Subgroups — C₂×C₄×#13050590 #24920643

Order 10624.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13050590	C₂	1	1	yes	maximal, characteristic
2	C₄×#13050590	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₁₆₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₃₃₂	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₈₃	C₂	1	4	yes	maximal, complemented
6	C₂×#14581751	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581749	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₈₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13050590	C₄	1	16	yes	direct product factor
11	C₂×#13050590	C₂²	1	8	yes	—
12	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
13	C₂²×D₃₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₈₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₈₃	C₂²	1	4	yes	—
16	C₂²×C₄×D₈₃	C₂²	1	2	yes	—
17	C₄×Dic₁₆₆	C₂²	1	16	yes	complemented
18	C₄×D₃₃₂	C₂²	1	16	yes	complemented
19	C₄×C₂²⋊D₈₃	C₂²	1	32	yes	complemented
20	#14581751	C₂²	1	8	yes	complemented
21	#14581749	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
23	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊Dic₈₃	C₂²	1	2	yes	—
25	C₂×(C₂×D₈₃)⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₈₃	C₂²	1	4	yes	—
28	C₄²×D₈₃	C₂²	1	16	yes	complemented
29	C₂³×C₄×C₈₃	C₂²	1	2	yes	—
30	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	C₂²	1	8	yes	complemented
35	C₂³×D₈₃	C₂³	1	2	yes	—
36	#13050590	C₂×C₄	1	64	yes	direct product factor
37	C₂×Dic₁₆₆	C₂×C₄	1	8	yes	complemented
38	C₂×Dic₁₆₆	C₂³	1	4	yes	—
39	C₂×D₃₃₂	C₂×C₄	1	8	yes	complemented
40	C₂×D₃₃₂	C₂³	1	4	yes	—
41	C₂×C₂²⋊D₈₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₂²⋊D₈₃	C₂³	1	8	yes	—
43	C₂²×Dic₈₃	C₂³	1	4	yes	—
44	C₂²×Dic₈₃	C₂³	1	2	yes	—
45	C₂×C₄×D₈₃	—	2	16	no	—
46	C₂×C₄×D₈₃	C₂×C₄	1	16	yes	complemented
47	C₂×C₄×D₈₃	C₂³	1	16	yes	—
48	C₂×C₄×D₈₃	C₂³	1	8	yes	—
49	C₄⋊Dic₈₃	C₂³	1	8	yes	—
50	Dic₈₃⋊C₄	C₂³	1	16	yes	—
51	C₂²⋊Dic₈₃	C₂³	1	8	yes	—
52	(C₂×D₈₃)⋊C₄	C₂³	1	16	yes	—
53	C₄×Dic₈₃	C₂³	1	8	yes	—
54	C₄×Dic₈₃	C₂³	1	16	yes	—
55	C₂⁴×C₈₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	—	2	8	no	—
57	C₂²×C₄×C₈₃	C₂×C₄	1	8	yes	complemented
58	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂³	1	8	yes	—
60	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
64	C₄²×C₈₃	C₂³	1	4	yes	—
65	C₄²×C₈₃	C₂³	1	4	yes	—
66	C₄²×C₈₃	C₂³	1	8	yes	—
67	C₂³×C₈₃	—	2	4	no	—
68	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₈₃	C₂⁴	1	2	yes	—
70	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₈₃	—	2	8	no	—
72	C₂²×D₈₃	C₂²×C₄	1	8	yes	—
73	C₂²×D₈₃	C₂⁴	1	4	yes	—
74	C₂×C₄×C₈₃	—	2	4	no	—
75	C₂×C₄×C₈₃	—	2	16	no	—
76	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₈₃	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₈₃	#41	1	8	yes	complemented
80	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
81	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
