Subgroups — C₂×#23761457 #34199550

Order 20224.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
3	#23761457	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	—	2	1	no	—
6	#19098968	—	2	1	no	—
7	#19098968	—	2	1	no	—
8	#19098968	C₄	1	2	yes	—
9	#19098968	C₂²	1	2	yes	—
10	C₂×#14579095	—	2	1	no	—
11	C₂×#14579095	C₄	1	2	yes	—
12	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
13	#19098316	—	2	1	no	—
14	#19098316	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₇₉	—	2	1	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₇₉	—	2	1	no	—
18	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
19	C₂²×Dic₁₅₈	—	4	1	no	—
20	C₂²×C₇₉⋊C₈	—	4	1	no	—
21	#14579161	—	2	2	no	—
22	#14579161	—	2	2	no	—
23	#14579161	—	4	1	no	—
24	#14579161	—	4	1	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	4	1	no	—
27	#14579095	—	2	2	no	—
28	#14579095	—	2	2	no	—
29	#14579095	—	2	2	no	—
30	#14579095	C₂×C₄	1	4	yes	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₇₉	—	2	2	no	—
37	C₂×C₁₆×C₇₉	—	4	2	no	—
38	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₅₈	—	4	2	no	—
43	C₂×Dic₁₅₈	—	4	2	no	—
44	C₂×Dic₁₅₈	—	8	4	no	—
45	C₂²×Dic₇₉	—	4	2	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	8	2	no	—
48	C₄⋊Dic₇₉	—	2	4	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₄⋊Dic₇₉	—	4	2	no	—
52	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₉	—	2	2	no	—
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₉	D₈	1	1	yes	characteristic
64	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₇₉	—	4	2	no	—
67	C₂³×C₇₉	#102	1	1	yes	characteristic
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	#101	1	2	yes	—
70	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#103	1	2	yes	—
74	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
75	C₂×C₄×C₇₉	Q₃₂	1	2	yes	—
76	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×Dic₇₉	—	4	4	no	—
80	C₂×Dic₇₉	—	8	4	no	—
81	Dic₁₅₈	—	8	4	no	—
82	Dic₁₅₈	—	8	4	no	—
83	C₈×C₇₉	—	2	1	no	—
84	C₈×C₇₉	—	2	1	no	—
85	C₈×C₇₉	#103	1	2	yes	—
86	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₇₉⋊C₈	—	8	2	no	—
89	C₂²×C₇₉	#332	1	2	yes	—
90	C₂²×C₇₉	#358	1	2	yes	—
91	C₂²×C₇₉	#361	1	2	yes	—
92	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	#332	1	2	yes	—
96	C₄×C₇₉	#358	1	2	yes	—
97	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
98	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
99	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
100	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
101	Dic₇₉	—	8	4	no	—
102	C₂×#1365	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	#1365	1	4	yes	complemented
104	C₂×C₇₉	C₂×#332	1	1	yes	characteristic
105	C₂×C₇₉	C₂×#358	1	1	yes	characteristic
106	C₂×C₇₉	C₂×#361	1	1	yes	characteristic
107	C₂×#500	—	79	1	no	—
108	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
109	C₂×#335	—	79	1	no	—
110	#1365	—	79	4	no	—
111	C₇₉	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	158	1	no	—
113	C₂²×C₁₆	—	2	1	no	—
114	#500	—	79	2	no	—
115	#500	—	79	2	no	—
116	#500	—	158	1	no	—
117	#500	—	158	1	no	—
118	C₂×#107	—	79	2	no	—
119	C₂×#107	—	158	1	no	—
120	C₂×C₄⋊C₈	—	158	1	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	C₄⋊C₁₆	—	2	1	no	—
123	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
124	#335	—	79	2	no	—
125	#335	—	158	1	no	—
126	C₂²×Q₈	—	316	1	no	—
127	C₂²×C₈	—	316	1	no	—
128	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
130	C₄⋊Q₈	—	316	1	no	—
131	C₄⋊Q₈	—	316	1	no	—
132	C₄⋊Q₈	—	158	2	no	—
133	C₄⋊Q₈	—	158	2	no	—
134	C₂×C₄⋊C₄	—	316	1	no	—
135	C₂×C₄⋊C₄	—	158	2	no	—
136	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	4	2	no	—
139	#107	—	79	4	no	—
140	#107	—	158	2	no	—
141	#107	—	158	2	no	—
142	#107	—	158	2	no	—
143	C₄⋊C₈	—	316	1	no	—
144	C₄⋊C₈	—	158	2	no	—
145	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
146	C₄×C₈	D₃₁₆	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
149	C₂×Q₈	—	316	2	no	—
150	C₂×Q₈	—	316	2	no	—
151	C₂×Q₈	—	632	4	no	—
152	C₂²×C₄	—	316	2	no	—
153	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
155	C₂²×C₄	#11929613	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	316	2	no	—
158	C₂×C₈	—	632	2	no	—
159	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
160	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
161	C₂×C₈	#11929612	1	2	yes	—
162	C₂×C₈	#11929614	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
165	C₄⋊C₄	—	316	2	no	—
166	C₄⋊C₄	—	316	2	no	—
167	C₄⋊C₄	—	316	2	no	—
168	C₄⋊C₄	—	158	4	no	—
169	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
170	C₄²	D₆₃₂	1	1	yes	characteristic
171	C₄²	#11929613	1	1	yes	characteristic
172	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
173	C₁₆	—	4	2	no	—
174	C₂³	#14579083	1	1	yes	characteristic
175	Q₈	—	632	4	no	—
176	Q₈	—	632	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	316	4	no	—
179	C₂×C₄	—	632	4	no	—
180	C₂×C₄	#14579079	1	2	yes	—
181	C₂×C₄	#14579083	1	1	yes	characteristic
182	C₂×C₄	#14579083	1	1	yes	characteristic
183	C₂×C₄	#14579083	1	1	yes	characteristic
184	C₂×C₄	#14579084	1	2	yes	—
185	C₂×C₄	#14579106	1	2	yes	—
186	C₂×C₄	Dic₆₃₂	1	2	yes	—
187	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
189	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	632	2	no	—
193	C₈	#14579084	1	2	yes	—
194	C₈	C₂×#11929612	1	1	yes	characteristic
195	C₈	C₂×#11929614	1	1	yes	characteristic
196	C₂²	#19098308	1	2	yes	—
197	C₂²	#19098368	1	2	yes	—
198	C₂²	#19098377	1	2	yes	—
199	C₂²	C₂×#14579083	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	632	4	no	—
203	C₄	#19098308	1	2	yes	—
204	C₄	#19098368	1	2	yes	—
205	C₄	C₂×#14579079	1	1	yes	characteristic
206	C₄	C₂×#14579084	1	1	yes	characteristic
207	C₄	C₂×#14579106	1	1	yes	characteristic
208	C₄	C₂×Dic₆₃₂	1	1	yes	characteristic
209	C₂	#23761457	1	4	yes	direct product factor
210	C₂	C₂×#19098308	1	1	yes	characteristic
211	C₂	C₂×#19098368	1	1	yes	characteristic
212	C₂	C₂×#19098377	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579083
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	C₂×#1365
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #23761457	C₂	direct (N × H)
103. C₂×C₇₉	#1365	semidirect
111. C₇₉	C₂×#1365	semidirect
209. C₂	#23761457	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
C₂×#19098316	2
#23761457	2
C₂×C₄⋊C₁₆×C₇₉	2
C₂×#1365	79

Subgroups — C2×#23761457 #34199550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761457 #34199550