Subgroups — #131×#595 #14545351

Order 2176.

572 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇⋊#131	C₂	1	1	yes	maximal, characteristic
2	#41×#595	C₂	1	1	yes	maximal, characteristic
3	#11763775	C₂	1	3	yes	maximal, complemented
4	#11763772	C₂	1	3	yes	maximal, complemented
5	M₁₆×#595	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×#595	C₂	1	3	yes	maximal, complemented
7	#131×D₁₇	C₂	1	1	yes	maximal, characteristic
8	#41×D₁₇	C₂²	1	1	yes	characteristic
9	Q₈×#595	C₄	1	2	yes	—
10	#10587680	C₄	1	2	yes	complemented
11	D₄×#595	C₄	1	6	yes	—
12	#10587678	C₄	1	6	yes	complemented
13	#10587677	C₂²	1	3	yes	—
14	C₂×C₄×#595	C₂²	1	3	yes	—
15	D₁₇⋊M₁₆	C₂²	1	3	yes	—
16	C₂×#63242	C₂²	1	3	yes	—
17	C₁₇×#131	C₄	1	1	yes	characteristic, complemented
18	#10587634	C₄	1	1	yes	characteristic, complemented
19	#10587593	C₄	1	3	yes	complemented
20	M₁₆×D₁₇	C₂²	1	3	yes	—
21	#10587587	C₄	1	3	yes	complemented
22	C₂×C₈×D₁₇	C₂²	1	3	yes	—
23	#10587532	C₂²	1	6	yes	—
24	C₈×#595	C₂²	1	1	yes	characteristic
25	C₈×#595	C₂²	1	3	yes	—
26	C₈×#595	C₂²	1	3	yes	—
27	C₈×#595	C₂²	1	3	yes	—
28	C₂²×#595	C₂×C₄	1	6	yes	—
29	#41×C₁₇	C₂×C₄	1	1	yes	characteristic
30	#63256	C₂×C₄	1	1	yes	characteristic
31	Q₈×D₁₇	C₂×C₄	1	1	yes	characteristic
32	#63254	C₂×C₄	1	3	yes	—
33	D₄×D₁₇	C₂×C₄	1	3	yes	—
34	#63252	C₂×C₄	1	3	yes	—
35	C₂×C₄×D₁₇	C₂³	1	3	yes	—
36	C₂²⋊#595	C₂×C₄	1	6	yes	—
37	#63247	C₂×C₄	1	6	yes	complemented
38	C₂×#3624	C₂×C₄	1	6	yes	complemented
39	C₄⋊#595	C₂×C₄	1	6	yes	—
40	C₄×#595	C₂×C₄	1	6	yes	—
41	C₄×#595	C₂³	1	1	yes	characteristic
42	C₄×#595	C₂³	1	3	yes	—
43	#63243	C₂×C₄	1	6	yes	complemented
44	#63242	C₂×C₄	1	6	yes	complemented
45	#63242	C₂³	1	1	yes	characteristic
46	#63242	C₂³	1	3	yes	—
47	M₁₆×C₁₇	C₂×C₄	1	3	yes	complemented
48	C₂×C₈×C₁₇	C₂×C₄	1	3	yes	complemented
49	#63223	C₂×C₄	1	3	yes	complemented
50	C₂×#3622	C₂×C₄	1	3	yes	complemented
51	#63218	C₂×C₄	1	6	yes	complemented
52	C₈×D₁₇	C₂×C₄	1	3	yes	complemented
53	C₈×D₁₇	C₂×C₄	1	3	yes	complemented
54	C₈×D₁₇	C₂³	1	1	yes	characteristic
55	C₈×D₁₇	C₂³	1	3	yes	—
56	C₂²×D₁₇	C₂²×C₄	1	3	yes	—
57	C₂×#595	—	2	12	no	—
58	C₂×#595	C₂²×C₄	1	6	yes	—
59	C₂×#595	C₂²×C₄	1	2	yes	—
60	Q₈×C₁₇	C₄²	1	1	yes	characteristic
61	D₄×C₁₇	C₄²	1	3	yes	—
