Subgroups — C₂×C₄×D₈₄ #13061563

Order 1344.

852 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₂₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#66631	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₈₄	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×D₂₈	—	3	1	no	maximal, Hall
8	C₂³×D₂₁	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₁	C₂²	1	2	yes	—
13	C₂×D₈₄	C₄	1	8	yes	direct product factor
14	C₂×D₈₄	C₂²	1	4	yes	—
15	C₂×C₄×D₂₁	—	2	8	no	—
16	C₂×C₄×D₂₁	C₂²	1	8	yes	complemented
17	C₃×C₄²×C₇	C₂²	1	4	yes	complemented
18	#66631	C₂²	1	8	yes	complemented
19	C₄⋊Dic₂₁	C₂²	1	4	yes	complemented
20	C₂²×D₂₈	—	3	1	no	—
21	C₂²×C₄×D₇	—	3	2	no	—
22	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
23	C₂×(C₂×D₇)⋊C₄	—	3	2	no	—
24	C₂×C₄⋊Dic₇	—	3	1	no	—
25	C₄×D₂₈	—	3	8	no	—
26	C₂×C₄×D₁₂	—	7	1	no	maximal, Hall
27	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
28	C₂²×D₂₁	—	2	4	no	—
29	C₂²×D₂₁	—	2	4	no	—
30	C₂²×D₂₁	C₂×C₄	1	8	yes	complemented
31	C₂²×D₂₁	C₂³	1	4	yes	—
32	C₂×C₃×C₄×C₇	—	2	2	no	—
33	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
35	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
38	C₂×Dic₂₁	—	2	4	no	—
39	C₂×Dic₂₁	C₂³	1	4	yes	—
40	D₈₄	C₂×C₄	1	16	yes	direct product factor
41	C₄×D₂₁	—	2	16	no	—
42	C₂³×D₇	—	3	2	no	—
43	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
46	C₂²×Dic₇	—	3	2	no	—
47	C₂×D₂₈	—	3	8	no	—
48	C₂×D₂₈	—	3	4	no	—
49	C₂×C₄×D₇	—	3	8	no	—
50	C₂×C₄×D₇	—	6	8	no	—
51	C₄²×C₇	C₂×D₃	1	4	yes	complemented
52	(C₂×D₇)⋊C₄	—	3	8	no	—
53	C₄⋊Dic₇	—	3	4	no	—
54	C₂²×D₁₂	—	7	1	no	—
55	C₂²×C₄×D₃	—	7	2	no	—
56	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
57	C₂×(C₂×D₃)⋊C₄	—	7	2	no	—
58	C₂×C₄⋊Dic₃	—	7	1	no	—
59	C₄×D₁₂	—	7	8	no	—
60	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
62	C₂²×C₃×C₇	#41	1	2	yes	—
63	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×D₂₁	—	2	16	no	—
65	C₂×D₂₁	—	2	4	no	—
66	C₂×D₂₁	—	2	4	no	—
67	C₂×D₂₁	C₂²×C₄	1	8	yes	complemented
68	C₃×C₄×C₇	—	2	2	no	—
69	C₃×C₄×C₇	C₂²×C₄	1	4	yes	complemented
70	C₃×C₄×C₇	C₂×D₄	1	4	yes	complemented
71	Dic₂₁	—	2	4	no	—
72	C₂×C₄×D₄	—	21	1	no	Sylow
73	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
74	C₂²×D₇	—	3	8	no	—
75	C₂²×D₇	—	3	4	no	—
76	C₂²×D₇	—	6	4	no	—
77	C₂²×D₇	—	6	4	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
80	C₂×C₄×C₇	D₁₂	1	4	yes	complemented
81	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
82	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
83	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
84	C₂×Dic₇	—	3	4	no	—
85	C₂×Dic₇	—	6	4	no	—
86	D₂₈	—	3	16	no	—
87	C₄×D₇	—	6	16	no	—
88	C₂³×D₃	—	7	2	no	—
89	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
