Subgroups — C₂³×#6381 #14217440

Order 1792.

3614 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6381	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×C₄×C₇	—	2	1	no	—
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂⁴×Dic₇	—	2	1	no	—
8	C₂²×C₇×C₄⋊C₄	—	2	7	no	—
9	C₂²×C₇×C₄⋊C₄	C₂²	1	14	yes	—
10	C₂²×C₄⋊Dic₇	—	2	7	no	—
11	C₂²×C₄⋊Dic₇	C₂²	1	14	yes	—
12	C₂³×C₇⋊C₈	C₄	1	16	yes	complemented
13	C₂³×C₇⋊C₈	C₂²	1	14	yes	—
14	C₂×#6381	C₂²	1	112	yes	direct product factor
15	C₂³×#107	—	7	1	no	maximal, Sylow
16	C₂⁵×C₇	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₇	—	2	14	no	—
18	C₂³×C₄×C₇	—	4	8	no	—
19	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₇	D₄	1	7	yes	—
21	C₂³×C₄×C₇	Q₈	1	8	yes	—
22	C₂³×C₄×C₇	C₂³	1	7	yes	—
23	C₂³×Dic₇	—	2	14	no	—
24	C₂³×Dic₇	—	4	8	no	—
25	C₂×C₇×C₄⋊C₄	—	2	42	no	—
26	C₂×C₇×C₄⋊C₄	C₂³	1	28	yes	—
27	C₂×C₄⋊Dic₇	—	2	42	no	—
28	C₂×C₄⋊Dic₇	C₂³	1	28	yes	—
29	C₂²×C₇⋊C₈	C₂×C₄	1	112	yes	complemented
30	C₂²×C₇⋊C₈	C₂³	1	28	yes	—
31	#6381	C₂³	1	64	yes	direct product factor
32	C₂⁴×C₈	—	7	1	no	—
33	C₂³×C₄⋊C₄	—	7	1	no	—
34	C₂³×C₄⋊C₄	D₇	1	1	yes	characteristic
35	C₂²×#107	—	7	28	no	—
36	C₂⁴×C₇	C₄⋊C₄	1	8	yes	—
37	C₂⁴×C₇	D₈	1	8	yes	—
38	C₂⁴×C₇	Q₁₆	1	8	yes	—
39	C₂⁴×C₇	C₂×D₄	1	7	yes	—
40	C₂²×C₄×C₇	—	2	28	no	—
41	C₂²×C₄×C₇	—	4	56	no	—
42	C₂²×C₄×C₇	C₄⋊C₄	1	56	yes	—
43	C₂²×C₄×C₇	C₂²×C₄	1	28	yes	—
44	C₂²×C₄×C₇	C₂×D₄	1	21	yes	—
45	C₂²×C₄×C₇	C₂×Q₈	1	28	yes	—
46	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
47	C₂²×Dic₇	—	2	28	no	—
48	C₂²×Dic₇	—	4	56	no	—
49	C₇×C₄⋊C₄	—	2	28	no	—
50	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
51	C₄⋊Dic₇	—	2	28	no	—
52	C₄⋊Dic₇	C₂⁴	1	8	yes	—
53	C₂×C₇⋊C₈	C₂²×C₄	1	112	yes	complemented
54	C₂×C₇⋊C₈	C₂⁴	1	8	yes	—
55	C₂⁴×C₄	—	2	1	no	—
56	C₂⁴×C₄	—	14	1	no	—
57	C₂⁴×C₄	C₂×D₇	1	1	yes	characteristic
58	C₂³×C₈	—	7	16	no	—
59	C₂³×C₈	—	7	14	no	—
60	C₂²×C₄⋊C₄	—	2	7	no	—
61	C₂²×C₄⋊C₄	—	7	14	no	—
62	C₂²×C₄⋊C₄	—	14	7	no	—
63	C₂²×C₄⋊C₄	C₂×D₇	1	14	yes	—
64	C₂×#107	—	7	112	no	—
65	C₂³×C₇	#107	1	64	yes	complemented
66	C₂³×C₇	C₂×C₄⋊C₄	1	28	yes	—
67	C₂³×C₇	C₂×D₈	1	28	yes	—
68	C₂³×C₇	C₂×Q₁₆	1	28	yes	—
69	C₂³×C₇	C₂²×D₄	1	7	yes	—
70	C₂×C₄×C₇	—	2	8	no	—
71	C₂×C₄×C₇	—	4	56	no	—
72	C₂×C₄×C₇	C₂×C₄⋊C₄	1	84	yes	—
73	C₂×C₄×C₇	C₂³×C₄	1	14	yes	—
74	C₂×C₄×C₇	C₂²×D₄	1	7	yes	—
75	C₂×C₄×C₇	C₂²×Q₈	1	14	yes	—
76	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
77	C₂×Dic₇	—	2	8	no	—
78	C₂×Dic₇	—	4	56	no	—
79	C₇⋊C₈	C₂³×C₄	1	16	yes	complemented
80	C₂⁵	C₂²⋊D₇	1	1	yes	characteristic
81	C₂³×C₄	—	2	14	no	—
82	C₂³×C₄	—	4	8	no	—
83	C₂³×C₄	—	14	14	no	—
84	C₂³×C₄	—	28	8	no	—
85	C₂³×C₄	Dic₁₄	1	8	yes	—
86	C₂³×C₄	C₄×D₇	1	8	yes	—
87	C₂³×C₄	C₂²⋊D₇	1	7	yes	—
88	C₂³×C₄	C₂²×D₇	1	7	yes	—
89	C₂²×C₈	—	7	112	no	—
90	C₂²×C₈	—	7	28	no	—
91	C₂×C₄⋊C₄	—	2	42	no	—
92	C₂×C₄⋊C₄	—	7	28	no	—
93	C₂×C₄⋊C₄	—	14	42	no	—
94	C₂×C₄⋊C₄	C₂²×D₇	1	28	yes	—
95	#107	—	7	64	no	—
96	C₂²×C₇	C₂×#107	1	112	yes	complemented
97	C₂²×C₇	C₂²×C₄⋊C₄	1	14	yes	—
98	C₂²×C₇	C₂²×D₈	1	14	yes	—