62	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
63	C₂²⋊D₁₇	C₄²	1	6	yes	—
64	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
65	D₆₈	C₄²	1	3	yes	—
66	C₄×D₁₇	C₂²×C₄	1	3	yes	—
67	C₄×D₁₇	C₂²×C₄	1	3	yes	—
68	C₄×D₁₇	C₂²×C₄	1	3	yes	—
69	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
70	Dic₃₄	C₄²	1	3	yes	—
71	#3624	C₂²×C₄	1	2	yes	—
72	#3624	C₂²×C₄	1	6	yes	—
73	C₈×C₁₇	C₂²×C₄	1	3	yes	—
74	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	#3622	C₂²×C₄	1	1	yes	characteristic
76	#3622	C₂²×C₄	1	3	yes	—
77	C₄×#131	—	17	1	no	maximal, Sylow
78	C₂²×C₁₇	C₂×C₄²	1	3	yes	—
79	C₂×D₁₇	—	2	6	no	—
80	C₂×D₁₇	C₂×C₄²	1	3	yes	—
81	C₂×D₁₇	C₂³×C₄	1	1	yes	characteristic
82	#595	—	2	6	no	—
83	#595	#131	1	4	yes	direct product factor
84	C₄×C₁₇	C₂×C₄²	1	3	yes	—
85	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
86	Dic₁₇	C₂×C₄²	1	3	yes	—
87	Dic₁₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×#131	—	17	1	no	—
89	C₂×#131	—	17	1	no	—
90	C₄×#41	—	17	1	no	—
91	#405	—	17	3	no	—
92	#405	—	17	3	no	—
93	C₄×M₁₆	—	17	3	no	—
94	C₂×C₄×C₈	—	17	3	no	—
95	C₂×C₁₇	—	2	3	no	—
96	C₂×C₁₇	C₂²×C₄²	1	1	yes	characteristic
97	D₁₇	—	2	3	no	—
98	D₁₇	C₂×#131	1	2	yes	—
99	C₂×#41	—	17	1	no	—
100	#131	—	17	6	no	—
101	#131	—	17	2	no	—
102	#131	—	17	1	no	—
103	#131	—	17	3	no	—
104	#131	—	17	3	no	—
105	#131	#595	1	1	yes	characteristic, direct product factor
106	C₂×M₁₆	—	17	3	no	—
107	C₂×M₁₆	—	17	3	no	—
108	C₂²×C₈	—	17	3	no	—
109	C₂²×C₈	—	17	3	no	—
110	C₄×Q₈	—	17	2	no	—
111	C₄×D₄	—	17	6	no	—
112	#117	—	17	3	no	—
113	C₂×C₄²	—	17	3	no	—
114	#97	—	17	6	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	3	no	—
117	C₄×C₈	—	17	3	no	—
118	C₄×C₈	—	17	3	no	—
119	C₁₇	C₄×#131	1	1	yes	characteristic, complemented, Sylow
120	#41	—	17	1	no	—
121	#41	—	17	3	no	—
122	#41	—	17	3	no	—
123	#41	C₂×#595	1	1	yes	characteristic
124	C₂×Q₈	—	17	1	no	—
125	C₂×D₄	—	17	3	no	—
126	C₂²×C₄	—	17	3	no	—
127	C₂²×C₄	—	17	6	no	—
128	M₁₆	—	17	6	no	—
129	M₁₆	—	17	6	no	—
130	M₁₆	—	17	3	no	—
131	M₁₆	—	17	6	no	—
132	M₁₆	C₂×#595	1	3	yes	complemented
133	C₂×C₈	—	17	6	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	3	no	—
136	C₂×C₈	—	17	6	no	—
137	C₂×C₈	—	17	3	no	—
138	C₂×C₈	—	17	3	no	—
139	C₂×C₈	—	17	3	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	3	no	—
