Subgroups — C₂×#23761456 #34199548

Order 20224.

350 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×#19098968	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#23761456	C₂	1	4	yes	maximal, direct product factor
4	C₂×#335×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
7	#19098968	—	2	1	no	—
8	#19098968	—	2	1	no	—
9	#19098968	C₄	1	2	yes	—
10	#19098968	C₂²	1	2	yes	—
11	C₂×#14579095	—	2	1	no	—
12	C₂×#14579095	C₄	1	2	yes	—
13	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#335×C₇₉	—	2	1	no	—
18	#335×C₇₉	C₂²	1	2	yes	—
19	C₂²×Dic₁₅₈	—	4	1	no	—
20	#14579161	—	2	2	no	—
21	#14579161	—	2	2	no	—
22	#14579161	—	4	1	no	—
23	#14579161	—	4	1	no	—
24	C₂×C₄⋊Dic₇₉	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	4	1	no	—
26	C₂×C₇₉⋊C₁₆	—	2	2	no	—
27	C₂×C₇₉⋊C₁₆	—	4	2	no	—
28	#14579095	—	2	2	no	—
29	#14579095	—	2	2	no	—
30	#14579095	—	2	2	no	—
31	#14579095	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₇₉	—	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	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄⋊C₈×C₇₉	—	4	1	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	2	no	—
45	C₂×Dic₁₅₈	—	8	2	no	—
46	C₂²×Dic₇₉	—	4	2	no	—
47	C₄⋊Dic₇₉	—	2	4	no	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₇₉⋊C₁₆	—	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	—	4	2	no	—
57	C₂×C₈×C₇₉	—	8	2	no	—
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
61	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
62	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₇₉	D₈	1	1	yes	characteristic
66	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
67	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₇₉	#102	1	1	yes	characteristic
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	#101	1	2	yes	—
71	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#103	1	2	yes	—
75	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
76	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
77	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₇₉	—	4	4	no	—
81	C₂×Dic₇₉	—	8	4	no	—
82	Dic₁₅₈	—	8	4	no	—
83	Dic₁₅₈	—	8	2	no	—
84	Dic₁₅₈	—	8	2	no	—
85	C₈×C₇₉	—	2	1	no	—
86	C₈×C₇₉	—	2	1	no	—
87	C₈×C₇₉	—	8	2	no	—
88	C₈×C₇₉	#103	1	2	yes	—
89	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂²×C₇₉	#332	1	2	yes	—
92	C₂²×C₇₉	#357	1	2	yes	—
93	C₂²×C₇₉	#362	1	2	yes	—
94	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	#332	1	2	yes	—
98	C₄×C₇₉	#357	1	2	yes	—
99	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
100	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
101	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
102	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
103	Dic₇₉	—	8	4	no	—
104	C₂×#1364	—	79	1	no	maximal, Sylow
105	C₂×C₇₉	#1364	1	4	yes	complemented
106	C₂×C₇₉	C₂×#332	1	1	yes	characteristic
107	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
108	C₂×C₇₉	C₂×#362	1	1	yes	characteristic
109	C₂×#500	—	79	1	no	—
110	C₂×C₄⋊C₁₆	—	79	1	no	—
111	C₂×#335	D₇₉	1	1	yes	characteristic
112	#1364	—	79	4	no	—
113	C₇₉	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	158	1	no	—
115	C₂²×C₁₆	—	158	1	no	—
116	#500	—	79	2	no	—
117	#500	—	79	2	no	—
118	#500	—	158	1	no	—
119	#500	—	158	1	no	—
120	C₂×#107	—	79	2	no	—
121	C₂×#107	—	158	1	no	—
122	C₂×C₄⋊C₈	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
124	C₄⋊C₁₆	—	79	2	no	—
125	C₄⋊C₁₆	—	158	1	no	—