99	C₂²×C₇	C₂²×Q₁₆	1	14	yes	—
100	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₇	—	4	8	no	—
102	C₄×C₇	C₂²×C₄⋊C₄	1	14	yes	—
103	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
104	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
105	Dic₇	—	4	8	no	—
106	C₂⁴	Dic₇⋊C₄	1	8	yes	—
107	C₂⁴	#1141	1	8	yes	—
108	C₂⁴	#1144	1	8	yes	—
109	C₂⁴	C₂×C₂²⋊D₇	1	7	yes	—
110	C₂²×C₄	—	2	28	no	—
111	C₂²×C₄	—	4	56	no	—
112	C₂²×C₄	—	14	28	no	—
113	C₂²×C₄	—	28	56	no	—
114	C₂²×C₄	Dic₇⋊C₄	1	56	yes	—
115	C₂²×C₄	C₂×Dic₁₄	1	28	yes	—
116	C₂²×C₄	C₂×C₄×D₇	1	28	yes	—
117	C₂²×C₄	C₂×C₂²⋊D₇	1	21	yes	—
118	C₂²×C₄	C₂³×D₇	1	7	yes	—
119	C₂×C₈	—	7	112	no	—
120	C₂×C₈	—	7	8	no	—
121	C₄⋊C₄	—	2	28	no	—
122	C₄⋊C₄	—	7	8	no	—
123	C₄⋊C₄	—	14	28	no	—
124	C₄⋊C₄	C₂³×D₇	1	8	yes	—
125	C₂×C₇	C₂²×#107	1	28	yes	complemented
126	C₂×C₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₇	C₂³×D₈	1	1	yes	characteristic
128	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
129	C₂³	#6381	1	64	yes	direct product factor
130	C₂³	C₂×Dic₇⋊C₄	1	28	yes	—
131	C₂³	C₂×#1141	1	28	yes	—
132	C₂³	C₂×#1144	1	28	yes	—
133	C₂³	C₂²×C₂²⋊D₇	1	7	yes	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	4	56	no	—
136	C₂×C₄	—	14	8	no	—
137	C₂×C₄	—	28	56	no	—
138	C₂×C₄	C₂×Dic₇⋊C₄	1	84	yes	—
139	C₂×C₄	C₂²×Dic₁₄	1	14	yes	—
140	C₂×C₄	C₂²×C₄×D₇	1	14	yes	—
141	C₂×C₄	C₂²×C₂²⋊D₇	1	7	yes	—
142	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
143	C₈	—	7	16	no	—
144	C₇	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
145	C₂²	C₂×#6381	1	112	yes	direct product factor
146	C₂²	C₂²×Dic₇⋊C₄	1	14	yes	—
147	C₂²	C₂²×#1141	1	14	yes	—
148	C₂²	C₂²×#1144	1	14	yes	—
149	C₂²	C₂³×C₂²⋊D₇	1	1	yes	characteristic
150	C₄	—	4	8	no	—
151	C₄	—	28	8	no	—
152	C₄	C₂²×Dic₇⋊C₄	1	14	yes	—
153	C₄	C₂³×Dic₁₄	1	1	yes	characteristic
154	C₄	C₂³×C₄×D₇	1	1	yes	characteristic
155	C₂	C₂²×#6381	1	28	yes	direct product factor
156	C₂	C₂³×Dic₇⋊C₄	1	1	yes	characteristic
157	C₂	C₂³×#1141	1	1	yes	characteristic
158	C₂	C₂³×#1144	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₇
commutator	C₄×C₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇
Fitting	C₂³×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₇	D₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₇
hypocenter	C₇	C₂³×#107
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂²×#6381	C₂	direct (N × H)
12. C₂³×C₇⋊C₈	C₄	semidirect
14. C₂×#6381	C₂²	direct (N × H)
29. C₂²×C₇⋊C₈	C₂×C₄	semidirect
31. #6381	C₂³	direct (N × H)
53. C₂×C₇⋊C₈	C₂²×C₄	semidirect
65. C₂³×C₇	#107	semidirect
79. C₇⋊C₈	C₂³×C₄	semidirect
96. C₂²×C₇	C₂×#107	semidirect
125. C₂×C₇	C₂²×#107	semidirect
129. C₂³	#6381	direct (N × H)
144. C₇	C₂³×#107	semidirect
145. C₂²	C₂×#6381	direct (N × H)
155. C₂	C₂²×#6381	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₇×C₄⋊C₄	2
C₂³×C₄⋊Dic₇	2
C₂⁴×C₇⋊C₈	2
C₂²×#6381	2
C₂³×#107	7

Subgroups — C23×#6381 #14217440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6381 #14217440