Finite Group Databasebeta

Subgroups — #14267697

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×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×#585	C₂	1	28	yes	maximal, complemented
3	#11744564	C₂	1	28	yes	maximal, complemented
4	#11744560	C₂	1	35	yes	maximal, complemented
5	C₂×#90618	C₂	1	35	yes	maximal, complemented
6	C₇×#585	C₂²	1	28	yes	complemented
7	C₂×C₇×H(5,𝔽₂)	C₂²	1	35	yes	complemented
8	#90620	C₂²	1	56	yes	complemented
9	D₇×2₋¹⁺⁴	C₂²	1	168	yes	complemented
10	#90618	C₂²	1	280	yes	complemented
11	#90618	C₂²	1	560	yes	complemented
12	D₇×H(5,𝔽₂)	C₂²	1	280	yes	complemented
13	#90616	C₂²	1	420	yes	complemented
14	C₂×#6554	C₂²	1	210	yes	complemented
15	C₂×D₇×#41	C₂²	1	210	yes	complemented
16	C₂×#6548	C₂²	1	315	yes	complemented
17	C₂²×#1159	C₂²	1	105	yes	complemented
18	#63103	—	7	1	no	maximal, Sylow
19	C₇×2₋¹⁺⁴	C₂³	1	56	yes	complemented
20	C₇×H(5,𝔽₂)	C₂³	1	280	yes	complemented
21	C₂×C₇×#41	C₂³	1	210	yes	complemented
22	C₂²×C₇×D₄	C₂³	1	105	yes	complemented
23	C₂²×C₂²⋊D₇	C₂³	1	210	yes	complemented
24	C₂³×Dic₇	C₂³	1	30	yes	complemented
25	#6554	C₂³	1	1680	yes	complemented
26	#6553	C₂³	1	560	yes	complemented
27	D₇×#41	C₂³	1	560	yes	complemented
28	D₇×#41	C₂³	1	1680	yes	complemented
29	#6551	C₂³	1	280	yes	complemented
30	C₂×#1161	C₂³	1	70	yes	complemented
31	C₂×Q₈×D₇	C₂³	1	210	yes	complemented
32	#6548	C₂³	1	2520	yes	complemented
33	C₂×#1159	C₂³	1	630	yes	complemented
34	C₂×#1159	C₂³	1	1260	yes	complemented
35	C₂×D₄×D₇	C₂³	1	630	yes	complemented
36	C₂×#1157	C₂³	1	630	yes	complemented
37	C₂²×C₄×D₇	C₂³	1	105	yes	complemented
38	C₂²×Dic₁₄	C₂³	1	105	yes	complemented
39	2₋¹⁺⁶	—	7	28	no	—
40	H(7,𝔽₂)	—	7	35	no	—
41	H(7,𝔽₂)	D₇	1	1	yes	characteristic, complemented
42	C₂×#585	—	7	28	no	—
43	C₂×#585	—	7	35	no	—
44	C₂⁴×C₇	C₂⁴	1	30	yes	—
45	C₂³×D₇	C₂⁴	1	105	yes	—
46	C₇×#41	C₂⁴	1	560	yes	—
47	C₂×C₇×Q₈	C₂⁴	1	70	yes	—
48	C₂×C₇×D₄	C₂⁴	1	630	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	105	yes	—
50	C₂×C₂²⋊D₇	C₂⁴	1	1260	yes	—
51	C₂²×Dic₇	C₂⁴	1	210	yes	—
52	C₂²×Dic₇	C₂⁴	1	105	yes	—
53	#1161	C₂⁴	1	56	yes	—
54	#1161	C₂⁴	1	280	yes	—
55	Q₈×D₇	C₂⁴	1	560	yes	—
56	#1159	C₂⁴	1	280	yes	—
