Subgroups — C₂×C₄×Dic₇₁₂ #36707704

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	2	yes	maximal
3	C₄×Dic₇₁₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151403	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₇₈	—	2	2	no	—
9	C₂×Dic₇₁₂	C₄	1	8	yes	direct product factor
10	C₂×Dic₇₁₂	C₂²	1	4	yes	—
11	C₄×Dic₃₅₆	—	2	4	no	—
12	C₄×Dic₃₅₆	C₂²	1	8	yes	—
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667937	—	2	2	no	—
15	#19151424	C₂²	1	4	yes	—
16	#19151403	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₈₉	C₂²	1	4	yes	—
21	C₂²×Dic₁₇₈	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	—	2	4	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	8	yes	—
25	C₂×Dic₃₅₆	C₂³	1	4	yes	—
26	C₄×Dic₁₇₈	—	2	8	no	—
27	C₄×Dic₁₇₈	—	4	4	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×#13083553	—	4	2	no	—
30	C₂×C₄×Dic₈₉	—	4	2	no	—
31	Dic₇₁₂	C₂×C₄	1	16	yes	direct product factor
32	#15667946	—	2	2	no	—
33	#15667946	C₂³	1	4	yes	—
34	#15667937	—	2	8	no	—
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₈₉	—	2	2	no	—
39	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
41	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
42	C₄×C₈×C₈₉	D₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂×Dic₁₇₈	—	2	8	no	—
45	C₂×Dic₁₇₈	—	2	4	no	—
46	C₂×Dic₁₇₈	—	4	2	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂²×Dic₈₉	—	4	2	no	—
49	C₂²×Dic₈₉	—	4	2	no	—
50	Dic₃₅₆	—	2	8	no	—
51	Dic₃₅₆	—	2	4	no	—
52	Dic₃₅₆	C₂²×C₄	1	8	yes	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	#13083553	—	4	8	no	—
56	C₄×Dic₈₉	—	4	8	no	—
57	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	—	2	2	no	—
61	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	#41	1	2	yes	—
65	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₄²×C₈₉	D₈	1	2	yes	—
67	C₄²×C₈₉	C₂×D₄	1	2	yes	—
68	C₁₆×C₈₉	—	2	2	no	—
69	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
70	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	Q₃₂	1	4	yes	complemented
73	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₈₉	#135	1	2	yes	—
78	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₈₉	—	4	4	no	—
80	C₂×Dic₈₉	—	4	8	no	—
81	C₂×Dic₈₉	—	4	4	no	—
82	C₂×Dic₈₉	—	8	4	no	—
83	Dic₁₇₈	—	2	8	no	—
84	Dic₁₇₈	—	4	4	no	—
85	Dic₁₇₈	—	4	8	no	—
86	C₈×C₈₉	—	2	2	no	—
87	C₈×C₈₉	C₄×D₄	1	2	yes	—
88	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
90	C₂²×C₈₉	C₄×D₈	1	2	yes	—
91	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
92	C₂²×C₈₉	#508	1	2	yes	—
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	C₄×D₈	1	2	yes	—
96	C₄×C₈₉	C₂×Q₃₂	1	4	yes	complemented
97	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
99	Dic₈₉	—	4	8	no	—
100	Dic₈₉	—	8	4	no	—
101	C₂×C₄×Q₃₂	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	C₄×Q₃₂	1	4	yes	complemented
103	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
104	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
106	C₂²×Q₃₂	—	89	1	no	—
107	C₂×C₄×Q₁₆	—	89	2	no	—
108	C₄×Q₃₂	—	89	8	no	—
109	C₂×#366	—	89	1	no	—
110	C₂×#358	—	89	2	no	—
111	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
112	C₈₉	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	89	2	no	—
114	C₂×C₄×Q₈	—	178	2	no	—
115	C₂×Q₃₂	—	89	8	no	—
116	C₂×Q₃₂	—	89	4	no	—
117	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
119	C₄×Q₁₆	—	89	8	no	—
