Finite Group Databasebeta

Subgroups — #14267699

Order 1792.

59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.

normal subgroups only

This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×2₋¹⁺⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	#11744566	C₂	1	36	yes	maximal, complemented
3	D₇×#585	C₂	1	36	yes	maximal, complemented
4	#11744563	C₂	1	27	yes	maximal, complemented
5	C₂×#90620	C₂	1	27	yes	maximal, complemented
6	C₇×#585	C₂²	1	36	yes	complemented
7	C₂×C₇×2₋¹⁺⁴	C₂²	1	27	yes	complemented
8	#90620	C₂²	1	216	yes	complemented
9	#90620	C₂²	1	432	yes	complemented
10	D₇×2₋¹⁺⁴	C₂²	1	216	yes	complemented
11	#90618	C₂²	1	120	yes	complemented
12	D₇×H(5,𝔽₂)	C₂²	1	360	yes	complemented
13	#90616	C₂²	1	540	yes	complemented
14	C₂×#6553	C₂²	1	270	yes	complemented
15	C₂×D₇×#41	C₂²	1	270	yes	complemented
16	C₂×#6551	C₂²	1	135	yes	complemented
17	C₂²×#1161	C₂²	1	45	yes	complemented
18	#63103	—	7	1	no	maximal, Sylow
19	C₇×2₋¹⁺⁴	C₂³	1	216	yes	complemented
20	C₇×H(5,𝔽₂)	C₂³	1	120	yes	complemented
21	C₂×C₇×#41	C₂³	1	270	yes	complemented
22	C₂²×C₇×Q₈	C₂³	1	45	yes	complemented
23	#6554	C₂³	1	720	yes	complemented
24	#6553	C₂³	1	2160	yes	complemented
25	D₇×#41	C₂³	1	720	yes	complemented
26	D₇×#41	C₂³	1	2160	yes	complemented
27	#6551	C₂³	1	1080	yes	complemented
28	C₂×#1161	C₂³	1	270	yes	complemented
29	C₂×#1161	C₂³	1	540	yes	complemented
30	C₂×Q₈×D₇	C₂³	1	270	yes	complemented
31	#6548	C₂³	1	1080	yes	complemented
32	C₂×#1159	C₂³	1	270	yes	complemented
33	C₂×D₄×D₇	C₂³	1	810	yes	complemented
34	C₂×#1157	C₂³	1	810	yes	complemented
35	C₂²×D₂₈	C₂³	1	135	yes	complemented
36	C₂²×C₄×D₇	C₂³	1	135	yes	complemented
37	2₋¹⁺⁶	—	7	27	no	—
38	2₋¹⁺⁶	D₇	1	1	yes	characteristic, complemented
39	H(7,𝔽₂)	—	7	36	no	—
40	C₂×#585	—	7	36	no	—
41	C₂×#585	—	7	27	no	—
42	C₂³×D₇	C₂⁴	1	135	yes	—
43	C₇×#41	C₂⁴	1	720	yes	—
44	C₂×C₇×Q₈	C₂⁴	1	270	yes	—
45	C₂×C₇×D₄	C₂⁴	1	270	yes	—
46	C₂²×C₄×C₇	C₂⁴	1	135	yes	—
47	C₂×C₂²⋊D₇	C₂⁴	1	540	yes	—
48	C₂²×Dic₇	C₂⁴	1	45	yes	—
49	#1161	C₂⁴	1	120	yes	—
50	#1161	C₂⁴	1	1080	yes	—
51	Q₈×D₇	C₂⁴	1	720	yes	—
52	#1159	C₂⁴	1	216	yes	—
53	#1159	C₂⁴	1	1080	yes	—
54	D₄×D₇	C₂⁴	1	2160	yes	—
55	#1157	C₂⁴	1	2160	yes	—
56	C₂×D₂₈	C₂⁴	1	810	yes	—
57	C₂×C₄×D₇	C₂⁴	1	270	yes	—
58	C₂×C₄×D₇	C₂⁴	1	810	yes	—
59	C₂×Dic₁₄	C₂⁴	1	270	yes	—
60	#585	—	7	216	no	—
61	#585	—	7	432	no	—
62	#585	—	7	120	no	—
63	#585	—	7	540	no	—
64	#585	C₂×D₇	1	36	yes	complemented
65	C₂×2₋¹⁺⁴	—	7	216	no	—
