Subgroups — C₃×C₄×Q₈×D₇ #13060913

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×C₄⋊C₄	C₂	1	3	yes	maximal
5	C₃×#6450	C₂	1	3	yes	maximal, complemented
6	C₃×C₄²×D₇	C₂	1	3	yes	maximal
7	C₃×C₄×Dic₁₄	C₂	1	3	yes	maximal, complemented
8	C₄×Q₈×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₃×Q₈×D₇	C₄	1	8	yes	direct product factor
11	C₂×C₃×C₄×D₇	C₂²	1	3	yes	—
12	C₂×C₃×C₄×D₇	C₂²	1	3	yes	—
13	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₄	C₂²	1	3	yes	—
15	C₃×C₇×C₄⋊C₄	C₂²	1	3	yes	—
16	C₃×C₄²×C₇	C₂²	1	3	yes	—
17	C₃×C₄⋊Dic₇	C₂²	1	3	yes	—
18	C₃×Dic₇⋊C₄	C₂²	1	6	yes	—
19	C₃×C₄×Dic₇	C₂²	1	3	yes	—
20	C₃×C₄×Dic₇	C₂²	1	3	yes	—
21	C₃×C₄×Dic₇	C₂²	1	3	yes	—
22	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic
23	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
24	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
25	D₇×C₄⋊C₄	C₂×C₃	1	3	yes	—
26	#6450	C₂×C₃	1	3	yes	complemented
27	C₄²×D₇	C₂×C₃	1	3	yes	—
28	C₄×Dic₁₄	C₂×C₃	1	3	yes	complemented
29	C₂×C₃×C₄×Q₈	—	7	1	no	maximal, Hall
30	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
31	C₃×C₇×Q₈	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
34	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₃×Dic₇	C₂³	1	3	yes	—
36	C₂×C₃×Dic₇	C₂³	1	3	yes	—
37	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
38	C₃×C₄×D₇	—	2	6	no	—
39	C₃×C₄×D₇	C₂×C₄	1	12	yes	—
40	C₃×C₄×D₇	Q₈	1	4	yes	direct product factor
41	C₃×Dic₁₄	C₂×C₄	1	12	yes	complemented
42	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
43	Q₈×D₇	C₃×C₄	1	8	yes	direct product factor
44	C₂×C₄×D₇	C₂²×C₃	1	3	yes	—
45	C₂×C₄×D₇	C₂²×C₃	1	3	yes	—
46	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
47	C₂×Dic₁₄	C₂²×C₃	1	3	yes	—
48	C₇×C₄⋊C₄	C₂²×C₃	1	3	yes	—
49	C₄²×C₇	C₂²×C₃	1	3	yes	—
50	C₄⋊Dic₇	C₂²×C₃	1	3	yes	—
51	Dic₇⋊C₄	C₂²×C₃	1	6	yes	—
52	C₄×Dic₇	C₂²×C₃	1	3	yes	—
53	C₄×Dic₇	C₂²×C₃	1	3	yes	—
54	C₄×Dic₇	C₂²×C₃	1	3	yes	—
55	C₂²×C₃×Q₈	—	7	1	no	—
56	C₃×C₄×Q₈	—	7	1	no	—
57	C₃×C₄×Q₈	—	7	3	no	—
58	C₃×C₄×Q₈	—	7	3	no	—
59	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, direct product factor
60	C₂×C₃×C₄⋊C₄	—	7	3	no	—
61	C₂×C₃×C₄²	—	7	3	no	—
62	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
64	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
65	C₂×C₃×D₇	#41	1	2	yes	—
66	C₃×C₄×C₇	—	2	3	no	—
67	C₃×C₄×C₇	C₂²×C₄	1	6	yes	—
68	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
69	C₃×Dic₇	—	2	3	no	—
70	C₃×Dic₇	C₂²×C₄	1	6	yes	—
71	C₃×Dic₇	C₂×Q₈	1	2	yes	complemented
72	C₂×C₄×Q₈	—	7	1	no	Sylow
73	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
74	C₇×Q₈	C₂×C₃×C₄	1	4	yes	complemented