84	C₂²⋊D₈₃	C₂²×C₄	1	32	yes	complemented
85	C₂×Dic₈₃	—	2	8	no	—
86	C₂×Dic₈₃	C₂²×C₄	1	8	yes	—
87	C₂×Dic₈₃	C₂⁴	1	8	yes	—
88	C₂×Dic₈₃	C₂⁴	1	4	yes	—
89	D₃₃₂	C₂²×C₄	1	16	yes	complemented
90	C₄×D₈₃	—	2	32	no	—
91	C₄×D₈₃	C₂²×C₄	1	32	yes	complemented
92	Dic₁₆₆	C₂²×C₄	1	16	yes	complemented
93	C₂²×C₈₃	—	2	8	no	—
94	C₂²×C₈₃	—	2	4	no	—
95	C₂²×C₈₃	C₂³×C₄	1	4	yes	—
96	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
97	C₂²×C₈₃	C₂×#41	1	4	yes	—
98	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₈₃	—	2	16	no	—
100	C₂×D₈₃	—	2	8	no	—
101	C₂×D₈₃	C₂³×C₄	1	8	yes	—
102	C₄×C₈₃	—	2	4	no	—
103	C₄×C₈₃	C₂³×C₄	1	4	yes	—
104	C₄×C₈₃	C₂³×C₄	1	4	yes	—
105	C₄×C₈₃	C₂×#41	1	8	yes	complemented
106	Dic₈₃	—	2	8	no	—
107	Dic₈₃	C₂³×C₄	1	8	yes	—
108	C₂×C₈₃	—	2	4	no	—
109	C₂×C₈₃	C₄×#41	1	4	yes	complemented
110	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
111	C₂×C₈₃	C₂²×#41	1	2	yes	—
112	D₈₃	—	2	8	no	—
113	C₂×C₄×#41	—	83	1	no	maximal, Sylow
114	C₈₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
115	C₂²×#41	—	83	1	no	—
116	C₄×#41	—	83	16	no	—
117	C₂×C₄×Q₈	—	83	2	no	—
118	C₂×C₄×D₄	—	83	4	no	—
119	C₂×C₄×D₄	—	83	2	no	—
120	C₂×#117	—	83	1	no	—
121	C₂×#117	—	83	2	no	—
122	C₂²×C₄²	—	83	2	no	—
123	C₂²×C₄²	D₈₃	1	1	yes	characteristic, complemented
124	C₂×#41	—	83	16	no	—
125	C₂×#41	—	83	8	no	—
126	C₂²×Q₈	—	83	1	no	—
127	C₂²×D₄	—	83	2	no	—
128	C₂²×D₄	—	83	1	no	—
129	C₂³×C₄	—	83	2	no	—
130	C₂³×C₄	—	83	4	no	—
131	C₂³×C₄	C₂×D₈₃	1	2	yes	—
132	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
133	C₄×Q₈	—	83	16	no	—
134	C₄×D₄	—	83	32	no	—
135	C₄×D₄	—	83	16	no	—
136	#117	—	83	8	no	—
137	#117	—	83	16	no	—
138	C₂×C₄⋊C₄	—	83	4	no	—
139	C₂×C₄⋊C₄	—	83	2	no	—
140	C₂×C₂²⋊C₄	—	83	2	no	—
141	C₂×C₂²⋊C₄	—	83	4	no	—
142	C₂×C₄²	—	83	2	no	—
143	C₂×C₄²	—	83	4	no	—
144	C₂×C₄²	—	83	16	no	—
145	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₈₃	1	2	yes	—
147	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₈₃	1	8	yes	complemented
149	C₂⁴	—	83	2	no	—
150	C₂⁴	C₂²×D₈₃	1	1	yes	characteristic
151	#41	—	83	64	no	—
152	C₂×Q₈	—	83	8	no	—
153	C₂×Q₈	—	83	4	no	—
154	C₂×D₄	—	83	16	no	—
155	C₂×D₄	—	83	8	no	—
156	C₂×D₄	—	83	8	no	—
157	C₂×D₄	—	83	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	83	2	no	—
160	C₂²×C₄	—	83	16	no	—
161	C₂²×C₄	—	83	8	no	—
162	C₂²×C₄	—	83	4	no	—
163	C₂²×C₄	—	83	16	no	—
164	C₂²×C₄	—	166	16	no	—
165	C₂²×C₄	C₄×D₈₃	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₈₃	1	8	yes	—
168	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
171	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
172	C₄⋊C₄	—	83	16	no	—
173	C₄⋊C₄	—	83	8	no	—
174	C₂²⋊C₄	—	83	8	no	—
175	C₂²⋊C₄	—	83	16	no	—
176	C₄²	—	83	8	no	—
177	C₄²	—	83	16	no	—
178	C₄²	C₂²×D₈₃	1	4	yes	—
179	C₄²	C₂²×D₈₃	1	4	yes	—
180	C₄²	C₂²×D₈₃	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	83	8	no	—
183	C₂³	—	83	4	no	—
184	C₂³	—	166	8	no	—
185	C₂³	C₂×C₄×D₈₃	1	4	yes	—