57	#1159	C₂⁴	1	2520	yes	—
58	D₄×D₇	C₂⁴	1	1680	yes	—
59	#1157	C₂⁴	1	1680	yes	—
60	C₂×D₂₈	C₂⁴	1	210	yes	—
61	C₂×C₄×D₇	C₂⁴	1	210	yes	—
62	C₂×C₄×D₇	C₂⁴	1	630	yes	—
63	C₂×Dic₁₄	C₂⁴	1	630	yes	—
64	#585	—	7	56	no	—
65	#585	—	7	280	no	—
66	#585	—	7	560	no	—
67	#585	—	7	420	no	—
68	#585	C₂×D₇	1	28	yes	complemented
69	C₂×2₋¹⁺⁴	—	7	168	no	—
70	C₂×2₋¹⁺⁴	—	7	210	no	—
71	C₂×H(5,𝔽₂)	—	7	280	no	—
72	C₂×H(5,𝔽₂)	—	7	315	no	—
73	C₂×H(5,𝔽₂)	C₂×D₇	1	35	yes	complemented
74	C₂²×#41	—	7	210	no	—
75	C₂²×#41	—	7	105	no	—
76	C₂³×C₇	—	8	30	no	—
77	C₂³×C₇	C₂⁵	1	105	yes	—
78	C₂²×D₇	—	8	105	no	—
79	C₂²×D₇	C₂⁵	1	210	yes	—
80	C₇×Q₈	C₂⁵	1	56	yes	—
81	C₇×D₄	C₂⁵	1	280	yes	—
82	C₂×C₄×C₇	C₂⁵	1	210	yes	—
83	C₂²⋊D₇	C₂⁵	1	560	yes	—
84	C₂×Dic₇	C₂⁵	1	315	yes	—
85	C₂×Dic₇	C₂⁵	1	35	yes	—
86	D₂₈	C₂⁵	1	168	yes	—
87	C₄×D₇	C₂⁵	1	28	yes	—
88	C₄×D₇	C₂⁵	1	420	yes	—
89	Dic₁₄	C₂⁵	1	280	yes	—
90	2₋¹⁺⁴	—	7	1680	no	—
91	2₋¹⁺⁴	—	7	280	no	—
92	2₋¹⁺⁴	C₂²×D₇	1	56	yes	complemented
93	H(5,𝔽₂)	—	7	560	no	—
94	H(5,𝔽₂)	—	7	2520	no	—
95	H(5,𝔽₂)	C₂²×D₇	1	280	yes	complemented
96	C₂×#41	—	7	560	no	—
97	C₂×#41	—	7	1680	no	—
98	C₂×#41	—	7	70	no	—
99	C₂×#41	—	7	630	no	—
100	C₂×#41	—	7	1260	no	—
101	C₂×#41	—	7	630	no	—
102	C₂×#41	C₂²×D₇	1	210	yes	complemented
103	C₂²×Q₈	—	7	210	no	—
104	C₂²×Q₈	—	7	105	no	—
105	C₂²×D₄	—	7	210	no	—
106	C₂²×D₄	—	7	630	no	—
107	C₂²×D₄	C₂²×D₇	1	105	yes	complemented
108	C₂³×C₄	—	7	30	no	—
109	C₂³×C₄	—	7	105	no	—
110	C₂²×C₇	—	4	105	no	—
111	C₂²×C₇	C₂⁶	1	35	yes	—
112	C₂×D₇	—	4	210	no	—
113	C₂×D₇	C₂⁶	1	28	yes	—
114	C₄×C₇	C₂⁶	1	28	yes	—
115	Dic₇	C₂⁶	1	35	yes	—
116	Dic₇	C₂⁶	1	1	yes	characteristic
117	C₂⁴	—	7	105	no	—
118	C₂⁴	C₂³×D₇	1	30	yes	—
119	#41	—	7	56	no	—
120	#41	—	7	280	no	—
121	#41	—	7	280	no	—
122	#41	—	7	2520	no	—
123	#41	—	7	1680	no	—
124	#41	C₂³×D₇	1	560	yes	—
125	C₂×Q₈	—	7	560	no	—
126	C₂×Q₈	—	7	630	no	—
127	C₂×Q₈	C₂³×D₇	1	70	yes	—
128	C₂×D₄	—	7	1260	no	—
129	C₂×D₄	—	7	1680	no	—
130	C₂×D₄	—	7	210	no	—
131	C₂×D₄	C₂³×D₇	1	630	yes	—
132	C₂²×C₄	—	7	105	no	—
133	C₂²×C₄	—	7	210	no	—
134	C₂²×C₄	—	7	210	no	—
135	C₂²×C₄	—	7	630	no	—
136	C₂²×C₄	C₂³×D₇	1	105	yes	—
137	C₂×C₇	—	2	35	no	—
138	C₂×C₇	C₂⁷	1	1	yes	characteristic