75	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
76	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
77	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
78	C₂×Dic₇	C₂³×C₃	1	3	yes	—
79	C₂×Dic₇	C₂³×C₃	1	3	yes	—
80	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
81	C₄×D₇	—	2	6	no	—
82	C₄×D₇	C₂×C₃×C₄	1	12	yes	—
83	C₄×D₇	C₃×Q₈	1	4	yes	direct product factor
84	Dic₁₄	C₂×C₃×C₄	1	12	yes	complemented
85	C₂×C₃×Q₈	—	7	8	no	—
86	C₂×C₃×Q₈	—	7	3	no	—
87	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	7	3	no	—
89	C₂²×C₃×C₄	—	7	1	no	—
90	C₂²×C₃×C₄	—	7	3	no	—
91	C₃×C₄⋊C₄	—	7	6	no	—
92	C₃×C₄⋊C₄	—	7	3	no	—
93	C₃×C₄⋊C₄	C₂×D₇	1	3	yes	—
94	C₃×C₄²	—	7	3	no	—
95	C₃×C₄²	—	7	3	no	—
96	C₃×C₄²	—	7	3	no	—
97	C₃×C₄²	C₂×D₇	1	3	yes	—
98	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
101	C₃×D₇	C₄×Q₈	1	4	yes	direct product factor
102	C₂²×Q₈	—	7	1	no	—
103	C₄×Q₈	—	7	1	no	—
104	C₄×Q₈	—	7	3	no	—
105	C₄×Q₈	—	7	3	no	—
106	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, direct product factor
107	C₂×C₄⋊C₄	—	7	3	no	—
108	C₂×C₄²	—	7	3	no	—
109	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
110	C₂×D₇	C₂²×C₃×C₄	1	2	yes	—
111	C₂×D₇	C₂×C₃×Q₈	1	2	yes	—
112	C₂×D₇	C₃×#41	1	2	yes	—
113	C₄×C₇	—	2	3	no	—
114	C₄×C₇	C₂²×C₃×C₄	1	6	yes	—
115	C₄×C₇	C₂×C₃×Q₈	1	2	yes	complemented
116	Dic₇	—	2	3	no	—
117	Dic₇	C₂²×C₃×C₄	1	6	yes	—
118	Dic₇	C₂×C₃×Q₈	1	2	yes	complemented
119	C₂³×C₃	—	7	1	no	—
120	C₃×Q₈	—	7	12	no	—
121	C₃×Q₈	C₄×D₇	1	4	yes	direct product factor
122	C₂×C₃×C₄	—	7	1	no	—
123	C₂×C₃×C₄	—	7	3	no	—
124	C₂×C₃×C₄	—	7	12	no	—
125	C₂×C₃×C₄	—	7	4	no	—
126	C₂×C₃×C₄	—	7	3	no	—
127	C₂×C₃×C₄	—	14	6	no	—
128	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
129	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
130	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
131	C₃×C₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Hall
132	C₂×Q₈	—	7	8	no	—
133	C₂×Q₈	—	7	3	no	—
134	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
135	C₂²×C₄	—	7	1	no	—
136	C₂²×C₄	—	7	3	no	—
137	C₂²×C₄	—	7	3	no	—
138	C₄⋊C₄	—	7	6	no	—
139	C₄⋊C₄	—	7	3	no	—
140	C₄⋊C₄	C₂×C₃×D₇	1	3	yes	—
141	C₄²	—	7	3	no	—
142	C₄²	—	7	3	no	—
143	C₄²	—	7	3	no	—
144	C₄²	C₂×C₃×D₇	1	3	yes	—
145	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
146	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
148	D₇	C₃×C₄×Q₈	1	4	yes	direct product factor
149	C₂²×C₃	—	7	2	no	—
150	C₂²×C₃	—	7	2	no	—
151	C₂²×C₃	—	7	2	no	—
152	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
153	C₃×C₄	—	2	3	no	—
154	C₃×C₄	—	7	6	no	—
155	C₃×C₄	—	7	2	no	—
156	C₃×C₄	—	14	3	no	—
