Subgroups — C₃×#89421 #13055083

Order 1344.

388 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₇⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×Dic₇	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#95	C₂	1	1	yes	maximal, characteristic
5	C₃×C₇⋊#95	C₂	1	1	yes	maximal, characteristic
6	#89421	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₃×Dic₇⋊C₄	C₄	1	8	yes	complemented
13	C₃×C₄×Dic₇	—	2	4	no	—
14	C₃×C₄×Dic₇	C₄	1	4	yes	complemented
15	C₂×Dic₇⋊C₄	C₂×C₃	1	2	yes	—
16	C₂×C₄×Dic₇	C₂×C₃	1	2	yes	—
17	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
18	C₇×#95	C₂×C₃	1	1	yes	characteristic
19	C₇⋊#95	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄×C₄⋊C₄	—	7	1	no	maximal, Hall
21	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	—	2	2	no	—
23	C₂×C₃×C₄×C₇	—	2	2	no	—
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
27	C₂×C₃×Dic₇	—	2	2	no	—
28	C₂×C₃×Dic₇	—	2	2	no	—
29	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
30	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
31	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
32	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×Dic₇	D₄	1	2	yes	—
34	C₂×C₃×Dic₇	Q₈	1	2	yes	—
35	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
37	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
38	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
39	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
40	Dic₇⋊C₄	C₃×C₄	1	8	yes	complemented
41	C₄×Dic₇	—	2	4	no	—
42	C₄×Dic₇	C₃×C₄	1	4	yes	complemented
43	C₂×C₃×C₄⋊C₄	—	7	2	no	—
44	C₂×C₃×C₄²	—	7	1	no	—
45	C₂×C₃×C₄²	—	7	2	no	—
46	C₃×#95	—	7	1	no	—
47	C₃×#95	D₇	1	1	yes	characteristic
48	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₃×C₇	#41	1	2	yes	—
53	C₃×C₄×C₇	—	2	2	no	—
54	C₃×C₄×C₇	—	2	4	no	—
55	C₃×Dic₇	—	2	4	no	—
56	C₃×Dic₇	C₄²	1	4	yes	complemented
57	C₃×Dic₇	C₄⋊C₄	1	4	yes	complemented
58	C₄×C₄⋊C₄	—	7	1	no	Sylow
59	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	—	2	2	no	—
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
64	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	—
65	C₂×Dic₇	—	2	2	no	—
66	C₂×Dic₇	—	2	2	no	—
67	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
68	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
69	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
70	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
71	C₂×Dic₇	C₃×D₄	1	2	yes	—
72	C₂×Dic₇	C₃×Q₈	1	2	yes	—
73	C₂²×C₃×C₄	—	7	1	no	—
74	C₂²×C₃×C₄	—	7	1	no	—
75	C₂²×C₃×C₄	—	7	2	no	—
76	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
78	C₃×C₄⋊C₄	—	7	8	no	—
79	C₃×C₄²	—	7	4	no	—
80	C₃×C₄²	—	14	4	no	—
81	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₇	#117	1	1	yes	characteristic
84	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
85	C₂×C₃×C₇	C₄×Q₈	1	2	yes	—
86	C₂×C₄⋊C₄	—	7	2	no	—
87	C₂×C₄²	—	7	1	no	—
88	C₂×C₄²	—	7	2	no	—
89	#95	—	7	1	no	—
90	#95	C₃×D₇	1	1	yes	characteristic
91	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
93	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
94	C₂²×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
95	C₂²×C₇	C₃×#41	1	2	yes	—
96	C₄×C₇	—	2	2	no	—
97	C₄×C₇	—	2	4	no	—
