Subgroups — C₂×C₄⋊C₄×D₂₁ #13061592

Order 1344.

836 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×C₄×D₂₁	C₂	1	2	yes	maximal
2	C₂²×C₄×D₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#66629	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×D₂₁	C₂	1	8	yes	maximal, direct product factor
7	C₂×D₇×C₄⋊C₄	—	3	1	no	maximal, Hall
8	C₂³×D₂₁	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₁	C₂²	1	2	yes	—
12	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₁	—	2	8	no	—
14	C₂×C₄×D₂₁	C₄	1	8	yes	complemented
15	C₂×C₄×D₂₁	C₂²	1	8	yes	—
16	C₂×C₄×D₂₁	C₂²	1	4	yes	—
17	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
18	C₄⋊Dic₂₁	C₂²	1	4	yes	complemented
19	#66629	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₇	—	3	1	no	—
21	C₂²×C₄×D₇	—	3	2	no	—
22	C₂×C₇×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
23	C₂×C₄⋊Dic₇	—	3	1	no	—
24	C₂×Dic₇⋊C₄	—	3	2	no	—
25	D₇×C₄⋊C₄	—	3	8	no	—
26	C₂×D₃×C₄⋊C₄	—	7	1	no	maximal, Hall
27	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
28	C₂²×D₂₁	C₂×C₄	1	4	yes	—
29	C₂²×D₂₁	D₄	1	4	yes	—
30	C₂²×D₂₁	Q₈	1	4	yes	—
31	C₂²×D₂₁	C₂³	1	2	yes	—
32	C₂×C₃×C₄×C₇	—	2	4	no	—
33	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
36	C₂×Dic₂₁	—	2	4	no	—
37	C₂×Dic₂₁	C₂×C₄	1	4	yes	complemented
38	C₂×Dic₂₁	C₂³	1	4	yes	—
39	C₂×Dic₂₁	C₂³	1	2	yes	—
40	C₄×D₂₁	—	2	16	no	—
41	C₄×D₂₁	C₂×C₄	1	16	yes	complemented
42	C₂³×D₇	—	3	1	no	—
43	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
44	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
45	C₂²×Dic₇	—	3	1	no	—
46	C₂²×Dic₇	—	3	2	no	—
47	C₂×C₄×D₇	—	3	8	no	—
48	C₂×C₄×D₇	—	3	4	no	—
49	C₂×C₄×D₇	—	3	8	no	—
50	C₂×C₄×D₇	—	6	8	no	—
51	C₇×C₄⋊C₄	C₂×D₃	1	4	yes	complemented
52	C₄⋊Dic₇	—	3	4	no	—
53	Dic₇⋊C₄	—	3	8	no	—
54	C₂²×C₄×D₃	—	7	1	no	—
55	C₂²×C₄×D₃	—	7	2	no	—
56	C₂×C₃×C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
57	C₂×C₄⋊Dic₃	—	7	1	no	—
58	C₂×Dic₃⋊C₄	—	7	2	no	—
59	D₃×C₄⋊C₄	—	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	C₂×Q₈	1	2	yes	—
63	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×D₂₁	C₄⋊C₄	1	16	yes	direct product factor
65	C₂×D₂₁	C₂²×C₄	1	4	yes	—
66	C₂×D₂₁	C₂×D₄	1	4	yes	—
67	C₂×D₂₁	C₂×Q₈	1	4	yes	—
68	C₃×C₄×C₇	—	2	4	no	—
69	C₃×C₄×C₇	C₂²×C₄	1	4	yes	complemented
70	Dic₂₁	—	2	4	no	—
71	Dic₂₁	C₂²×C₄	1	4	yes	complemented
72	C₂²×C₄⋊C₄	—	21	1	no	Sylow
73	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
74	C₂²×D₇	—	3	4	no	—
75	C₂²×D₇	—	3	4	no	—
76	C₂²×D₇	—	3	4	no	—
77	C₂²×D₇	—	3	2	no	—
78	C₂×C₄×C₇	—	2	4	no	—
79	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
80	C₂×C₄×C₇	C₂²×D₃	1	4	yes	—
81	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
