Subgroups — C₂×#19128177 #24920652

Order 10624.

1488 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic
2	#19128177	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×Dic₁₆₆	C₂	1	3	yes	maximal, complemented
5	C₂×#14581757	C₂	1	3	yes	maximal, complemented
6	C₂×#14581750	C₂	1	3	yes	maximal
7	C₂×#14581749	C₂	1	3	yes	maximal
8	C₂×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₆₆	C₂²	1	3	yes	—
10	C₂×Q₈×D₈₃	C₄	1	16	yes	complemented
11	C₂×Q₈×D₈₃	C₂²	1	8	yes	—
12	C₂²×C₄×D₈₃	C₂²	1	3	yes	—
13	Q₈×Dic₈₃	C₂²	1	8	yes	complemented
14	C₄×Dic₁₆₆	C₂²	1	24	yes	complemented
15	#14581757	C₂²	1	24	yes	complemented
16	#14581750	C₂²	1	24	yes	—
17	#14581749	C₂²	1	24	yes	—
18	C₂×C₄⋊Dic₈₃	C₂²	1	3	yes	—
19	C₂×Dic₈₃⋊C₄	C₂²	1	3	yes	—
20	C₂×Dic₈₃⋊C₄	C₂²	1	3	yes	—
21	C₂×(C₂×D₈₃)⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₈₃)⋊C₄	C₂²	1	3	yes	—
23	C₂×C₄×Dic₈₃	C₂²	1	3	yes	—
24	C₂×C₄×Dic₈₃	C₂²	1	6	yes	—
25	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₈₃	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
28	C₂×C₄²×C₈₃	C₂²	1	3	yes	—
29	C₂³×D₈₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₆₆	C₂×C₄	1	24	yes	—
31	C₂×Dic₁₆₆	C₂³	1	12	yes	—
32	Q₈×D₈₃	C₂×C₄	1	64	yes	complemented
33	C₂²×Dic₈₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₈₃	C₂³	1	3	yes	—
35	C₂²×Dic₈₃	C₂³	1	3	yes	—
36	C₂×C₄×D₈₃	C₂×C₄	1	24	yes	—
37	C₂×C₄×D₈₃	C₂³	1	12	yes	—
38	C₄⋊Dic₈₃	C₂³	1	12	yes	—
39	Dic₈₃⋊C₄	C₂³	1	12	yes	—
40	Dic₈₃⋊C₄	C₂³	1	12	yes	—
41	(C₂×D₈₃)⋊C₄	C₂³	1	4	yes	—
42	(C₂×D₈₃)⋊C₄	C₂³	1	12	yes	—
43	C₄×Dic₈₃	C₂³	1	12	yes	—
44	C₄×Dic₈₃	C₂³	1	24	yes	—
45	C₂×Q₈×C₈₃	C₂×C₄	1	8	yes	—
46	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
47	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
49	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
50	C₄⋊C₄×C₈₃	C₂³	1	12	yes	—
51	C₄²×C₈₃	C₂³	1	12	yes	—
52	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
53	C₂²×D₈₃	—	2	2	no	—
54	C₂²×D₈₃	—	2	2	no	—
55	C₂²×D₈₃	C₂²×C₄	1	4	yes	—
56	C₂²×D₈₃	C₂⁴	1	2	yes	—
57	Q₈×C₈₃	C₂²×C₄	1	16	yes	—
58	C₂×C₄×C₈₃	—	2	6	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	C₂²×C₄	1	12	yes	—
61	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
62	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
63	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
64	C₂×Dic₈₃	—	2	2	no	—
65	C₂×Dic₈₃	—	2	6	no	—
66	C₂×Dic₈₃	C₂²×C₄	1	12	yes	—
67	C₂×Dic₈₃	C₂⁴	1	2	yes	—
68	C₂×Dic₈₃	C₂⁴	1	6	yes	—
69	C₂×Dic₈₃	C₂⁴	1	6	yes	—
70	C₄×D₈₃	C₂²×C₄	1	48	yes	—
71	Dic₁₆₆	C₂²×C₄	1	48	yes	—
72	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
73	C₂²×C₈₃	2₋¹⁺⁴	1	2	yes	—
74	C₂²×C₈₃	2₋¹⁺⁴	1	2	yes	—
75	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
76	C₂×D₈₃	—	2	8	no	—
77	C₂×D₈₃	—	2	2	no	—
78	C₂×D₈₃	—	2	2	no	—
79	C₂×D₈₃	C₂³×C₄	1	4	yes	—
80	C₄×C₈₃	—	2	6	no	—
81	C₄×C₈₃	—	2	2	no	—
82	C₄×C₈₃	C₂³×C₄	1	12	yes	—
83	Dic₈₃	—	2	2	no	—
84	Dic₈₃	—	2	6	no	—
85	Dic₈₃	C₂³×C₄	1	12	yes	—
86	C₂×C₈₃	#519	1	4	yes	complemented
87	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
88	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
89	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
90	D₈₃	—	2	4	no	—
91	C₂×#519	—	83	1	no	maximal, Sylow
92	C₈₃	C₂×#519	1	1	yes	characteristic, complemented, Sylow
93	C₂³×Q₈	—	83	1	no	—
94	#519	—	83	16	no	—
95	C₂×C₄×Q₈	—	83	1	no	—
96	C₂×C₄×Q₈	—	83	3	no	—
97	C₂×C₄×Q₈	—	83	3	no	—
98	C₂×C₄×Q₈	D₈₃	1	1	yes	characteristic, complemented