186	C₂³	C₂³×D₈₃	1	2	yes	—
187	C₂³	C₂³×D₈₃	1	1	yes	characteristic
188	Q₈	—	83	16	no	—
189	D₄	—	83	32	no	—
190	D₄	—	83	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	83	8	no	—
194	C₂×C₄	—	83	4	no	—
195	C₂×C₄	—	83	32	no	—
196	C₂×C₄	—	83	8	no	—
197	C₂×C₄	—	166	8	no	—
198	C₂×C₄	—	166	32	no	—
199	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₈₃	1	16	yes	complemented
202	C₂×C₄	#13050590	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₈₃	1	4	yes	—
204	C₂×C₄	C₂³×D₈₃	1	2	yes	—
205	C₂×C₄	C₂³×D₈₃	1	2	yes	—
206	C₂×C₄	C₂³×D₈₃	1	4	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	83	8	no	—
210	C₂²	—	166	16	no	—
211	C₂²	—	166	8	no	—
212	C₂²	C₂²×C₄×D₈₃	1	4	yes	—
213	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
214	C₂²	C₂×#13050590	1	4	yes	—
215	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	83	8	no	—
218	C₄	—	166	8	no	—
219	C₄	C₂²×C₄×D₈₃	1	4	yes	—
220	C₄	C₂²×C₄×D₈₃	1	4	yes	—
221	C₄	C₂×#13050590	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	166	8	no	—
224	C₂	C₄×#13050590	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic
226	C₂	C₂²×#13050590	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₈₃
commutator	C₂×C₈₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₈₃
Fitting	C₂²×C₄²×C₈₃	C₂
socle	C₂³×C₈₃	C₂⁴
prime radical	C₂²×#13050590	C₂
cosocle	C₂²×C₈₃	C₂⁵
hypercenter	C₂²×C₄²	D₈₃
hypocenter	C₈₃	C₂×C₄×#41
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#13050590	C₂	direct (N × H)
3. C₂×C₄×Dic₁₆₆	C₂	semidirect
4. C₂×C₄×D₃₃₂	C₂	semidirect
5. C₂×C₄×C₂²⋊D₈₃	C₂	semidirect
6. C₂×#14581751	C₂	semidirect
7. C₂×#14581749	C₂	semidirect
8. C₂×C₄²×D₈₃	C₂	semidirect
9. C₂²×C₄²×C₈₃	C₂	semidirect
10. C₂×#13050590	C₄	direct (N × H)
17. C₄×Dic₁₆₆	C₂²	semidirect
18. C₄×D₃₃₂	C₂²	semidirect
19. C₄×C₂²⋊D₈₃	C₂²	semidirect
20. #14581751	C₂²	semidirect
21. #14581749	C₂²	semidirect
28. C₄²×D₈₃	C₂²	semidirect
34. C₂×C₄²×C₈₃	C₂²	semidirect
36. #13050590	C₂×C₄	direct (N × H)
37. C₂×Dic₁₆₆	C₂×C₄	semidirect
39. C₂×D₃₃₂	C₂×C₄	semidirect
41. C₂×C₂²⋊D₈₃	C₂×C₄	semidirect
46. C₂×C₄×D₈₃	C₂×C₄	semidirect
57. C₂²×C₄×C₈₃	C₂×C₄	semidirect
78. C₂×C₄×C₈₃	C₂²×C₄	semidirect
79. C₂×C₄×C₈₃	#41	semidirect
84. C₂²⋊D₈₃	C₂²×C₄	semidirect
89. D₃₃₂	C₂²×C₄	semidirect
91. C₄×D₈₃	C₂²×C₄	semidirect
92. Dic₁₆₆	C₂²×C₄	semidirect
105. C₄×C₈₃	C₂×#41	semidirect
109. C₂×C₈₃	C₄×#41	semidirect
114. C₈₃	C₂×C₄×#41	semidirect
123. C₂²×C₄²	D₈₃	semidirect
148. C₂×C₄²	C₂×D₈₃	semidirect
165. C₂²×C₄	C₄×D₈₃	semidirect
201. C₂×C₄	C₂×C₄×D₈₃	semidirect
202. C₂×C₄	#13050590	direct (N × H)
221. C₄	C₂×#13050590	direct (N × H)
224. C₂	C₄×#13050590	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×#13050590	2
C₄×#13050590	2
C₂×C₄×Dic₁₆₆	2
C₂×C₄×D₃₃₂	2
C₂×C₄×C₂²⋊D₈₃	2
C₂×#14581751	2
C₂×#14581749	2
C₂×C₄²×D₈₃	2
C₂²×C₄²×C₈₃	2
C₂×C₄×#41	83

Subgroups — C2×C4×#13050590 #24920643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13050590 #24920643