139	D₇	—	2	28	no	—
140	C₂³	—	7	210	no	—
141	C₂³	—	8	30	no	—
142	C₂³	—	56	105	no	—
143	C₂³	C₂⁴×D₇	1	105	yes	—
144	Q₈	—	7	280	no	—
145	Q₈	C₂⁴×D₇	1	56	yes	—
146	D₄	—	7	560	no	—
147	D₄	—	7	168	no	—
148	D₄	C₂⁴×D₇	1	280	yes	—
149	C₂×C₄	—	7	35	no	—
150	C₂×C₄	—	7	315	no	—
151	C₂×C₄	—	7	28	no	—
152	C₂×C₄	—	7	420	no	—
153	C₂×C₄	C₂⁴×D₇	1	210	yes	—
154	C₇	#63103	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	4	105	no	—
156	C₂²	—	7	28	no	—
157	C₂²	—	28	210	no	—
158	C₂²	C₂⁵×D₇	1	35	yes	—
159	C₄	—	7	1	no	—
160	C₄	—	7	35	no	—
161	C₄	C₂⁵×D₇	1	28	yes	—
162	C₂	—	2	35	no	—
163	C₂	—	14	28	no	—
164	C₂	C₂⁶×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁶×D₇
commutator, socle, cosocle	C₂×C₇	C₂⁷
Fitting	C₇×H(7,𝔽₂)	C₂
hypercenter	H(7,𝔽₂)	D₇
hypocenter	C₇	#63103
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₇×H(7,𝔽₂)	C₂	semidirect
2. D₇×#585	C₂	semidirect
3. #11744564	C₂	semidirect
4. #11744560	C₂	semidirect
5. C₂×#90618	C₂	semidirect
6. C₇×#585	C₂²	semidirect
7. C₂×C₇×H(5,𝔽₂)	C₂²	semidirect
8. #90620	C₂²	semidirect
9. D₇×2₋¹⁺⁴	C₂²	semidirect
10. #90618	C₂²	semidirect
11. #90618	C₂²	semidirect
12. D₇×H(5,𝔽₂)	C₂²	semidirect
13. #90616	C₂²	semidirect
14. C₂×#6554	C₂²	semidirect
15. C₂×D₇×#41	C₂²	semidirect
16. C₂×#6548	C₂²	semidirect
17. C₂²×#1159	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×D₄	C₂³	semidirect
23. C₂²×C₂²⋊D₇	C₂³	semidirect
24. C₂³×Dic₇	C₂³	semidirect
25. #6554	C₂³	semidirect
26. #6553	C₂³	semidirect
27. D₇×#41	C₂³	semidirect
28. D₇×#41	C₂³	semidirect
29. #6551	C₂³	semidirect
30. C₂×#1161	C₂³	semidirect
31. C₂×Q₈×D₇	C₂³	semidirect
32. #6548	C₂³	semidirect
33. C₂×#1159	C₂³	semidirect
34. C₂×#1159	C₂³	semidirect
35. C₂×D₄×D₇	C₂³	semidirect
36. C₂×#1157	C₂³	semidirect
37. C₂²×C₄×D₇	C₂³	semidirect
38. C₂²×Dic₁₄	C₂³	semidirect
41. H(7,𝔽₂)	D₇	semidirect
68. #585	C₂×D₇	semidirect
73. C₂×H(5,𝔽₂)	C₂×D₇	semidirect
92. 2₋¹⁺⁴	C₂²×D₇	semidirect
95. H(5,𝔽₂)	C₂²×D₇	semidirect
102. C₂×#41	C₂²×D₇	semidirect
107. C₂²×D₄	C₂²×D₇	semidirect
154. 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×H(7,𝔽₂)	2
D₇×#585	2
#11744564	2
#11744560	2
C₂×#90618	2
#63103	7