92	C₂²×Dic₃	—	7	2	no	—
93	C₂×D₁₂	—	7	8	no	—
94	C₂×D₁₂	—	7	4	no	—
95	C₂×C₄×D₃	—	7	8	no	—
96	C₂×C₄×D₃	—	14	8	no	—
97	C₃×C₄²	C₂×D₇	1	4	yes	complemented
98	(C₂×D₃)⋊C₄	—	7	8	no	—
99	C₄⋊Dic₃	—	7	4	no	—
100	C₂×C₃×C₇	C₄×D₄	1	4	yes	complemented
101	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
104	D₂₁	—	2	8	no	—
105	C₂²×D₄	—	21	1	no	—
106	C₂³×C₄	—	21	2	no	—
107	C₄×D₄	—	21	8	no	—
108	C₂×C₄⋊C₄	—	21	1	no	—
109	C₂×C₂²⋊C₄	—	21	2	no	—
110	C₂×C₄²	D₂₁	1	1	yes	characteristic, complemented
111	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
112	C₂²×C₇	C₂×D₁₂	1	2	yes	—
113	C₂²×C₇	#235	1	2	yes	—
114	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
115	C₂×D₇	—	3	8	no	—
116	C₂×D₇	—	6	16	no	—
117	C₂×D₇	—	6	4	no	—
118	C₂×D₇	—	6	4	no	—
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	C₂×C₄×D₃	1	4	yes	complemented
121	C₄×C₇	C₂×D₁₂	1	4	yes	complemented
122	Dic₇	—	6	4	no	—
123	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
124	C₂²×D₃	—	7	8	no	—
125	C₂²×D₃	—	7	4	no	—
126	C₂²×D₃	—	14	4	no	—
127	C₂²×D₃	—	14	4	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
130	C₂×C₃×C₄	D₂₈	1	4	yes	complemented
131	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
132	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
133	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
134	C₂×Dic₃	—	7	4	no	—
135	C₂×Dic₃	—	14	4	no	—
136	D₁₂	—	7	16	no	—
137	C₄×D₃	—	14	16	no	—
138	C₃×C₇	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Hall
139	C₂⁴	—	21	2	no	—
140	C₂×D₄	—	21	8	no	—
141	C₂×D₄	—	21	4	no	—
142	C₂²×C₄	—	21	2	no	—
143	C₂²×C₄	—	21	8	no	—
144	C₂²×C₄	—	42	8	no	—
145	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
148	C₄⋊C₄	—	21	4	no	—
149	C₂²⋊C₄	—	21	8	no	—
150	C₄²	C₂×D₂₁	1	4	yes	complemented
151	C₂×C₇	C₄×D₁₂	1	4	yes	complemented
152	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
153	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
154	C₂×C₇	C₂×#235	1	1	yes	characteristic
155	D₇	—	6	8	no	—
156	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
157	C₂²×C₃	C₂×D₂₈	1	2	yes	—
158	C₂²×C₃	#1157	1	2	yes	—
159	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
160	C₂×D₃	—	7	8	no	—
161	C₂×D₃	—	14	16	no	—
162	C₂×D₃	—	14	4	no	—
163	C₂×D₃	—	14	4	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	C₂×C₄×D₇	1	4	yes	complemented
166	C₃×C₄	C₂×D₂₈	1	4	yes	complemented
167	Dic₃	—	14	4	no	—
168	C₂³	—	21	8	no	—
169	C₂³	—	21	4	no	—
170	C₂³	—	42	4	no	—
171	C₂³	—	42	4	no	—
172	C₂³	C₂²×D₂₁	1	1	yes	characteristic
173	D₄	—	21	16	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	21	4	no	—
176	C₂×C₄	—	42	4	no	—
177	C₂×C₄	—	42	16	no	—
178	C₂×C₄	C₄×D₂₁	1	4	yes	complemented
