×C₇₉	—	2	1	no	—
18	C₂×#107×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₅₈	—	4	1	no	—
21	#14579161	—	2	4	no	—
22	#14579161	—	4	2	no	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₂×C₄⋊Dic₇₉	—	4	1	no	—
25	C₂×C₇₉⋊C₁₆	Q₈	1	8	yes	complemented
26	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
27	#14579095	—	2	2	no	—
28	#14579095	—	2	4	no	—
29	#14579095	C₂³	1	4	yes	—
30	C₂²×Q₈×C₇₉	—	4	1	no	—
31	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₉	—	2	4	no	—
34	C₄⋊Q₈×C₇₉	—	4	2	no	—
35	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	#107×C₇₉	—	2	4	no	—
39	#107×C₇₉	—	2	2	no	—
40	#107×C₇₉	C₂³	1	4	yes	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	C₄×C₈×C₇₉	C₂³	1	2	yes	—
43	C₂×Dic₁₅₈	—	4	4	no	—
44	C₂×Dic₁₅₈	—	8	4	no	—
45	C₂²×Dic₇₉	—	4	2	no	—
46	C₄⋊Dic₇₉	—	2	4	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	C₄⋊Dic₇₉	—	4	4	no	—
49	C₇₉⋊C₁₆	C₂×Q₈	1	8	yes	complemented
50	C₂×Q₈×C₇₉	—	4	4	no	—
51	C₂×Q₈×C₇₉	—	8	4	no	—
52	C₂²×C₄×C₇₉	—	4	2	no	—
53	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
57	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	—	2	4	no	—
62	C₄⋊C₄×C₇₉	—	4	2	no	—
63	C₄⋊C₄×C₇₉	—	4	4	no	—
64	C₄²×C₇₉	D₈	1	2	yes	—
65	C₄²×C₇₉	C₂×D₄	1	2	yes	—
66	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₇₉	—	8	8	no	—
68	C₂×C₄×C₇₉	—	4	4	no	—
69	C₂×C₄×C₇₉	—	8	4	no	—
70	C₂×C₄×C₇₉	SD₃₂	1	4	yes	—
71	C₂×C₄×C₇₉	SD₃₂	1	4	yes	—
72	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₇₉	—	4	4	no	—
79	C₂×Dic₇₉	—	8	4	no	—
80	Dic₁₅₈	—	8	8	no	—
81	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
82	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
83	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
84	C₂²×C₇₉	#500	1	2	yes	—
85	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
86	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
87	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₇₉	—	8	4	no	—
89	C₄×C₇₉	#500	1	2	yes	—
90	C₄×C₇₉	C₂×SD₃₂	1	4	yes	—
91	C₄×C₇₉	C₂×SD₃₂	1	4	yes	—
92	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₇₉	—	8	4	no	—
95	C₂×#2254	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	#2254	1	4	yes	complemented
97	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
98	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
99	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
100	C₂×#500	—	79	1	no	—
101	C₂×#500	D₇₉	1	1	yes	characteristic
102	#2254	—	79	8	no	—
103	C₂×#367	—	79	4	no	—
104	C₂×C₄×C₁₆	—	79	1	no	—
105	C₇₉	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×C₄⋊Q₈	—	158	1	no	—
108	C₂²×C₁₆	—	79	2	no	—
109	#500	—	2	2	no	—
110	#500	—	79	4	no	—
111	#500	—	158	2	no	—
112	#500	C₂×D₇₉	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	79	2	no	—
115	C₂×#107	—	158	1	no	—
116	C₂×#107	C₂×D₇₉	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	#367	—	79	16	no	—
119	C₄×C₁₆	—	79	4	no	—
120	C₂²×Q₈	—	4	1	no	—
121	C₂²×Q₈	—	316	1	no	—
122	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₄⋊Q₈	—	316	2	no	—
127	C₄⋊Q₈	—	158	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	316	1	no	—
131	C₂×C₄⋊C₄	—	158	2	no	—
132	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
133	C₂×C₁₆	—	79	8	no	—
134	C₂×C₁₆	—	79	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	79	4	no	—
138	#107	—	158	2	no	—
139	#107	—	158	4	no	—
140	#107	C₂²×D₇₉	1	4	yes	—
141	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
142	C₄×C₈	C₂²×D₇₉	1	2	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	316	4	no	—
146	C₂×Q₈	—	632	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	316	2	no	—
149	C₂²×C₄	#11929609	1	1	yes	characteristic
150	C₂²×C₄	#11929609	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂×C₈	#11929614	1	4	yes	—
153	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
155	C₂×C₈	Q₈×D₇₉	1	4	yes	—
156	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	316	4	no	—
161	C₄⋊C₄	—	316	2	no	—
162	C₄⋊C₄	—	158	4	no	—
163	C₄²	#11929609	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
165	C₁₆	—	79	8	no	—
166	C₂³	C₂×#11929609	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	632	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	316	4	no	—
172	C₂×C₄	—	632	4	no	—
173	C₂×C₄	#14579104	1	4	yes	—
174	C₂×C₄	#14579105	1	4	yes	—
175	C₂×C₄	#14579159	1	2	yes	—
176	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
177	C₂×C₄	C₂×#11929609	1	2	yes	—
178	C₂×C₄	C₂×#11929609	1	2	yes	—
179	C₂×C₄	C₂×#11929614	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
181	C₈	#14579159	1	2	yes	—
182	C₈	C₂×#11929614	1	4	yes	—
183	C₈	C₂×Q₈×D₇₉	1	2	yes	—
184	C₂²	#19098966	1	2	yes	—
185	C₂²	C₂×#14579104	1	2	yes	—
186	C₂²	C₂×#14579105	1	2	yes	—
187	C₂²	C₂²×#11929609	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	632	4	no	—
190	C₄	#19098966	1	2	yes	—
191	C₄	C₂×#14579104	1	4	yes	—
192	C₄	C₂×#14579105	1	4	yes	—
193	C₄	C₂×#14579159	1	1	yes	characteristic
194	C₄	C₂²×#11929614	1	1	yes	characteristic
195	C₂	#23765680	1	4	yes	direct product factor
196	C₂	C₂×#19098966	1	1	yes	characteristic
197	C₂	C₂²×#14579104	1	1	yes	characteristic
198	C₂	C₂²×#14579105	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	C₂×#2254
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #23765680	C₂	direct (N × H)
25. C₂×C₇₉⋊C₁₆	Q₈	semidirect
49. C₇₉⋊C₁₆	C₂×Q₈	semidirect
96. C₂×C₇₉	#2254	semidirect
105. C₇₉	C₂×#2254	semidirect
195. C₂	#23765680	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×#19098968	2
#23765680	2
C₂×#19098390	2
C₂×C₄×C₇₉⋊C₁₆	2
C₂×#500×C₇₉	2
C₂×#2254	79

Subgroups — C2×#23765680 #34443753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765680 #34443753