157	C₃×C₄	C₂×C₄×D₇	1	6	yes	—
158	C₃×C₄	Q₈×D₇	1	2	yes	direct product factor
159	C₂³	—	7	1	no	—
160	Q₈	—	7	12	no	—
161	Q₈	C₃×C₄×D₇	1	4	yes	direct product factor
162	C₂×C₄	—	7	1	no	—
163	C₂×C₄	—	7	3	no	—
164	C₂×C₄	—	7	12	no	—
165	C₂×C₄	—	7	4	no	—
166	C₂×C₄	—	7	3	no	—
167	C₂×C₄	—	14	6	no	—
168	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
169	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
170	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
171	C₇	C₂×C₃×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
172	C₂×C₃	—	7	4	no	—
173	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
174	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
175	C₂×C₃	D₇×#41	1	1	yes	characteristic
176	C₂²	—	7	2	no	—
177	C₂²	—	7	2	no	—
178	C₂²	—	7	2	no	—
179	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
180	C₄	—	2	3	no	—
181	C₄	—	7	6	no	—
182	C₄	—	7	2	no	—
183	C₄	—	14	3	no	—
184	C₄	C₂×C₃×C₄×D₇	1	6	yes	—
185	C₄	C₃×Q₈×D₇	1	2	yes	direct product factor
186	C₃	C₄×Q₈×D₇	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	—	7	4	no	—
188	C₂	C₂²×C₃×C₄×D₇	1	1	yes	characteristic
189	C₂	C₂×C₃×Q₈×D₇	1	1	yes	characteristic
190	C₂	C₃×D₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂²×C₃×D₇	C₂³
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₂×C₃×C₄×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×Q₈	C₂	semidirect
3. C₃×Q₈×Dic₇	C₂	semidirect
5. C₃×#6450	C₂	semidirect
7. C₃×C₄×Dic₁₄	C₂	semidirect
8. C₄×Q₈×D₇	C₃	direct (N × H)
10. C₃×Q₈×D₇	C₄	direct (N × H)
23. C₄×C₇×Q₈	C₂×C₃	semidirect
24. Q₈×Dic₇	C₂×C₃	semidirect
26. #6450	C₂×C₃	semidirect
28. C₄×Dic₁₄	C₂×C₃	semidirect
31. C₃×C₇×Q₈	C₂×C₄	semidirect
40. C₃×C₄×D₇	Q₈	direct (N × H)
41. C₃×Dic₁₄	C₂×C₄	semidirect
43. Q₈×D₇	C₃×C₄	direct (N × H)
59. C₃×C₄×Q₈	D₇	direct (N × H)
68. C₃×C₄×C₇	C₂×Q₈	semidirect
71. C₃×Dic₇	C₂×Q₈	semidirect
74. C₇×Q₈	C₂×C₃×C₄	semidirect
83. C₄×D₇	C₃×Q₈	direct (N × H)
84. Dic₁₄	C₂×C₃×C₄	semidirect
101. C₃×D₇	C₄×Q₈	direct (N × H)
106. C₄×Q₈	C₃×D₇	direct (N × H)
115. C₄×C₇	C₂×C₃×Q₈	semidirect
118. Dic₇	C₂×C₃×Q₈	semidirect
121. C₃×Q₈	C₄×D₇	direct (N × H)
131. C₃×C₇	C₂×C₄×Q₈	semidirect
148. D₇	C₃×C₄×Q₈	direct (N × H)
158. C₃×C₄	Q₈×D₇	direct (N × H)
161. Q₈	C₃×C₄×D₇	direct (N × H)
171. C₇	C₂×C₃×C₄×Q₈	semidirect
185. C₄	C₃×Q₈×D₇	direct (N × H)
186. C₃	C₄×Q₈×D₇	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×Q₈×D₇	2
C₃×C₄×C₇×Q₈	2
C₃×Q₈×Dic₇	2
C₃×D₇×C₄⋊C₄	2
C₃×#6450	2
C₃×C₄²×D₇	2
C₃×C₄×Dic₁₄	2
C₄×Q₈×D₇	3
C₂×C₃×C₄×Q₈	7

Subgroups — C3×C4×Q8×D7 #13060913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×Q₈×D₇ #13060913