99	C₂×#117	—	83	3	no	—
100	C₂×#117	—	83	3	no	—
101	C₂²×Q₈	—	83	16	no	—
102	C₂²×Q₈	—	83	8	no	—
103	C₂²×Q₈	—	83	3	no	—
104	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
105	C₂³×C₄	—	83	3	no	—
106	C₄×Q₈	—	83	8	no	—
107	C₄×Q₈	—	83	24	no	—
108	C₄×Q₈	—	83	24	no	—
109	C₄×Q₈	C₂×D₈₃	1	8	yes	complemented
110	#117	—	83	24	no	—
111	#117	—	83	24	no	—
112	C₂×C₄⋊C₄	—	83	3	no	—
113	C₂×C₄⋊C₄	—	83	3	no	—
114	C₂×C₄⋊C₄	—	83	3	no	—
115	C₂×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
116	C₂×C₂²⋊C₄	—	83	3	no	—
117	C₂×C₂²⋊C₄	—	83	1	no	—
118	C₂×C₄²	—	83	6	no	—
119	C₂×C₄²	—	83	3	no	—
120	C₂×C₄²	C₂×D₈₃	1	3	yes	—
121	C₂⁴	—	83	1	no	—
122	C₂×Q₈	—	83	64	no	—
123	C₂×Q₈	—	83	24	no	—
124	C₂×Q₈	—	83	12	no	—
125	C₂×Q₈	C₄×D₈₃	1	8	yes	—
126	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
127	C₂²×C₄	—	83	3	no	—
128	C₂²×C₄	—	83	24	no	—
129	C₂²×C₄	—	83	12	no	—
130	C₂²×C₄	—	83	1	no	—
131	C₂²×C₄	—	83	3	no	—
132	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
133	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
135	C₄⋊C₄	—	83	12	no	—
136	C₄⋊C₄	—	83	12	no	—
137	C₄⋊C₄	—	83	12	no	—
138	C₄⋊C₄	C₂²×D₈₃	1	12	yes	—
139	C₂²⋊C₄	—	83	12	no	—
140	C₂²⋊C₄	—	83	4	no	—
141	C₄²	—	83	24	no	—
142	C₄²	—	83	12	no	—
143	C₄²	C₂²×D₈₃	1	12	yes	—
144	C₂³	—	83	4	no	—
145	C₂³	—	83	2	no	—
146	C₂³	—	166	2	no	—
147	C₂³	—	166	2	no	—
148	C₂³	C₂³×D₈₃	1	1	yes	characteristic
149	Q₈	—	83	48	no	—
150	Q₈	C₂×C₄×D₈₃	1	16	yes	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	83	12	no	—
154	C₂×C₄	—	83	6	no	—
155	C₂×C₄	—	83	48	no	—
156	C₂×C₄	—	83	2	no	—
157	C₂×C₄	—	83	6	no	—
158	C₂×C₄	—	166	2	no	—
159	C₂×C₄	—	166	6	no	—
160	C₂×C₄	C₂×C₄×D₈₃	1	12	yes	—
161	C₂×C₄	C₂³×D₈₃	1	6	yes	—
162	C₂×C₄	C₂³×D₈₃	1	2	yes	—
163	C₂×C₄	C₂³×D₈₃	1	6	yes	—
164	C₂²	—	83	4	no	—
165	C₂²	—	166	8	no	—
166	C₂²	—	166	2	no	—
167	C₂²	—	166	2	no	—
168	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
169	C₂²	#14581839	1	2	yes	—
170	C₂²	#14581840	1	2	yes	—
171	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
172	C₄	—	2	6	no	—
173	C₄	—	2	2	no	—
174	C₄	—	83	12	no	—
175	C₄	—	166	2	no	—
176	C₄	—	166	6	no	—
177	C₄	C₂²×C₄×D₈₃	1	12	yes	—
178	C₂	—	166	4	no	—
179	C₂	#19128177	1	4	yes	direct product factor
180	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic
181	C₂	C₂×#14581839	1	1	yes	characteristic
182	C₂	C₂×#14581840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₈₃
commutator	C₂×C₈₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₈₃
Fitting	C₂×C₄×Q₈×C₈₃	C₂
socle	C₂³×C₈₃	C₂⁴
prime radical	C₂³×D₈₃	C₂³
cosocle	C₂²×C₈₃	C₂⁵
hypercenter	C₂×C₄×Q₈	D₈₃
hypocenter	C₈₃	C₂×#519
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #19128177	C₂	direct (N × H)
3. C₂×Q₈×Dic₈₃	C₂	semidirect
4. C₂×C₄×Dic₁₆₆	C₂	semidirect
5. C₂×#14581757	C₂	semidirect
8. C₂×C₄×Q₈×C₈₃	C₂	semidirect
10. C₂×Q₈×D₈₃	C₄	semidirect
13. Q₈×Dic₈₃	C₂²	semidirect
14. C₄×Dic₁₆₆	C₂²	semidirect
15. #14581757	C₂²	semidirect
26. C₄×Q₈×C₈₃	C₂²	semidirect
32. Q₈×D₈₃	C₂×C₄	semidirect
86. C₂×C₈₃	#519	semidirect
92. C₈₃	C₂×#519	semidirect
98. C₂×C₄×Q₈	D₈₃	semidirect
109. C₄×Q₈	C₂×D₈₃	semidirect
179. C₂	#19128177	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×Q₈×D₈₃	2
#19128177	2
C₂×Q₈×Dic₈₃	2
C₂×C₄×Dic₁₆₆	2
C₂×#14581757	2
C₂×#14581750	2
C₂×#14581749	2
C₂×C₄×Q₈×C₈₃	2
C₂×#519	83

Subgroups — C2×#19128177 #24920652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128177 #24920652