Finite Group Databasebeta

Subgroups — #11719475

Order 832.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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	#88257	C₂	1	1	yes	maximal, characteristic, complemented
2	#88256	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	#88205	C₂	1	1	yes	maximal, characteristic, complemented
5	#88203	C₂	1	1	yes	maximal, characteristic, complemented
6	#88172	C₂	1	2	yes	maximal, complemented
7	#88170	C₂	1	2	yes	maximal, complemented
8	SD₁₆×D₁₃	C₂	1	2	yes	maximal, complemented
9	#88165	C₂	1	2	yes	maximal, complemented
10	C₂×#6095	C₂	1	1	yes	maximal, characteristic, complemented
11	#88155	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₁₃×#41	C₂²	1	1	yes	characteristic, complemented
14	#6131	C₂²	1	2	yes	—
15	Q₈×D₁₃	C₂²	1	2	yes	complemented
16	#6129	—	2	2	no	—
17	#6129	C₂²	1	2	yes	complemented
18	D₄×D₁₃	—	2	2	no	—
19	D₄×D₁₃	C₂²	1	2	yes	—
20	#6127	—	2	1	no	—
21	#6127	—	2	1	no	—
22	#6127	C₂²	1	1	yes	characteristic
23	C₂×D₅₂	—	2	1	no	—
24	C₂×D₅₂	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₆	—	2	1	no	—
26	C₂×Dic₂₆	C₂²	1	1	yes	characteristic, complemented
27	C₁₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₁₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
29	C₁₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₁₃×D₈	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
32	#6107	C₂²	1	2	yes	complemented
33	#6106	C₂²	1	2	yes	complemented
34	#6105	C₂²	1	2	yes	complemented
35	#6104	C₂²	1	2	yes	complemented
36	#6099	C₂²	1	1	yes	characteristic, complemented
37	#6095	C₂²	1	2	yes	complemented
38	#6095	C₂²	1	2	yes	complemented
39	#6094	C₂²	1	2	yes	complemented
40	C₈×D₁₃	C₂²	1	2	yes	complemented
41	C₂²×D₁₃	—	2	1	no	—
42	C₂²×D₁₃	—	2	2	no	—
43	Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	D₄×C₁₃	—	2	1	no	—
46	D₄×C₁₃	—	2	1	no	—
47	D₄×C₁₃	C₂³	1	1	yes	characteristic
48	D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²⋊D₁₃	—	2	2	no	—
53	C₂²⋊D₁₃	—	2	2	no	—
54	C₂²⋊D₁₃	D₄	1	2	yes	—
55	C₂×Dic₁₃	—	2	1	no	—
56	C₂×Dic₁₃	—	2	2	no	—
57	D₅₂	—	2	2	no	—
58	D₅₂	—	2	1	no	—
59	D₅₂	D₄	1	1	yes	characteristic
60	D₅₂	C₂³	1	2	yes	—
61	C₄×D₁₃	—	2	2	no	—
62	C₄×D₁₃	—	2	2	no	—
63	C₄×D₁₃	C₂³	1	2	yes	—
64	Dic₂₆	—	2	1	no	—
65	Dic₂₆	—	2	2	no	—
66	Dic₂₆	D₄	1	1	yes	characteristic
67	Dic₂₆	C₂³	1	2	yes	—
68	C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₈×C₁₃	C₂³	1	1	yes	characteristic
70	#1055	C₂³	1	2	yes	—
71	#577	—	13	1	no	maximal, Sylow
72	C₂²×C₁₃	—	2	1	no	—
73	C₂²×C₁₃	—	2	1	no	—
74	C₂²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×D₁₃	—	2	2	no	—
76	C₂×D₁₃	—	8	2	no	—
77	C₂×D₁₃	—	8	1	no	—
78	C₂×D₁₃	C₂×D₄	1	2	yes	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	Dic₁₃	—	2	2	no	—
84	Dic₁₃	C₂×D₄	1	2	yes	—
85	2₋¹⁺⁴	—	13	1	no	—
86	H(5,𝔽₂)	—	13	1	no	—
87	#137	—	13	1	no	—
88	#137	—	13	2	no	—
89	AGL(1,ℤ/8)	—	13	1	no	—
90	AGL(1,ℤ/8)	—	13	2	no	—
91	#135	—	13	2	no	—
92	#135	D₁₃	1	1	yes	characteristic, complemented
93	C₂×SD₁₆	—	13	2	no	—
94	C₂×SD₁₆	—	13	1	no	—
95	#131	—	13	1	no	—
96	C₂×C₁₃	—	2	1	no	—
97	C₂×C₁₃	—	4	1	no	—
98	C₂×C₁₃	—	4	1	no	—
99	C₂×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	D₁₃	—	2	2	no	—
101	D₁₃	—	4	2	no	—
102	#41	—	13	1	no	—
103	#41	—	13	2	no	—
104	#41	—	13	2	no	—
105	#41	—	26	1	no	—
106	#41	—	26	1	no	—
107	#41	—	26	2	no	—