66	C₂×2₋¹⁺⁴	—	7	135	no	—
67	C₂×2₋¹⁺⁴	C₂×D₇	1	27	yes	complemented
68	C₂×H(5,𝔽₂)	—	7	360	no	—
69	C₂×H(5,𝔽₂)	—	7	270	no	—
70	C₂²×#41	—	7	270	no	—
71	C₂²×#41	—	7	45	no	—
72	C₂³×C₇	C₂⁵	1	45	yes	—
73	C₂²×D₇	—	8	135	no	—
74	C₂²×D₇	C₂⁵	1	270	yes	—
75	C₇×Q₈	C₂⁵	1	120	yes	—
76	C₇×D₄	C₂⁵	1	216	yes	—
77	C₂×C₄×C₇	C₂⁵	1	270	yes	—
78	C₂²⋊D₇	C₂⁵	1	432	yes	—
79	C₂×Dic₇	C₂⁵	1	135	yes	—
80	C₂×Dic₇	C₂⁵	1	27	yes	—
81	D₂₈	C₂⁵	1	360	yes	—
82	C₄×D₇	C₂⁵	1	36	yes	—
83	C₄×D₇	C₂⁵	1	540	yes	—
84	Dic₁₄	C₂⁵	1	216	yes	—
85	2₋¹⁺⁴	—	7	720	no	—
86	2₋¹⁺⁴	—	7	1080	no	—
87	2₋¹⁺⁴	C₂²×D₇	1	216	yes	complemented
88	H(5,𝔽₂)	—	7	2160	no	—
89	H(5,𝔽₂)	—	7	1080	no	—
90	H(5,𝔽₂)	C₂²×D₇	1	120	yes	complemented
91	C₂×#41	—	7	720	no	—
92	C₂×#41	—	7	2160	no	—
93	C₂×#41	—	7	270	no	—
94	C₂×#41	—	7	540	no	—
95	C₂×#41	—	7	270	no	—
96	C₂×#41	—	7	810	no	—
97	C₂×#41	C₂²×D₇	1	270	yes	complemented
98	C₂²×Q₈	—	7	270	no	—
99	C₂²×Q₈	C₂²×D₇	1	45	yes	complemented
100	C₂²×D₄	—	7	810	no	—
101	C₂²×D₄	—	7	135	no	—
102	C₂³×C₄	—	7	135	no	—
103	C₂²×C₇	—	4	45	no	—
104	C₂²×C₇	C₂⁶	1	27	yes	—
105	C₂×D₇	—	4	270	no	—
106	C₂×D₇	C₂⁶	1	36	yes	—
107	C₄×C₇	C₂⁶	1	36	yes	—
108	Dic₇	C₂⁶	1	27	yes	—
109	Dic₇	C₂⁶	1	1	yes	characteristic
110	C₂⁴	—	7	135	no	—
111	#41	—	7	120	no	—
112	#41	—	7	1080	no	—
113	#41	—	7	216	no	—
114	#41	—	7	1080	no	—
115	#41	—	7	2160	no	—
116	#41	C₂³×D₇	1	720	yes	—
117	C₂×Q₈	—	7	720	no	—
118	C₂×Q₈	—	7	270	no	—
119	C₂×Q₈	C₂³×D₇	1	270	yes	—
120	C₂×D₄	—	7	540	no	—
121	C₂×D₄	—	7	2160	no	—
122	C₂×D₄	—	7	810	no	—
123	C₂×D₄	C₂³×D₇	1	270	yes	—
124	C₂²×C₄	—	7	45	no	—
125	C₂²×C₄	—	7	270	no	—
126	C₂²×C₄	—	7	810	no	—
127	C₂²×C₄	C₂³×D₇	1	135	yes	—
128	C₂×C₇	—	2	27	no	—
129	C₂×C₇	C₂⁷	1	1	yes	characteristic
130	D₇	—	2	36	no	—
131	C₂³	—	7	270	no	—
132	C₂³	—	56	135	no	—
133	C₂³	C₂⁴×D₇	1	45	yes	—
134	Q₈	—	7	216	no	—
135	Q₈	C₂⁴×D₇	1	120	yes	—
136	D₄	—	7	432	no	—
137	D₄	—	7	360	no	—
138	D₄	C₂⁴×D₇	1	216	yes	—
139	C₂×C₄	—	7	27	no	—
140	C₂×C₄	—	7	135	no	—
141	C₂×C₄	—	7	36	no	—
142	C₂×C₄	—	7	540	no	—
143	C₂×C₄	C₂⁴×D₇	1	270	yes	—
144	C₇	#63103	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	4	45	no	—
146	C₂²	—	7	36	no	—
147	C₂²	—	28	270	no	—
148	C₂²	C₂⁵×D₇	1	27	yes	—
149	C₄	—	7	1	no	—
150	C₄	—	7	27	no	—
151	C₄	C₂⁵×D₇	1	36	yes	—
152	C₂	—	2	27	no	—
153	C₂	—	14	36	no	—
154	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁷
Fitting	C₇×2₋¹⁺⁶	C₂
hypercenter	2₋¹⁺⁶	D₇
hypocenter	C₇	#63103
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁶

Complements

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

N	H ≅ G/N	type
1. C₇×2₋¹⁺⁶	C₂	semidirect
2. #11744566	C₂	semidirect
3. D₇×#585	C₂	semidirect
4. #11744563	C₂	semidirect
5. C₂×#90620	C₂	semidirect
6. C₇×#585	C₂²	semidirect
7. C₂×C₇×2₋¹⁺⁴	C₂²	semidirect
8. #90620	C₂²	semidirect
9. #90620	C₂²	semidirect
10. D₇×2₋¹⁺⁴	C₂²	semidirect
11. #90618	C₂²	semidirect
12. D₇×H(5,𝔽₂)	C₂²	semidirect
13. #90616	C₂²	semidirect
14. C₂×#6553	C₂²	semidirect
15. C₂×D₇×#41	C₂²	semidirect
16. C₂×#6551	C₂²	semidirect
17. C₂²×#1161	C₂²	semidirect
19. C₇×2₋¹⁺⁴	C₂³	semidirect
20. C₇×H(5,𝔽₂)	C₂³	semidirect
21. C₂×C₇×#41	C₂³	semidirect
22. C₂²×C₇×Q₈	C₂³	semidirect
23. #6554	C₂³	semidirect
24. #6553	C₂³	semidirect
25. D₇×#41	C₂³	semidirect
26. D₇×#41	C₂³	semidirect
27. #6551	C₂³	semidirect
28. C₂×#1161	C₂³	semidirect
29. C₂×#1161	C₂³	semidirect
30. C₂×Q₈×D₇	C₂³	semidirect
31. #6548	C₂³	semidirect
32. C₂×#1159	C₂³	semidirect
33. C₂×D₄×D₇	C₂³	semidirect
34. C₂×#1157	C₂³	semidirect
35. C₂²×D₂₈	C₂³	semidirect
36. C₂²×C₄×D₇	C₂³	semidirect
38. 2₋¹⁺⁶	D₇	semidirect
64. #585	C₂×D₇	semidirect
67. C₂×2₋¹⁺⁴	C₂×D₇	semidirect
87. 2₋¹⁺⁴	C₂²×D₇	semidirect
90. H(5,𝔽₂)	C₂²×D₇	semidirect
97. C₂×#41	C₂²×D₇	semidirect
99. C₂²×Q₈	C₂²×D₇	semidirect
144. C₇	#63103	semidirect

Maximal subgroups

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

subgroup	index
C₇×2₋¹⁺⁶	2
#11744566	2
D₇×#585	2
#11744563	2
C₂×#90620	2
#63103	7