82	C₂×Dic₇	—	3	4	no	—
83	C₂×Dic₇	—	3	2	no	—
84	C₂×Dic₇	—	3	4	no	—
85	C₂×Dic₇	—	6	4	no	—
86	C₄×D₇	—	3	16	no	—
87	C₄×D₇	—	6	16	no	—
88	C₂³×D₃	—	7	1	no	—
89	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
90	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
91	C₂²×Dic₃	—	7	1	no	—
92	C₂²×Dic₃	—	7	2	no	—
93	C₂×C₄×D₃	—	7	8	no	—
94	C₂×C₄×D₃	—	7	4	no	—
95	C₂×C₄×D₃	—	7	8	no	—
96	C₂×C₄×D₃	—	14	8	no	—
97	C₃×C₄⋊C₄	C₂×D₇	1	4	yes	complemented
98	C₄⋊Dic₃	—	7	4	no	—
99	Dic₃⋊C₄	—	7	8	no	—
100	C₂×C₃×C₇	C₂×C₄⋊C₄	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	C₂²×Q₈	1	1	yes	characteristic
104	D₂₁	C₂×C₄⋊C₄	1	8	yes	direct product factor
105	C₂³×C₄	—	21	1	no	—
106	C₂³×C₄	—	21	2	no	—
107	C₂×C₄⋊C₄	—	21	2	no	—
108	C₂×C₄⋊C₄	—	21	1	no	—
109	C₂×C₄⋊C₄	—	21	8	no	—
110	C₂×C₄⋊C₄	D₂₁	1	1	yes	characteristic, direct product factor
111	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
112	C₂²×C₇	D₃×D₄	1	2	yes	—
113	C₂²×C₇	D₃×Q₈	1	2	yes	—
114	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
115	C₂×D₇	—	3	16	no	—
116	C₂×D₇	—	3	4	no	—
117	C₂×D₇	—	3	4	no	—
118	C₂×D₇	—	3	4	no	—
119	C₄×C₇	—	2	4	no	—
120	C₄×C₇	C₂×C₄×D₃	1	4	yes	complemented
121	Dic₇	—	3	4	no	—
122	Dic₇	—	6	4	no	—
123	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
124	C₂²×D₃	—	7	4	no	—
125	C₂²×D₃	—	7	4	no	—
126	C₂²×D₃	—	7	4	no	—
127	C₂²×D₃	—	7	2	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
130	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
131	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
132	C₂×Dic₃	—	7	4	no	—
133	C₂×Dic₃	—	7	2	no	—
134	C₂×Dic₃	—	7	4	no	—
135	C₂×Dic₃	—	14	4	no	—
136	C₄×D₃	—	7	16	no	—
137	C₄×D₃	—	14	16	no	—
138	C₃×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
139	C₂⁴	—	21	1	no	—
140	C₂²×C₄	—	21	1	no	—
141	C₂²×C₄	—	21	8	no	—
142	C₂²×C₄	—	21	4	no	—
143	C₂²×C₄	—	21	2	no	—
144	C₂²×C₄	—	21	8	no	—
145	C₂²×C₄	—	42	8	no	—
146	C₂²×C₄	C₂×D₂₁	1	2	yes	—
147	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
148	C₄⋊C₄	—	21	8	no	—
149	C₄⋊C₄	—	21	4	no	—
150	C₄⋊C₄	C₂×D₂₁	1	4	yes	direct product factor
151	C₂×C₇	D₃×C₄⋊C₄	1	4	yes	complemented
152	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
153	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
154	C₂×C₇	C₂×D₃×Q₈	1	1	yes	characteristic
155	D₇	—	3	8	no	—
156	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
157	C₂²×C₃	D₄×D₇	1	2	yes	—
158	C₂²×C₃	Q₈×D₇	1	2	yes	—
159	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
160	C₂×D₃	—	7	16	no	—
161	C₂×D₃	—	7	4	no	—
162	C₂×D₃	—	7	4	no	—
163	C₂×D₃	—	7	4	no	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	C₂×C₄×D₇	1	4	yes	complemented