120	C₄×Q₁₆	—	178	4	no	—
121	C₂×#107	—	89	2	no	—
122	C₂×#103	—	178	2	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	#366	—	89	4	no	—
125	#358	—	89	8	no	—
126	C₄×C₁₆	C₂×D₈₉	1	4	yes	—
127	C₂²×Q₈	—	178	2	no	—
128	C₂×Q₁₆	—	89	8	no	—
129	C₂×Q₁₆	—	89	4	no	—
130	C₂×Q₁₆	—	178	2	no	—
131	C₂×Q₁₆	—	178	4	no	—
132	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
134	C₄×Q₈	—	356	4	no	—
135	C₄×Q₈	—	178	8	no	—
136	C₂×C₄⋊C₄	—	356	2	no	—
137	C₂×C₄⋊C₄	—	178	2	no	—
138	C₂×C₄²	—	356	2	no	—
139	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
140	Q₃₂	—	89	16	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
143	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
144	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
145	#107	—	89	4	no	—
146	#107	—	178	2	no	—
147	#103	—	178	8	no	—
148	C₄×C₈	D₃₅₆	1	2	yes	—
149	C₄×C₈	C₂²×D₈₉	1	2	yes	—
150	C₂×Q₈	—	356	2	no	—
151	C₂×Q₈	—	356	4	no	—
152	C₂×Q₈	—	178	8	no	—
153	C₂×Q₈	—	178	4	no	—
154	C₂²×C₄	—	356	2	no	—
155	C₂²×C₄	—	356	2	no	—
156	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
157	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
159	Q₁₆	—	89	8	no	—
160	Q₁₆	—	178	8	no	—
161	Q₁₆	—	178	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
164	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
165	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
166	C₂×C₈	#13083574	1	2	yes	—
167	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	356	8	no	—
169	C₄⋊C₄	—	356	2	no	—
170	C₄⋊C₄	—	178	4	no	—
171	C₄²	—	356	8	no	—
172	C₄²	D₇₁₂	1	2	yes	—
173	C₄²	C₂×D₃₅₆	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
176	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
177	Q₈	—	356	8	no	—
178	Q₈	—	356	4	no	—
179	Q₈	—	178	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	356	8	no	—
182	C₂×C₄	—	356	4	no	—
183	C₂×C₄	—	356	4	no	—
184	C₂×C₄	—	712	4	no	—
185	C₂×C₄	Dic₇₁₂	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
187	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
188	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
190	C₂×C₄	#15668034	1	2	yes	—
191	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₃₅₆	1	2	yes	—
194	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
195	C₈	C₂×#13083574	1	1	yes	characteristic
196	C₂²	C₄×D₇₁₂	1	2	yes	—
197	C₂²	C₂×Dic₇₁₂	1	2	yes	—
198	C₂²	#19152030	1	2	yes	—
199	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	356	8	no	—
202	C₄	—	712	4	no	—
203	C₄	C₄×D₇₁₂	1	2	yes	—
204	C₄	C₂×Dic₇₁₂	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
206	C₄	C₂×#15668034	1	1	yes	characteristic
207	C₂	C₄×Dic₇₁₂	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
209	C₂	C₂²×Dic₇₁₂	1	1	yes	characteristic
210	C₂	C₂×#19152030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₇₁₂	C₂	direct (N × H)
9. C₂×Dic₇₁₂	C₄	direct (N × H)
31. Dic₇₁₂	C₂×C₄	direct (N × H)
72. C₂×C₄×C₈₉	Q₃₂	semidirect
96. C₄×C₈₉	C₂×Q₃₂	semidirect
102. C₂×C₈₉	C₄×Q₃₂	semidirect
112. C₈₉	C₂×C₄×Q₃₂	semidirect
185. C₂×C₄	Dic₇₁₂	direct (N × H)
204. C₄	C₂×Dic₇₁₂	direct (N × H)
207. C₂	C₄×Dic₇₁₂	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×Dic₇₁₂	2
C₂×C₄×Dic₃₅₆	2
C₄×Dic₇₁₂	2
C₂×#19151424	2
C₂×#19151403	2
C₂×C₄×C₁₆×C₈₉	2
C₂×C₄×Q₃₂	89

Subgroups — C2×C4×Dic712 #36707704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₇₁₂ #36707704