Subgroups — C₂×#23765272 #34441567

Order 20224.

670 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	C₂²×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₁₆	C₂	1	2	yes	maximal
3	#23765272	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098391	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098368	C₂	1	2	yes	maximal
6	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₅₈	—	2	2	no	—
9	C₂×Dic₆₃₂	C₄	1	8	yes	complemented
10	C₂×Dic₆₃₂	C₂²	1	4	yes	—
11	C₄×Dic₃₁₆	—	2	4	no	—
12	C₄×Dic₃₁₆	C₂²	1	8	yes	—
13	C₂×#14579095	C₂²	1	2	yes	—
14	C₂×#14579086	—	2	2	no	—
15	#19098391	C₂²	1	4	yes	—
16	#19098368	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	#346×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	C₄×Dic₁₅₈	—	2	8	no	—
27	C₄×Dic₁₅₈	—	4	4	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×Dic₇₉⋊C₄	—	4	2	no	—
30	C₂×C₄×Dic₇₉	—	4	2	no	—
31	Dic₆₃₂	C₂×C₄	1	16	yes	complemented
32	#14579095	—	2	2	no	—
33	#14579095	C₂³	1	4	yes	—
34	#14579086	—	2	8	no	—
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₇₉	—	2	2	no	—
39	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
41	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
42	C₄×C₈×C₇₉	D₄	1	2	yes	—
43	C₄×C₈×C₇₉	C₂³	1	2	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	—	4	2	no	—
50	Dic₃₁₆	—	2	8	no	—
51	Dic₃₁₆	—	2	4	no	—
52	Dic₃₁₆	C₂²×C₄	1	8	yes	—
53	C₄⋊Dic₇₉	—	2	4	no	—
54	C₄⋊Dic₇₉	—	4	2	no	—
55	Dic₇₉⋊C₄	—	4	8	no	—
56	C₄×Dic₇₉	—	4	8	no	—
57	C₂²×C₄×C₇₉	D₈	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	—	2	2	no	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	#41	1	2	yes	—
64	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₄²×C₇₉	D₈	1	2	yes	—
66	C₄²×C₇₉	C₂×D₄	1	2	yes	—
67	C₁₆×C₇₉	—	2	2	no	—
68	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
69	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	4	no	—
71	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
73	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#135	1	2	yes	—
75	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₇₉	—	4	4	no	—
77	C₂×Dic₇₉	—	4	8	no	—
78	C₂×Dic₇₉	—	4	4	no	—
79	C₂×Dic₇₉	—	8	4	no	—
80	Dic₁₅₈	—	2	8	no	—
81	Dic₁₅₈	—	4	4	no	—
82	Dic₁₅₈	—	4	8	no	—
83	C₈×C₇₉	—	2	2	no	—
84	C₈×C₇₉	C₄×D₄	1	2	yes	—
85	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
87	C₂²×C₇₉	C₄×D₈	1	2	yes	—
88	C₂²×C₇₉	#510	1	4	yes	—
89	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₇₉	—	2	4	no	—
91	C₄×C₇₉	C₄×D₈	1	2	yes	—
92	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
93	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
94	Dic₇₉	—	4	8	no	—
95	Dic₇₉	—	8	4	no	—
96	C₂×#2176	—	79	1	no	maximal, Sylow
97	C₂×C₇₉	#2176	1	4	yes	complemented
98	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
99	C₂×C₇₉	C₂×#510	1	2	yes	—
100	C₂²×Q₃₂	—	79	1	no	—
101	C₂×C₄×Q₁₆	—	79	2	no	—
102	#2176	—	79	8	no	—
103	C₂×#367	—	79	1	no	—
104	C₂×#358	—	79	2	no	—
105	C₂×#346	D₇₉	1	1	yes	characteristic
106	C₇₉	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₁₆	—	79	2	no	—
108	C₂×C₄×Q₈	—	158	2	no	—
109	C₂×Q₃₂	—	79	4	no	—
110	C₂×Q₃₂	—	79	8	no	—
111	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