166	Dic₃	—	7	4	no	—
167	Dic₃	—	14	4	no	—
168	C₂³	—	21	4	no	—
169	C₂³	—	21	4	no	—
170	C₂³	—	21	4	no	—
171	C₂³	—	21	2	no	—
172	C₂³	C₂²×D₂₁	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	21	4	no	—
175	C₂×C₄	—	21	2	no	—
176	C₂×C₄	—	21	16	no	—
177	C₂×C₄	—	21	4	no	—
178	C₂×C₄	—	42	16	no	—
179	C₂×C₄	—	42	4	no	—
180	C₂×C₄	C₄×D₂₁	1	4	yes	complemented
181	C₂×C₄	C₂²×D₂₁	1	4	yes	—
182	C₂×C₄	C₂²×D₂₁	1	2	yes	—
183	C₇	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₃	D₇×C₄⋊C₄	1	4	yes	complemented
185	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
186	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
187	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
188	D₃	—	7	8	no	—
189	C₂²	—	21	16	no	—
190	C₂²	—	21	4	no	—
191	C₂²	—	21	4	no	—
192	C₂²	—	21	4	no	—
193	C₂²	C₂×C₄×D₂₁	1	2	yes	—
194	C₂²	D₄×D₂₁	1	2	yes	—
195	C₂²	Q₈×D₂₁	1	2	yes	—
196	C₂²	C₂³×D₂₁	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	21	4	no	—
199	C₄	—	42	4	no	—
200	C₄	C₂×C₄×D₂₁	1	4	yes	complemented
201	C₃	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	21	8	no	—
203	C₂	C₄⋊C₄×D₂₁	1	4	yes	direct product factor
204	C₂	C₂²×C₄×D₂₁	1	1	yes	characteristic
205	C₂	C₂×D₄×D₂₁	1	1	yes	characteristic
206	C₂	C₂×Q₈×D₂₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₇×C₄⋊C₄	C₂	semidirect
4. C₂×C₄⋊Dic₂₁	C₂	semidirect
5. C₂×#66629	C₂	semidirect
6. C₄⋊C₄×D₂₁	C₂	direct (N × H)
14. C₂×C₄×D₂₁	C₄	semidirect
17. C₃×C₇×C₄⋊C₄	C₂²	semidirect
18. C₄⋊Dic₂₁	C₂²	semidirect
19. #66629	C₂²	semidirect
22. C₂×C₇×C₄⋊C₄	D₃	semidirect
33. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
37. C₂×Dic₂₁	C₂×C₄	semidirect
41. C₄×D₂₁	C₂×C₄	semidirect
51. C₇×C₄⋊C₄	C₂×D₃	semidirect
56. C₂×C₃×C₄⋊C₄	D₇	semidirect
64. C₂×D₂₁	C₄⋊C₄	direct (N × H)
69. C₃×C₄×C₇	C₂²×C₄	semidirect
71. Dic₂₁	C₂²×C₄	semidirect
79. C₂×C₄×C₇	C₄×D₃	semidirect
97. C₃×C₄⋊C₄	C₂×D₇	semidirect
100. C₂×C₃×C₇	C₂×C₄⋊C₄	semidirect
104. D₂₁	C₂×C₄⋊C₄	direct (N × H)
110. C₂×C₄⋊C₄	D₂₁	direct (N × H)
120. C₄×C₇	C₂×C₄×D₃	semidirect
129. C₂×C₃×C₄	C₄×D₇	semidirect
138. C₃×C₇	C₂²×C₄⋊C₄	semidirect
150. C₄⋊C₄	C₂×D₂₁	direct (N × H)
151. C₂×C₇	D₃×C₄⋊C₄	semidirect
165. C₃×C₄	C₂×C₄×D₇	semidirect
180. C₂×C₄	C₄×D₂₁	semidirect
183. C₇	C₂×D₃×C₄⋊C₄	semidirect
184. C₂×C₃	D₇×C₄⋊C₄	semidirect
200. C₄	C₂×C₄×D₂₁	semidirect
201. C₃	C₂×D₇×C₄⋊C₄	semidirect
203. C₂	C₄⋊C₄×D₂₁	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×C₄×D₂₁	2
C₂×C₃×C₇×C₄⋊C₄	2
C₂×C₄⋊Dic₂₁	2
C₂×#66629	2
C₄⋊C₄×D₂₁	2
C₂×D₇×C₄⋊C₄	3
C₂×D₃×C₄⋊C₄	7

Subgroups — C2×C4⋊C4×D21 #13061592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×D₂₁ #13061592