Subgroups — #16924749

Order 4352.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528530	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
3	#14526611	C₂	1	4	yes	maximal
4	C₄×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#3622	C₄	1	4	yes	—
6	C₂×C₄×#3622	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	#11762564	C₄	1	4	yes	—
9	C₂×C₁₇×#97	C₄	1	2	yes	—
10	C₂×C₁₇×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
12	C₄³×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×#3622	C₂×C₄	1	4	yes	—
20	C₂²×#3622	C₂×C₄	1	4	yes	—
21	C₁₇×#97	—	2	4	no	—
22	C₁₇×#97	C₈	1	8	yes	complemented
23	C₄×C₈×C₁₇	—	2	4	no	—
24	C₄×#3622	—	2	8	no	—
25	C₄×#3622	—	2	4	no	—
26	C₄×#3622	—	2	4	no	—
27	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
28	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
29	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇	—	2	4	no	—
33	C₂×C₈×C₁₇	—	2	4	no	—
34	C₂×C₈×C₁₇	C₂×C₈	1	8	yes	—
35	C₄²×C₁₇	—	2	4	no	—
36	C₄²×C₁₇	—	2	4	no	—
37	C₄²×C₁₇	C₂×C₈	1	4	yes	—
38	C₄²×C₁₇	M₁₆	1	4	yes	—
39	C₄²×C₁₇	#41	1	4	yes	—
40	C₂×#3622	—	2	4	no	—
41	C₂×#3622	—	2	4	no	—
42	C₂×#3622	—	2	4	no	—
43	C₂×#3622	—	2	8	no	—
44	C₂×#3622	M₁₆	1	8	yes	—
45	#8132	—	17	1	no	maximal, Sylow
46	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
52	C₂×C₄×C₁₇	#97	1	4	yes	—
53	C₂×C₄×C₁₇	#117	1	2	yes	—
54	C₂×C₄×C₁₇	#117	1	2	yes	—
55	C₂×C₄×C₁₇	#117	1	2	yes	—
56	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₁₇	#131	1	4	yes	—
60	C₈×C₁₇	—	2	8	no	—
61	#3622	—	2	8	no	—
62	#3622	—	4	4	no	—
63	#1744	—	17	1	no	—
64	C₄×#97	D₁₇	1	1	yes	characteristic
65	C₄²×C₈	—	17	1	no	—
66	#1273	—	17	4	no	—
67	C₂²×C₁₇	#376	1	1	yes	characteristic
68	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
70	C₂²×C₁₇	C₄×M₁₆	1	2	yes	—
71	C₂²×C₁₇	#405	1	2	yes	—
72	C₄×C₁₇	—	2	4	no	—
73	C₄×C₁₇	—	2	4	no	—
74	C₄×C₁₇	#431	1	4	yes	—
75	C₄×C₁₇	#432	1	4	yes	—
76	C₄×C₁₇	#433	1	4	yes	—
77	C₂×#97	Dic₁₇	1	2	yes	—
78	C₂×#97	Dic₁₇	1	2	yes	—
79	C₂×C₄×C₈	—	17	4	no	—
80	C₂×C₄×C₈	—	17	2	no	—
81	C₂×C₄×C₈	—	17	2	no	—
82	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
83	C₄³	C₂×D₁₇	1	1	yes	characteristic
84	#336	—	17	4	no	—
85	C₂×C₁₇	C₈×M₁₆	1	2	yes	—
86	C₂×C₁₇	#1451	1	2	yes	—
87	C₂×C₁₇	#1744	1	1	yes	characteristic
88	C₂×C₁₇	#1745	1	1	yes	characteristic
89	C₂×C₁₇	#1746	1	1	yes	characteristic
90	C₂²×C₈	—	17	4	no	—
91	C₂²×C₈	—	17	4	no	—
92	C₂²×C₈	C₂×Dic₁₇	1	4	yes	—
93	C₂×C₄²	C₄×D₁₇	1	2	yes	—
94	C₂×C₄²	C₄×D₁₇	1	2	yes	—
95	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
98	#97	—	2	4	no	—
99	#97	#3622	1	8	yes	complemented
100	C₄×C₈	—	2	4	no	—
101	C₄×C₈	—	34	8	no	—
102	C₄×C₈	—	34	4	no	—
103	C₄×C₈	—	34	4	no	—
104	C₁₇	#8132	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
106	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	17	8	no	—
113	C₂×C₈	—	34	4	no	—
114	C₂×C₈	—	34	4	no	—
115	C₂×C₈	—	34	4	no	—
116	C₂×C₈	—	34	8	no	—
117	C₂×C₈	C₂×#3622	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×#3622	1	4	yes	—
121	C₄²	#63223	1	4	yes	—
122	C₄²	#63252	1	4	yes	—
123	C₂³	C₂×C₄×Dic₁₇	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×#3622	1	4	yes	—
129	C₂×C₄	#10587474	1	4	yes	—
130	C₂×C₄	#10587558	1	2	yes	—
131	C₂×C₄	#10587558	1	2	yes	—
132	C₂×C₄	M₁₆×D₁₇	1	4	yes	—
133	C₂×C₄	#10587593	1	4	yes	—
134	C₂×C₄	C₂²×#3622	1	2	yes	—
135	C₂×C₄	C₂×#63223	1	2	yes	—
136	C₂×C₄	#10587612	1	2	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	34	8	no	—
139	C₈	—	68	4	no	—
140	C₂²	#11762655	1	1	yes	characteristic
141	C₂²	C₂×C₄×#3622	1	1	yes	characteristic
142	C₂²	C₂×#10587474	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₁₇	1	2	yes	—
144	C₂²	#11762767	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#11762862	1	4	yes	—
148	C₄	#11762864	1	4	yes	—
149	C₄	#11762871	1	4	yes	—
150	C₂	M₁₆×#3622	1	2	yes	—
151	C₂	#14527071	1	2	yes	—
152	C₂	#14528530	1	1	yes	characteristic
153	C₂	#14528532	1	1	yes	characteristic
154	C₂	#14528537	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₄×C₁₇×#97	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#97	D₁₇
hypocenter	C₁₇	#8132
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
22. C₁₇×#97	C₈	semidirect
99. #97	#3622	semidirect
104. C₁₇	#8132	semidirect

Maximal subgroups

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

subgroup	index
#14528530	2
C₄²×#3622	2
#14526611	2
C₄×C₁₇×#97	2
#8132	17