98	Dic₇	—	2	4	no	—
99	Dic₇	C₃×C₄²	1	4	yes	complemented
100	Dic₇	C₃×C₄⋊C₄	1	4	yes	complemented
101	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	7	2	no	—
106	C₂×C₃×C₄	—	7	2	no	—
107	C₂×C₃×C₄	—	7	4	no	—
108	C₂×C₃×C₄	—	7	2	no	—
109	C₂×C₃×C₄	—	7	2	no	—
110	C₂×C₃×C₄	—	7	4	no	—
111	C₂×C₃×C₄	—	14	2	no	—
112	C₂×C₃×C₄	—	14	2	no	—
113	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
114	C₂×C₃×C₄	C₄×D₇	1	4	yes	—
115	C₃×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
116	C₂²×C₄	—	7	1	no	—
117	C₂²×C₄	—	7	1	no	—
118	C₂²×C₄	—	7	2	no	—
119	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
121	C₄⋊C₄	—	7	8	no	—
122	C₄²	—	7	4	no	—
123	C₄²	—	14	4	no	—
124	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
125	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₇	C₃×#117	1	1	yes	characteristic
127	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
128	C₂×C₇	C₃×C₄×Q₈	1	2	yes	—
129	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
130	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
131	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
132	C₂²×C₃	#1159	1	2	yes	—
133	C₂²×C₃	Q₈×D₇	1	1	yes	characteristic
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	—	7	4	no	—
137	C₃×C₄	—	7	4	no	—
138	C₃×C₄	—	14	4	no	—
139	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	7	2	no	—
144	C₂×C₄	—	7	2	no	—
145	C₂×C₄	—	7	4	no	—
146	C₂×C₄	—	7	2	no	—
147	C₂×C₄	—	7	2	no	—
148	C₂×C₄	—	7	4	no	—
149	C₂×C₄	—	14	2	no	—
150	C₂×C₄	—	14	2	no	—
151	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
152	C₂×C₄	C₃×C₄×D₇	1	4	yes	—
153	C₇	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₃	C₄²×D₇	1	1	yes	characteristic
155	C₂×C₃	#6440	1	1	yes	characteristic
156	C₂×C₃	(C₂²⋊D₇)⋊C₄	1	2	yes	—
157	C₂×C₃	#6450	1	2	yes	—
158	C₂×C₃	D₇×C₄⋊C₄	1	1	yes	characteristic
159	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
160	C₂²	C₂×C₃×C₄×D₇	1	2	yes	—
161	C₂²	C₃×D₄×D₇	1	1	yes	characteristic
162	C₂²	C₃×#1159	1	2	yes	—
163	C₂²	C₃×Q₈×D₇	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	4	no	—
166	C₄	—	7	4	no	—
167	C₄	—	7	4	no	—
168	C₄	—	14	4	no	—
169	C₃	#89421	1	1	yes	characteristic, direct product factor, Sylow
170	C₂	C₃×C₄²×D₇	1	1	yes	characteristic
171	C₂	C₃×#6440	1	1	yes	characteristic
172	C₂	C₃×(C₂²⋊D₇)⋊C₄	1	2	yes	—
173	C₂	C₃×#6450	1	2	yes	—
174	C₂	C₃×D₇×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#95	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₃×#95	D₇
hypocenter	C₇	C₃×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

Complements

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

N	H ≅ G/N	type
6. #89421	C₃	direct (N × H)
12. C₃×Dic₇⋊C₄	C₄	semidirect
14. C₃×C₄×Dic₇	C₄	semidirect
40. Dic₇⋊C₄	C₃×C₄	semidirect
42. C₄×Dic₇	C₃×C₄	semidirect
56. C₃×Dic₇	C₄²	semidirect
57. C₃×Dic₇	C₄⋊C₄	semidirect
99. Dic₇	C₃×C₄²	semidirect
100. Dic₇	C₃×C₄⋊C₄	semidirect
115. C₃×C₇	C₄×C₄⋊C₄	semidirect
153. C₇	C₃×C₄×C₄⋊C₄	semidirect
169. C₃	#89421	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×Dic₇⋊C₄	2
C₂×C₃×C₄×Dic₇	2
C₃×C₇×#95	2
C₃×C₇⋊#95	2
#89421	3
C₃×C₄×C₄⋊C₄	7

Subgroups — C3×#89421 #13055083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89421 #13055083