179	C₂×C₄	D₈₄	1	4	yes	direct product factor
180	C₂×C₄	C₂²×D₂₁	1	2	yes	—
181	C₂×C₄	C₂²×D₂₁	1	2	yes	—
182	C₂×C₄	C₂²×D₂₁	1	2	yes	—
183	C₇	C₂×C₄×D₁₂	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	C₄×D₂₈	1	4	yes	complemented
185	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
186	C₂×C₃	C₂²×D₂₈	1	1	yes	characteristic
187	C₂×C₃	C₂×#1157	1	1	yes	characteristic
188	D₃	—	14	8	no	—
189	C₂²	—	21	8	no	—
190	C₂²	—	42	16	no	—
191	C₂²	—	42	4	no	—
192	C₂²	—	42	4	no	—
193	C₂²	C₂×C₄×D₂₁	1	2	yes	—
194	C₂²	C₂×D₈₄	1	2	yes	—
195	C₂²	#66728	1	2	yes	—
196	C₂²	C₂³×D₂₁	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	42	4	no	—
199	C₄	C₂×C₄×D₂₁	1	4	yes	complemented
200	C₄	C₂×D₈₄	1	4	yes	direct product factor
201	C₃	C₂×C₄×D₂₈	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	42	8	no	—
203	C₂	C₄×D₈₄	1	4	yes	direct product factor
204	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
205	C₂	C₂²×D₈₄	1	1	yes	characteristic
206	C₂	C₂×#66728	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂³×D₂₁
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂²×D₈₄	C₂
cosocle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	C₂×C₄×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₁	C₂	semidirect
3. C₂×C₃×C₄²×C₇	C₂	semidirect
4. C₂×#66631	C₂	semidirect
5. C₂×C₄⋊Dic₂₁	C₂	semidirect
6. C₄×D₈₄	C₂	direct (N × H)
13. C₂×D₈₄	C₄	direct (N × H)
16. C₂×C₄×D₂₁	C₂²	semidirect
17. C₃×C₄²×C₇	C₂²	semidirect
18. #66631	C₂²	semidirect
19. C₄⋊Dic₂₁	C₂²	semidirect
22. C₂×C₄²×C₇	D₃	semidirect
30. C₂²×D₂₁	C₂×C₄	semidirect
33. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
34. C₂×C₃×C₄×C₇	D₄	semidirect
40. D₈₄	C₂×C₄	direct (N × H)
51. C₄²×C₇	C₂×D₃	semidirect
56. C₂×C₃×C₄²	D₇	semidirect
67. C₂×D₂₁	C₂²×C₄	semidirect
69. C₃×C₄×C₇	C₂²×C₄	semidirect
70. C₃×C₄×C₇	C₂×D₄	semidirect
79. C₂×C₄×C₇	C₄×D₃	semidirect
80. C₂×C₄×C₇	D₁₂	semidirect
97. C₃×C₄²	C₂×D₇	semidirect
100. C₂×C₃×C₇	C₄×D₄	semidirect
110. C₂×C₄²	D₂₁	semidirect
120. C₄×C₇	C₂×C₄×D₃	semidirect
121. C₄×C₇	C₂×D₁₂	semidirect
129. C₂×C₃×C₄	C₄×D₇	semidirect
130. C₂×C₃×C₄	D₂₈	semidirect
138. C₃×C₇	C₂×C₄×D₄	semidirect
150. C₄²	C₂×D₂₁	semidirect
151. C₂×C₇	C₄×D₁₂	semidirect
165. C₃×C₄	C₂×C₄×D₇	semidirect
166. C₃×C₄	C₂×D₂₈	semidirect
178. C₂×C₄	C₄×D₂₁	semidirect
179. C₂×C₄	D₈₄	direct (N × H)
183. C₇	C₂×C₄×D₁₂	semidirect
184. C₂×C₃	C₄×D₂₈	semidirect
199. C₄	C₂×C₄×D₂₁	semidirect
200. C₄	C₂×D₈₄	direct (N × H)
201. C₃	C₂×C₄×D₂₈	semidirect
203. C₂	C₄×D₈₄	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×D₈₄	2
C₂²×C₄×D₂₁	2
C₂×C₃×C₄²×C₇	2
C₂×#66631	2
C₂×C₄⋊Dic₂₁	2
C₄×D₈₄	2
C₂×C₄×D₂₈	3
C₂×C₄×D₁₂	7

Subgroups — C2×C4×D84 #13061563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₈₄ #13061563