113	C₄×Q₁₆	—	79	8	no	—
114	C₄×Q₁₆	—	158	4	no	—
115	C₂×#107	—	79	2	no	—
116	C₂×#103	—	158	2	no	—
117	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	#367	—	79	4	no	—
119	#358	—	79	8	no	—
120	#346	C₂×D₇₉	1	4	yes	—
121	C₂²×Q₈	—	158	2	no	—
122	C₂×Q₁₆	—	79	8	no	—
123	C₂×Q₁₆	—	79	4	no	—
124	C₂×Q₁₆	—	158	2	no	—
125	C₂×Q₁₆	—	158	4	no	—
126	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
128	C₄×Q₈	—	316	4	no	—
129	C₄×Q₈	—	158	8	no	—
130	C₂×C₄⋊C₄	—	316	2	no	—
131	C₂×C₄⋊C₄	—	158	2	no	—
132	C₂×C₄²	—	316	2	no	—
133	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
134	Q₃₂	—	79	16	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
137	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
138	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
139	#107	—	79	4	no	—
140	#107	—	158	2	no	—
141	#103	—	158	8	no	—
142	C₄×C₈	D₃₁₆	1	2	yes	—
143	C₄×C₈	C₂²×D₇₉	1	2	yes	—
144	C₂×Q₈	—	316	2	no	—
145	C₂×Q₈	—	316	4	no	—
146	C₂×Q₈	—	158	8	no	—
147	C₂×Q₈	—	158	4	no	—
148	C₂²×C₄	—	316	2	no	—
149	C₂²×C₄	—	316	2	no	—
150	C₂²×C₄	D₆₃₂	1	2	yes	—
151	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
152	Q₁₆	—	79	8	no	—
153	Q₁₆	—	158	4	no	—
154	Q₁₆	—	158	8	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
157	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
158	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
159	C₂×C₈	#11929624	1	2	yes	—
160	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
161	C₄⋊C₄	—	316	8	no	—
162	C₄⋊C₄	—	316	2	no	—
163	C₄⋊C₄	—	158	4	no	—
164	C₄²	—	316	8	no	—
165	C₄²	D₆₃₂	1	2	yes	—
166	C₄²	C₂×D₃₁₆	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂×C₄×D₇₉	1	4	yes	—
169	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
170	Q₈	—	316	8	no	—
171	Q₈	—	316	4	no	—
172	Q₈	—	158	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	316	8	no	—
175	C₂×C₄	—	316	4	no	—
176	C₂×C₄	—	316	4	no	—
177	C₂×C₄	—	632	4	no	—
178	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
179	C₂×C₄	C₂×D₆₃₂	1	4	yes	—
180	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
181	C₂×C₄	#14579183	1	2	yes	—
182	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₃₁₆	1	2	yes	—
185	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
186	C₈	C₂×#11929624	1	1	yes	characteristic
187	C₂²	C₄×D₆₃₂	1	2	yes	—
188	C₂²	#19099006	1	4	yes	—
189	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	316	8	no	—
192	C₄	—	632	4	no	—
193	C₄	C₄×D₆₃₂	1	2	yes	—
194	C₄	C₂×C₄×D₃₁₆	1	1	yes	characteristic
195	C₄	C₂×#14579183	1	1	yes	characteristic
196	C₂	#23765272	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₆₃₂	1	1	yes	characteristic
198	C₂	C₂×#19099006	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	C₂×#2176
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #23765272	C₂	direct (N × H)
9. C₂×Dic₆₃₂	C₄	semidirect
31. Dic₆₃₂	C₂×C₄	semidirect
97. C₂×C₇₉	#2176	semidirect
106. C₇₉	C₂×#2176	semidirect
196. C₂	#23765272	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×C₄×Dic₃₁₆	2
#23765272	2
C₂×#19098391	2
C₂×#19098368	2
C₂×#346×C₇₉	2
C₂×#2176	79

Subgroups — C2×#23765272 #34441567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765272 #34441567