108	#41	C₂×D₁₃	1	1	yes	characteristic
109	#41	C₂×D₁₃	1	1	yes	characteristic, complemented
110	C₂×Q₈	—	13	2	no	—
111	C₂×Q₈	—	13	1	no	—
112	C₂×Q₈	—	26	1	no	—
113	C₂×D₄	—	13	2	no	—
114	C₂×D₄	—	13	1	no	—
115	C₂×D₄	—	26	1	no	—
116	C₂×D₄	—	26	2	no	—
117	Q₁₆	—	13	2	no	—
118	Q₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
119	SD₁₆	—	13	2	no	—
120	SD₁₆	—	13	2	no	—
121	SD₁₆	—	13	2	no	—
122	SD₁₆	—	13	2	no	—
123	SD₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
124	SD₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
125	D₈	—	13	2	no	—
126	D₈	C₂×D₁₃	1	1	yes	characteristic, complemented
127	M₁₆	—	13	1	no	—
128	M₁₆	—	13	2	no	—
129	C₂×C₈	—	13	2	no	—
130	C₂×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
131	C₁₃	#577	1	1	yes	characteristic, complemented, Sylow
132	C₂³	—	26	2	no	—
133	C₂³	—	26	1	no	—
134	Q₈	—	13	1	no	—
135	Q₈	—	13	2	no	—
136	Q₈	—	26	2	no	—
137	Q₈	—	26	1	no	—
138	Q₈	C₂²×D₁₃	1	1	yes	characteristic
139	Q₈	C₂²×D₁₃	1	1	yes	characteristic
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	13	2	no	—
143	D₄	—	13	1	no	—
144	D₄	—	13	2	no	—
145	D₄	—	26	2	no	—
146	D₄	—	26	2	no	—
147	D₄	—	26	2	no	—
148	D₄	—	26	1	no	—
149	D₄	C₂²×D₁₃	1	1	yes	characteristic
150	D₄	C₂²×D₁₃	1	1	yes	characteristic
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	13	2	no	—
154	C₂×C₄	—	26	2	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	1	no	—
158	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₈	—	13	2	no	—
160	C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₈	C₂²×D₁₃	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	13	2	no	—
165	C₂²	—	26	2	no	—
166	C₂²	—	104	1	no	—
167	C₂²	—	104	2	no	—
168	C₂²	D₄×D₁₃	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	13	2	no	—
172	C₄	—	26	2	no	—
173	C₄	D₄×D₁₃	1	1	yes	characteristic
174	C₄	C₂³×D₁₃	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	26	2	no	—
179	C₂	—	52	2	no	—
180	C₂	C₂×D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂⁴
Frattini	C₄	C₂³×D₁₃
Fitting	C₁₃×#135	C₂
socle	C₂×C₁₃	C₂²×D₄
hypercenter	#135	D₁₃
hypocenter	C₁₃	#577
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #88257	C₂	semidirect
2. #88256	C₂	semidirect
3. C₁₃×#135	C₂	semidirect
4. #88205	C₂	semidirect
5. #88203	C₂	semidirect
6. #88172	C₂	semidirect
7. #88170	C₂	semidirect
8. SD₁₆×D₁₃	C₂	semidirect
9. #88165	C₂	semidirect
10. C₂×#6095	C₂	semidirect
11. #88155	C₂	semidirect
13. C₁₃×#41	C₂²	semidirect
15. Q₈×D₁₃	C₂²	semidirect
17. #6129	C₂²	semidirect
26. C₂×Dic₂₆	C₂²	semidirect
27. C₁₃×Q₁₆	C₂²	semidirect
28. C₁₃×SD₁₆	C₂²	semidirect
29. C₁₃×SD₁₆	C₂²	semidirect
30. C₁₃×D₈	C₂²	semidirect
31. C₂×C₈×C₁₃	C₂²	semidirect
32. #6107	C₂²	semidirect
33. #6106	C₂²	semidirect
34. #6105	C₂²	semidirect
35. #6104	C₂²	semidirect
36. #6099	C₂²	semidirect
37. #6095	C₂²	semidirect
38. #6095	C₂²	semidirect
39. #6094	C₂²	semidirect
40. C₈×D₁₃	C₂²	semidirect
92. #135	D₁₃	semidirect
109. #41	C₂×D₁₃	semidirect
118. Q₁₆	C₂×D₁₃	semidirect
123. SD₁₆	C₂×D₁₃	semidirect
124. SD₁₆	C₂×D₁₃	semidirect
126. D₈	C₂×D₁₃	semidirect
130. C₂×C₈	C₂×D₁₃	semidirect
131. C₁₃	#577	semidirect

Maximal subgroups

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

subgroup	index
#88257	2
#88256	2
C₁₃×#135	2
#88205	2
#88203	2
#88172	2
#88170	2
SD₁₆×D₁₃	2
#88165	2
C₂×#6095	2
#88155	2
#577	13