Subgroups — C₂×#23765687 #34443767

Order 20224.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 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	#23765687	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098390	C₂	1	2	yes	maximal
4	C₂×#19098388	C₂	1	2	yes	maximal
5	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
6	C₂×#499×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579161	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#19098968	—	2	2	no	—
10	#19098968	C₂²	1	4	yes	—
11	C₂×#14579095	—	2	1	no	—
12	C₂×#14579095	C₂²	1	2	yes	—
13	#19098390	C₂²	1	8	yes	—
14	#19098388	C₂²	1	8	yes	—
15	#19098339	C₂²	1	4	yes	—
16	C₂×#125×C₇₉	—	2	1	no	—
17	#499×C₇₉	—	2	2	no	—
18	#499×C₇₉	C₂²	1	4	yes	—
19	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₇₉	—	2	1	no	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₅₈	—	4	1	no	—
24	#14579161	—	2	4	no	—
25	#14579161	—	4	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	4	1	no	—
28	C₂×C₇₉⋊C₁₆	Q₈	1	8	yes	complemented
29	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
30	#14579095	—	2	2	no	—
31	#14579095	—	2	4	no	—
32	#14579095	C₂³	1	4	yes	—
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	#125×C₇₉	—	2	4	no	—
36	#125×C₇₉	—	4	2	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
40	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	#107×C₇₉	—	2	1	no	—
43	#107×C₇₉	—	2	1	no	—
44	#107×C₇₉	C₂³	1	2	yes	—
45	#107×C₇₉	C₂³	1	2	yes	—
46	#106×C₇₉	—	2	4	no	—
47	C₄×C₈×C₇₉	D₄	1	2	yes	—
48	C₄×C₈×C₇₉	C₂³	1	2	yes	—
49	C₂×Dic₁₅₈	—	4	4	no	—
50	C₂×Dic₁₅₈	—	8	4	no	—
51	C₂²×Dic₇₉	—	4	2	no	—
52	C₄⋊Dic₇₉	—	2	4	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₄⋊Dic₇₉	—	4	4	no	—
55	C₇₉⋊C₁₆	C₂×Q₈	1	8	yes	complemented
56	C₂²×C₄×C₇₉	—	4	1	no	—
57	C₂²×C₄×C₇₉	—	4	1	no	—
58	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
62	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₇₉	—	2	2	no	—
67	C₄⋊C₄×C₇₉	—	2	2	no	—
68	C₄⋊C₄×C₇₉	—	4	1	no	—
69	C₄⋊C₄×C₇₉	—	4	1	no	—
70	C₄⋊C₄×C₇₉	—	4	4	no	—
71	C₄⋊C₄×C₇₉	—	4	4	no	—
72	C₄²×C₇₉	D₈	1	2	yes	—
73	C₄²×C₇₉	C₂×D₄	1	2	yes	—
74	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	—	8	2	no	—
80	C₂×C₄×C₇₉	—	8	2	no	—
81	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₇₉	—	4	4	no	—
88	C₂×Dic₇₉	—	8	4	no	—
89	Dic₁₅₈	—	8	8	no	—
90	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
91	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
92	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
93	C₂²×C₇₉	#500	1	2	yes	—
94	C₂²×C₇₉	#509	1	2	yes	—
95	C₂²×C₇₉	#510	1	2	yes	—
96	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₇₉	—	2	2	no	—
98	C₄×C₇₉	—	2	2	no	—
99	C₄×C₇₉	—	8	2	no	—
100	C₄×C₇₉	—	8	2	no	—
101	C₄×C₇₉	#500	1	2	yes	—
102	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₇₉	—	8	4	no	—
105	C₂×#2255	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	#2255	1	4	yes	complemented
107	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
108	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
109	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
110	C₂×#500	—	79	1	no	—
111	C₂×#499	D₇₉	1	1	yes	characteristic
112	#2255	—	79	8	no	—
113	C₂×#367	—	79	2	no	—
114	C₂×#366	—	79	2	no	—
115	C₂×#346	—	79	1	no	—
