Subgroups — #17186986

Order 4352.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541887	C₂	1	2	yes	maximal
2	#14530389	C₂	1	6	yes	maximal, complemented
3	C₂×#11762587	C₂	1	6	yes	maximal, complemented
4	C₁₇×#2875	C₂	1	1	yes	maximal, characteristic
5	#11763226	C₄	1	16	yes	complemented
6	C₂×#10587483	C₂²	1	12	yes	—
7	#11762587	C₂²	1	4	yes	—
8	#11762587	C₂²	1	12	yes	—
9	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
10	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
11	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
12	C₁₇×#405	C₄	1	3	yes	—
13	C₁₇×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₁₇	C₄	1	3	yes	—
15	C₂×C₁₇×#97	C₂²	1	3	yes	—
16	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₁₇×#131	—	2	4	no	—
18	C₁₇×#131	C₂×C₄	1	8	yes	—
19	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
20	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
23	C₄×Q₈×C₁₇	—	2	1	no	—
24	C₄×D₄×C₁₇	—	2	3	no	—
25	C₁₇×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
27	C₁₇×#97	C₂×C₄	1	3	yes	—
28	C₁₇×#97	C₂×C₄	1	3	yes	—
29	C₁₇×#97	C₂³	1	1	yes	characteristic
30	C₁₇×#97	C₂³	1	3	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	6	yes	—
32	#10587483	C₂×C₄	1	24	yes	complemented
33	#10587483	C₂³	1	8	yes	—
34	C₂×#63214	C₂×C₄	1	24	yes	complemented
35	#41×C₁₇	C₄²	1	4	yes	—
36	#41×C₁₇	M₁₆	1	4	yes	—
37	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
38	C₂×D₄×C₁₇	M₁₆	1	3	yes	—
39	C₂²×C₄×C₁₇	—	2	3	no	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
41	M₁₆×C₁₇	—	2	6	no	—
42	M₁₆×C₁₇	C₄²	1	12	yes	—
43	C₂×C₈×C₁₇	—	2	6	no	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	12	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₁₇	—	2	3	no	—
50	C₂²⋊C₄×C₁₇	—	2	3	no	—
51	C₄²×C₁₇	—	2	3	no	—
52	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
53	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	#63214	C₂²×C₄	1	16	yes	—
55	#33329	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₂×M₁₆	1	3	yes	—
57	Q₈×C₁₇	#97	1	4	yes	—
58	D₄×C₁₇	#97	1	12	yes	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	3	no	—
61	C₂×C₄×C₁₇	—	8	3	no	—
62	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
64	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
65	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	—	2	3	no	—
68	C₈×C₁₇	C₂×C₄²	1	6	yes	—
69	C₈×C₁₇	C₂³×C₄	1	2	yes	—
70	#3407	—	17	2	no	—
71	#2875	D₁₇	1	1	yes	characteristic
72	#2110	—	17	6	no	—
73	C₂×#347	—	17	6	no	—
74	C₂²×C₁₇	—	4	3	no	—
75	C₂²×C₁₇	C₂×#97	1	6	yes	—
76	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
77	C₄×C₁₇	—	4	3	no	—
78	C₄×C₁₇	—	4	1	no	—
79	C₄×C₁₇	C₂×#97	1	6	yes	—
80	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
81	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
82	C₂×#131	Dic₁₇	1	1	yes	characteristic
83	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
84	C₄×#41	Dic₁₇	1	1	yes	characteristic
85	#504	—	17	16	no	—
86	C₂×M₃₂	—	17	12	no	—
87	#405	Dic₁₇	1	3	yes	—
88	#405	C₂×D₁₇	1	3	yes	—
89	C₄×M₁₆	Dic₁₇	1	3	yes	—
90	C₂×#97	C₂×D₁₇	1	3	yes	—
91	#347	—	17	4	no	—
92	#347	—	17	12	no	—
93	C₂×C₁₇	—	2	1	no	—
94	C₂×C₁₇	—	2	6	no	—
95	C₂×C₁₇	C₂²×#97	1	1	yes	characteristic
96	C₂×#41	C₂×Dic₁₇	1	1	yes	characteristic
97	#131	—	2	4	no	—
98	#131	C₄×D₁₇	1	8	yes	—
99	C₂×M₁₆	C₂×Dic₁₇	1	3	yes	—
100	C₂×M₁₆	C₂×Dic₁₇	1	3	yes	—
101	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
102	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×D₄	—	2	3	no	—
105	#117	C₂×Dic₁₇	1	3	yes	—
106	C₂×C₄²	C₂×Dic₁₇	1	3	yes	—
107	M₃₂	—	17	24	no	—
108	M₃₂	—	17	8	no	—
109	C₂×C₁₆	—	17	24	no	—
110	#97	C₂×Dic₁₇	1	3	yes	—
111	#97	C₂×Dic₁₇	1	3	yes	—
112	#97	C₂²×D₁₇	1	1	yes	characteristic
113	#97	C₂²×D₁₇	1	3	yes	—
114	C₄×C₈	C₂×Dic₁₇	1	6	yes	—
115	C₁₇	#33329	1	1	yes	characteristic, complemented, Sylow
116	#41	#63223	1	4	yes	—
117	#41	C₄×Dic₁₇	1	4	yes	—
118	C₂×Q₈	#63223	1	1	yes	characteristic
119	C₂×D₄	#63223	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
122	M₁₆	—	2	6	no	—
123	M₁₆	C₄×Dic₁₇	1	12	yes	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂×C₄×D₁₇	1	12	yes	—
126	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
127	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
129	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
130	C₄⋊C₄	—	2	3	no	—
131	C₂²⋊C₄	—	2	3	no	—
132	C₄²	—	2	3	no	—
133	C₄²	C₂²×Dic₁₇	1	3	yes	—
134	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
135	C₁₆	—	17	16	no	—
136	C₂³	C₂×#63223	1	3	yes	—
137	Q₈	#10587474	1	4	yes	—
138	D₄	#10587474	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×#63223	1	3	yes	—
143	C₂×C₄	C₂×#63223	1	6	yes	—
144	C₂×C₄	C₂×C₄×Dic₁₇	1	6	yes	—
145	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄×Dic₁₇	1	6	yes	—
149	C₈	C₂²×C₄×D₁₇	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#10587474	1	6	yes	—
152	C₂²	C₂²×#63223	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×#10587474	1	6	yes	—
156	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
157	C₄	C₂²×#63223	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#10587474	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₁₇
commutator	C₄×C₁₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2875	C₂
socle	C₂×C₁₇	C₂²×#97
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2875	D₁₇
hypocenter	C₁₇	#33329
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
2. #14530389	C₂	semidirect
3. C₂×#11762587	C₂	semidirect
5. #11763226	C₄	semidirect
32. #10587483	C₂×C₄	semidirect
34. C₂×#63214	C₂×C₄	semidirect
115. C₁₇	#33329	semidirect

Maximal subgroups

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

subgroup	index
#14541887	2
#14530389	2
C₂×#11762587	2
C₁₇×#2875	2
#33329	17