142	C₂×C₈	C₂×#595	1	3	yes	complemented
143	C₄⋊C₄	—	17	6	no	—
144	C₂²⋊C₄	—	17	6	no	—
145	C₄²	—	17	6	no	—
146	C₄²	—	17	1	no	—
147	C₄²	—	17	3	no	—
148	C₂³	—	17	3	no	—
149	Q₈	—	17	3	no	—
150	Q₈	C₄×#595	1	1	yes	characteristic
151	D₄	—	17	6	no	—
152	D₄	—	17	3	no	—
153	D₄	C₄×#595	1	3	yes	—
154	C₂×C₄	—	17	2	no	—
155	C₂×C₄	—	17	3	no	—
156	C₂×C₄	—	17	3	no	—
157	C₂×C₄	—	17	3	no	—
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	17	3	no	—
160	C₂×C₄	—	17	6	no	—
161	C₂×C₄	—	34	12	no	—
162	C₂×C₄	C₂²×#595	1	3	yes	—
163	C₈	—	17	2	no	—
164	C₈	—	17	6	no	—
165	C₈	—	17	1	no	—
166	C₈	—	17	3	no	—
167	C₈	C₂²×#595	1	3	yes	—
168	C₈	C₂²×#595	1	1	yes	characteristic
169	C₂²	—	17	1	no	—
170	C₂²	—	17	3	no	—
171	C₂²	—	34	6	no	—
172	C₂²	C₂×C₄×#595	1	3	yes	—
173	C₄	—	17	4	no	—
174	C₄	—	17	1	no	—
175	C₄	—	17	3	no	—
176	C₄	—	34	6	no	—
177	C₄	C₂×C₄×#595	1	3	yes	—
178	C₄	C₂³×#595	1	1	yes	characteristic
179	C₂	—	2	3	no	—
180	C₂	—	17	2	no	—
181	C₂	—	34	3	no	—
182	C₂	C₂²×C₄×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#595
commutator, socle	C₂×C₁₇	C₂²×C₄²
Frattini	C₄	C₂³×#595
Fitting	C₁₇×#131	C₄
prime radical	#41×D₁₇	C₂²
cosocle	C₄×D₁₇	C₂⁴
hypercenter	#131	#595
hypocenter	C₁₇	C₄×#131
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
3. #11763775	C₂	semidirect
4. #11763772	C₂	semidirect
5. M₁₆×#595	C₂	semidirect
6. C₂×C₈×#595	C₂	semidirect
10. #10587680	C₄	semidirect
12. #10587678	C₄	semidirect
17. C₁₇×#131	C₄	semidirect
18. #10587634	C₄	semidirect
19. #10587593	C₄	semidirect
21. #10587587	C₄	semidirect
37. #63247	C₂×C₄	semidirect
38. C₂×#3624	C₂×C₄	semidirect
43. #63243	C₂×C₄	semidirect
44. #63242	C₂×C₄	semidirect
47. M₁₆×C₁₇	C₂×C₄	semidirect
48. C₂×C₈×C₁₇	C₂×C₄	semidirect
49. #63223	C₂×C₄	semidirect
50. C₂×#3622	C₂×C₄	semidirect
51. #63218	C₂×C₄	semidirect
52. C₈×D₁₇	C₂×C₄	semidirect
53. C₈×D₁₇	C₂×C₄	semidirect
83. #595	#131	direct (N × H)
105. #131	#595	direct (N × H)
119. C₁₇	C₄×#131	semidirect
132. M₁₆	C₂×#595	semidirect
142. C₂×C₈	C₂×#595	semidirect

Maximal subgroups

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

subgroup	index
D₁₇⋊#131	2
#41×#595	2
#11763775	2
#11763772	2
M₁₆×#595	2
C₂×C₈×#595	2
#131×D₁₇	2
C₄×#131	17