Subgroups — #17978843

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545208	C₂	1	6	yes	maximal, complemented
2	#14545207	C₂	1	2	yes	maximal, complemented
3	C₂×#11763742	C₂	1	3	yes	maximal, complemented
4	#14545205	C₂	1	3	yes	maximal, complemented
5	D₁₇×#504	C₂	1	1	yes	maximal, characteristic
6	#11763742	C₂²	1	3	yes	—
7	#11763742	C₂²	1	1	yes	characteristic
8	#11763742	C₂²	1	6	yes	complemented
9	#11763741	C₂²	1	6	yes	complemented
10	#11763740	C₂²	1	6	yes	complemented
11	C₂×#10587467	C₂²	1	6	yes	complemented
12	#131×D₁₇	C₄	1	1	yes	characteristic
13	#11763227	C₂²	1	3	yes	—
14	#11763226	C₂²	1	1	yes	characteristic
15	#11763225	C₄	1	3	yes	—
16	M₃₂×D₁₇	C₄	1	3	yes	—
17	C₂×C₁₆×D₁₇	C₂²	1	3	yes	—
18	C₁₇×#504	C₄	1	1	yes	characteristic
19	#41×D₁₇	C₈	1	1	yes	characteristic
20	C₁₇×#131	C₂×C₄	1	1	yes	characteristic
21	#10587634	C₈	1	1	yes	characteristic
22	#10587593	C₈	1	3	yes	—
23	M₁₆×D₁₇	C₈	1	3	yes	—
24	#10587587	C₂×C₄	1	3	yes	—
25	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
26	C₁₇×M₃₂	C₂×C₄	1	3	yes	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	3	yes	—
28	#10587483	C₂×C₄	1	3	yes	—
29	C₂×#63214	C₂³	1	3	yes	—
30	#10587469	C₂×C₄	1	6	yes	—
31	C₁₆×D₁₇	C₂×C₄	1	3	yes	—
32	C₁₆×D₁₇	C₂×C₄	1	3	yes	—
33	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
34	C₁₆×D₁₇	C₂³	1	3	yes	—
35	#10587467	C₂³	1	2	yes	—
36	#10587467	C₂³	1	6	yes	—
37	#41×C₁₇	C₂×C₈	1	1	yes	characteristic
38	#63256	C₁₆	1	1	yes	characteristic
39	Q₈×D₁₇	C₁₆	1	1	yes	characteristic
40	#63254	C₁₆	1	3	yes	—
41	D₄×D₁₇	C₁₆	1	3	yes	—
42	#63252	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
44	M₁₆×C₁₇	C₂×C₈	1	3	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
46	#63223	C₂×C₈	1	3	yes	—
47	C₂×#3622	C₂×C₈	1	3	yes	—
48	#63218	C₂×C₈	1	6	yes	—
49	C₈×D₁₇	C₂×C₈	1	3	yes	—
50	C₈×D₁₇	C₂×C₈	1	3	yes	—
51	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₈×D₁₇	C₂²×C₄	1	3	yes	—
53	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₁₇	C₂²×C₄	1	3	yes	—
55	#63214	C₂²×C₄	1	3	yes	—
56	#63214	C₂⁴	1	1	yes	characteristic
57	#33974	—	17	1	no	maximal, Sylow
58	C₂²×D₁₇	C₂×C₁₆	1	3	yes	—
59	Q₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
60	D₄×C₁₇	C₂×C₁₆	1	3	yes	—
61	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
62	C₂²⋊D₁₇	C₂×C₁₆	1	6	yes	—
63	C₂×Dic₁₇	C₂×C₁₆	1	3	yes	—
64	D₆₈	C₂×C₁₆	1	3	yes	—
65	C₄×D₁₇	C₂×C₁₆	1	3	yes	—
66	C₄×D₁₇	C₂×C₁₆	1	3	yes	—
67	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₄×D₁₇	C₂²×C₈	1	3	yes	—
69	Dic₃₄	C₂×C₁₆	1	3	yes	—
70	C₈×C₁₇	C₂²×C₈	1	3	yes	—
71	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
72	#3622	C₂²×C₈	1	1	yes	characteristic
73	#3622	C₂²×C₈	1	3	yes	—
74	C₂×#504	—	17	1	no	—
75	#2258	—	17	2	no	—
76	#2258	—	17	6	no	—
77	C₂×M₆₄	—	17	3	no	—
78	C₂×M₆₄	—	17	3	no	—
79	C₂²×C₁₇	C₂²×C₁₆	1	3	yes	—
80	C₂×D₁₇	—	4	3	no	—
81	C₂×D₁₇	C₂²×C₁₆	1	1	yes	characteristic
82	C₂×D₁₇	C₂²×C₁₆	1	3	yes	—
83	C₄×C₁₇	C₂²×C₁₆	1	3	yes	—