126	#335	—	2	1	no	—
127	#335	C₂×D₇₉	1	2	yes	—
128	C₂²×Q₈	—	316	1	no	—
129	C₂²×C₈	—	4	1	no	—
130	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
131	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
132	C₄⋊Q₈	—	316	1	no	—
133	C₄⋊Q₈	—	316	1	no	—
134	C₄⋊Q₈	—	158	2	no	—
135	C₄⋊Q₈	—	158	2	no	—
136	C₂×C₄⋊C₄	—	316	1	no	—
137	C₂×C₄⋊C₄	—	158	2	no	—
138	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
139	C₂×C₁₆	—	316	2	no	—
140	C₂×C₁₆	—	158	2	no	—
141	#107	—	79	4	no	—
142	#107	—	158	2	no	—
143	#107	—	158	2	no	—
144	#107	—	158	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
148	C₄×C₈	D₃₁₆	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
151	C₂×Q₈	—	316	2	no	—
152	C₂×Q₈	—	316	2	no	—
153	C₂×Q₈	—	632	2	no	—
154	C₂×Q₈	—	632	2	no	—
155	C₂²×C₄	—	316	2	no	—
156	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	#11929609	1	1	yes	characteristic
158	C₂²×C₄	#11929611	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	8	2	no	—
162	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
163	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
164	C₂×C₈	#11929613	1	2	yes	—
165	C₂×C₈	Dic₃₁₆	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
167	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
168	C₄⋊C₄	—	316	2	no	—
169	C₄⋊C₄	—	316	2	no	—
170	C₄⋊C₄	—	316	2	no	—
171	C₄⋊C₄	—	158	4	no	—
172	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	C₄²	#11929609	1	1	yes	characteristic
174	C₄²	#11929611	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
176	C₁₆	—	316	2	no	—
177	C₂³	#14579081	1	1	yes	characteristic
178	Q₈	—	632	2	no	—
179	Q₈	—	632	2	no	—
180	Q₈	—	632	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	316	4	no	—
183	C₂×C₄	—	632	4	no	—
184	C₂×C₄	#14579079	1	2	yes	—
185	C₂×C₄	#14579081	1	1	yes	characteristic
186	C₂×C₄	#14579081	1	1	yes	characteristic
187	C₂×C₄	#14579081	1	1	yes	characteristic
188	C₂×C₄	#14579086	1	2	yes	—
189	C₂×C₄	#14579102	1	2	yes	—
190	C₂×C₄	#14579104	1	2	yes	—
191	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
193	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	8	2	no	—
197	C₈	#14579086	1	2	yes	—
198	C₈	C₂×#11929613	1	1	yes	characteristic
199	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
200	C₂²	#19098310	1	2	yes	—
201	C₂²	#19098363	1	2	yes	—
202	C₂²	#19098378	1	2	yes	—
203	C₂²	C₂×#14579081	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	632	4	no	—
207	C₄	#19098310	1	2	yes	—
208	C₄	#19098363	1	2	yes	—
209	C₄	C₂×#14579079	1	1	yes	characteristic
210	C₄	C₂×#14579086	1	1	yes	characteristic
211	C₄	C₂×#14579102	1	1	yes	characteristic
212	C₄	C₂×#14579104	1	1	yes	characteristic
213	C₂	#23761456	1	4	yes	direct product factor
214	C₂	C₂×#19098310	1	1	yes	characteristic
215	C₂	C₂×#19098363	1	1	yes	characteristic
216	C₂	C₂×#19098378	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579081
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#335×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#335	D₇₉
hypocenter	C₇₉	C₂×#1364
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #23761456	C₂	direct (N × H)
105. C₂×C₇₉	#1364	semidirect
113. C₇₉	C₂×#1364	semidirect
213. C₂	#23761456	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×C₄⋊(C₇₉⋊C₁₆)	2
#23761456	2
C₂×#335×C₇₉	2
C₂×#1364	79

Subgroups — C2×#23761456 #34199548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761456 #34199548