Subgroups — C₇×#12024 #13189240

Order 1792.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#1568	C₂	1	2	yes	maximal
3	C₇×#1372	C₂	1	2	yes	maximal, complemented
4	C₇×#1371	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
7	C₇×#432	—	2	2	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₇×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₇×#335	C₂²	1	4	yes	—
14	C₇×#334	—	2	2	no	—
15	#12024	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
19	C₂×C₇×C₁₆	—	2	2	no	—
20	C₂×C₇×C₁₆	—	2	2	no	—
21	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₇×C₄⋊C₈	—	2	4	no	—
26	C₇×C₂²⋊C₈	—	4	2	no	—
27	C₇×#97	—	4	2	no	—
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic
33	#1568	C₂×C₇	1	2	yes	—
34	#1372	C₂×C₇	1	2	yes	complemented
35	#1371	C₂×C₇	1	2	yes	complemented
36	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇	Q₁₆	1	1	yes	characteristic
39	C₂×C₇×C₈	—	2	2	no	—
40	C₂×C₇×C₈	—	2	1	no	—
41	C₂×C₇×C₈	—	2	1	no	—
42	C₂×C₇×C₈	—	4	4	no	—
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇	D₈	1	1	yes	characteristic
49	C₄²×C₇	Q₁₆	1	1	yes	characteristic
50	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
51	C₇×C₁₆	—	2	1	no	—
52	C₇×C₁₆	—	2	1	no	—
53	C₇×C₁₆	M₁₆	1	2	yes	—
54	C₇×C₁₆	M₁₆	1	2	yes	—
55	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
56	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
57	#432	—	2	2	no	—
58	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
59	C₄×C₁₆	C₄×C₇	1	1	yes	characteristic
60	C₄×C₁₆	C₄×C₇	1	1	yes	characteristic
61	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
62	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
63	#335	C₂²×C₇	1	4	yes	—
64	#334	—	2	2	no	—
65	C₂³×C₇	#107	1	1	yes	characteristic
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	1	no	—
68	C₂×C₄×C₇	—	2	1	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	#107	1	1	yes	characteristic
71	C₂×C₄×C₇	#107	1	1	yes	characteristic
72	C₂×C₄×C₇	#107	1	1	yes	characteristic
73	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₇×C₈	—	2	1	no	—
77	C₇×C₈	—	2	1	no	—
78	C₇×C₈	—	8	4	no	—
79	C₇×C₈	#108	1	2	yes	—
80	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
81	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
83	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
84	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
88	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
89	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
90	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
91	C₄⋊C₈	—	2	4	no	—
92	C₂²⋊C₈	—	4	2	no	—
93	#97	—	4	2	no	—
94	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
95	C₄×C₈	C₇×D₄	1	1	yes	characteristic
96	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
98	C₂²×C₇	—	2	1	no	—
99	C₂²×C₇	—	2	1	no	—
100	C₂²×C₇	#368	1	2	yes	—
101	C₂²×C₇	C₂×#107	1	1	yes	characteristic
102	C₄×C₇	—	2	1	no	—
103	C₄×C₇	—	2	1	no	—
104	C₄×C₇	—	2	1	no	—
105	C₄×C₇	—	2	1	no	—
106	C₄×C₇	#423	1	1	yes	characteristic
107	C₄×C₇	C₂×#108	1	1	yes	characteristic
108	C₄×C₇	#508	1	2	yes	—
109	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
111	C₂²×C₄	C₇×Q₁₆	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
119	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
120	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
121	C₄²	C₇×D₈	1	1	yes	characteristic
122	C₄²	C₇×Q₁₆	1	1	yes	characteristic
123	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	C₇×M₁₆	1	2	yes	—
127	C₁₆	C₇×M₁₆	1	2	yes	—
128	C₂×C₇	—	2	2	no	—
129	C₂×C₇	#1567	1	1	yes	characteristic
130	C₂×C₇	#2158	1	1	yes	characteristic
131	C₂×C₇	C₂×#368	1	1	yes	characteristic
132	C₂³	C₇×#107	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	C₇×#107	1	1	yes	characteristic
138	C₂×C₄	C₇×#107	1	1	yes	characteristic
139	C₂×C₄	C₇×#107	1	1	yes	characteristic
140	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	4	no	—
146	C₈	C₇×#108	1	2	yes	—
147	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
148	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
149	C₇	#12024	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₇×#368	1	2	yes	—
153	C₂²	C₂×C₇×#107	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	C₇×#423	1	1	yes	characteristic
159	C₄	C₂×C₇×#108	1	1	yes	characteristic
160	C₄	C₇×#508	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₇×#1567	1	1	yes	characteristic
163	C₂	C₇×#2158	1	1	yes	characteristic
164	C₂	C₂×C₇×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#107
prime radical	C₂³×C₇	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1372	C₂	semidirect
4. C₇×#1371	C₂	semidirect
15. #12024	C₇	direct (N × H)
34. #1372	C₂×C₇	semidirect
35. #1371	C₂×C₇	semidirect
149. C₇	#12024	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×C₄×C₇×C₁₆	2
C₇×#1568	2
C₇×#1372	2
C₇×#1371	2
#12024	7

Subgroups — C7×#12024 #13189240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12024 #13189240