116	C₇₉	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	158	1	no	—
118	C₂×#125	—	2	1	no	—
119	C₂²×C₁₆	—	79	2	no	—
120	#500	—	79	4	no	—
121	#500	—	158	2	no	—
122	#499	—	2	2	no	—
123	#499	C₂×D₇₉	1	4	yes	—
124	C₂×#107	—	79	2	no	—
125	C₂×#107	—	158	1	no	—
126	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
127	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
128	C₂×#106	—	2	1	no	—
129	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
130	#367	—	79	8	no	—
131	#366	—	79	8	no	—
132	#346	—	79	4	no	—
133	C₂²×Q₈	—	316	1	no	—
134	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	316	2	no	—
137	C₄⋊Q₈	—	158	4	no	—
138	#125	—	2	4	no	—
139	#125	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	316	1	no	—
145	C₂×C₄⋊C₄	—	158	2	no	—
146	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂×C₁₆	—	79	8	no	—
148	C₂×C₁₆	—	79	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	79	4	no	—
152	#107	—	158	2	no	—
153	#107	—	158	4	no	—
154	#107	C₂²×D₇₉	1	2	yes	—
155	#107	C₂²×D₇₉	1	2	yes	—
156	#106	—	2	4	no	—
157	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
158	C₄×C₈	C₂²×D₇₉	1	2	yes	—
159	C₂×Q₈	—	316	4	no	—
160	C₂×Q₈	—	632	4	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	316	2	no	—
164	C₂²×C₄	#11929609	1	1	yes	characteristic
165	C₂²×C₄	#11929609	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
167	C₂×C₈	#11929614	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
170	C₂×C₈	Q₈×D₇₉	1	4	yes	—
171	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	316	4	no	—
179	C₄⋊C₄	—	316	2	no	—
180	C₄⋊C₄	—	158	4	no	—
181	C₄²	#11929609	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
183	C₁₆	—	79	8	no	—
184	C₂³	C₂×#11929609	1	1	yes	characteristic
185	Q₈	—	632	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	316	4	no	—
193	C₂×C₄	—	632	4	no	—
194	C₂×C₄	#14579159	1	2	yes	—
195	C₂×C₄	C₂×#11929609	1	2	yes	—
196	C₂×C₄	C₂×#11929609	1	2	yes	—
197	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
198	C₂×C₄	C₂×#11929614	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
200	C₈	#14579159	1	2	yes	—
201	C₈	C₂×#11929614	1	4	yes	—
202	C₈	C₂×Q₈×D₇₉	1	2	yes	—
203	C₂²	#19098966	1	2	yes	—
204	C₂²	#19098998	1	2	yes	—
205	C₂²	#19099003	1	2	yes	—
206	C₂²	C₂²×#11929609	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	632	4	no	—
212	C₄	#19098966	1	2	yes	—
213	C₄	C₂×#14579159	1	1	yes	characteristic
214	C₄	C₂²×#11929614	1	1	yes	characteristic
215	C₂	#23765687	1	4	yes	direct product factor
216	C₂	C₂×#19098966	1	1	yes	characteristic
217	C₂	C₂×#19098998	1	1	yes	characteristic
218	C₂	C₂×#19099003	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#499×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#499	D₇₉
hypocenter	C₇₉	C₂×#2255
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #23765687	C₂	direct (N × H)
28. C₂×C₇₉⋊C₁₆	Q₈	semidirect
55. C₇₉⋊C₁₆	C₂×Q₈	semidirect
106. C₂×C₇₉	#2255	semidirect
116. C₇₉	C₂×#2255	semidirect
215. C₂	#23765687	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
#23765687	2
C₂×#19098390	2
C₂×#19098388	2
C₂×#19098339	2
C₂×#499×C₇₉	2
C₂×#2255	79

Subgroups — C2×#23765687 #34443767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765687 #34443767