84	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
85	Dic₁₇	C₂²×C₁₆	1	3	yes	—
86	Dic₁₇	C₂²×C₁₆	1	1	yes	characteristic
87	C₂×#131	—	17	1	no	—
88	#504	—	17	1	no	—
89	#504	—	17	3	no	—
90	#504	—	17	3	no	—
91	#504	#595	1	1	yes	characteristic
92	C₂×M₃₂	—	17	3	no	—
93	C₂²×C₁₆	—	17	3	no	—
94	M₆₄	—	17	6	no	—
95	M₆₄	—	17	3	no	—
96	M₆₄	—	17	1	no	—
97	M₆₄	—	17	6	no	—
98	C₂×C₃₂	—	17	6	no	—
99	C₂×C₃₂	—	17	6	no	—
100	C₂×C₁₇	—	2	3	no	—
101	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
102	D₁₇	—	2	1	no	—
103	D₁₇	—	2	3	no	—
104	C₂×#41	—	17	1	no	—
105	#131	—	17	1	no	—
106	#131	—	17	3	no	—
107	#131	—	17	3	no	—
108	#131	C₂×#595	1	1	yes	characteristic
109	C₂×M₁₆	—	17	3	no	—
110	C₂²×C₈	—	17	3	no	—
111	M₃₂	—	17	3	no	—
112	M₃₂	—	17	6	no	—
113	M₃₂	C₂×#595	1	3	yes	—
114	C₂×C₁₆	—	17	3	no	—
115	C₂×C₁₆	—	17	3	no	—
116	C₂×C₁₆	—	17	3	no	—
117	C₂×C₁₆	—	17	3	no	—
118	C₂×C₁₆	—	17	1	no	—
119	C₂×C₁₆	C₂×#595	1	3	yes	—
120	C₃₂	—	17	2	no	—
121	C₃₂	—	17	6	no	—
122	C₁₇	#33974	1	1	yes	characteristic, complemented, Sylow
123	#41	—	17	1	no	—
124	#41	—	17	3	no	—
125	#41	—	17	3	no	—
126	#41	#63242	1	1	yes	characteristic
127	C₂×Q₈	—	17	1	no	—
128	C₂×D₄	—	17	3	no	—
129	C₂²×C₄	—	17	3	no	—
130	M₁₆	—	17	3	no	—
131	M₁₆	—	17	6	no	—
132	M₁₆	#63242	1	3	yes	—
133	C₂×C₈	—	17	3	no	—
134	C₂×C₈	—	17	3	no	—
135	C₂×C₈	—	17	3	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	3	no	—
138	C₂×C₈	C₂²×#595	1	3	yes	—
139	C₁₆	—	17	1	no	—
140	C₁₆	—	17	3	no	—
141	C₁₆	C₂²×#595	1	1	yes	characteristic
142	C₁₆	C₂²×#595	1	3	yes	—
143	C₂³	—	17	3	no	—
144	Q₈	—	17	3	no	—
145	Q₈	#10587529	1	1	yes	characteristic
146	D₄	—	17	6	no	—
147	D₄	—	17	3	no	—
148	D₄	#10587529	1	3	yes	—
149	C₂×C₄	—	17	3	no	—
150	C₂×C₄	—	17	3	no	—
151	C₂×C₄	—	17	3	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	3	no	—
154	C₂×C₄	C₂×#63242	1	3	yes	—
155	C₈	—	17	1	no	—
156	C₈	—	17	3	no	—
157	C₈	C₂×#63242	1	3	yes	—
158	C₈	C₂³×#595	1	1	yes	characteristic
159	C₂²	—	17	1	no	—
160	C₂²	—	17	3	no	—
161	C₂²	—	68	3	no	—
162	C₂²	C₂×#10587529	1	3	yes	—
163	C₄	—	17	1	no	—
164	C₄	—	17	3	no	—
165	C₄	C₂×#10587529	1	3	yes	—
166	C₄	C₂²×#63242	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	34	1	no	—
169	C₂	—	34	3	no	—
170	C₂	C₂²×#10587529	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#595
commutator, socle	C₂×C₁₇	C₂³×C₁₆
Fitting	C₁₇×#504	C₄
prime radical	#41×D₁₇	C₈
cosocle	#63214	C₂⁴
hypercenter	#504	#595
hypocenter	C₁₇	#33974
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #14545208	C₂	semidirect
2. #14545207	C₂	semidirect
3. C₂×#11763742	C₂	semidirect
4. #14545205	C₂	semidirect
8. #11763742	C₂²	semidirect
9. #11763741	C₂²	semidirect
10. #11763740	C₂²	semidirect
11. C₂×#10587467	C₂²	semidirect
122. C₁₇	#33974	semidirect

Maximal subgroups

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

subgroup	index
#14545208	2
#14545207	2
C₂×#11763742	2
#14545205	2
D₁₇×#504	2
#33974	17