Subgroups — #17186984

Order 4352.

716 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541887	C₂	1	1	yes	maximal, characteristic
2	#14537210	C₂	1	1	yes	maximal, characteristic
3	#14530388	C₂	1	6	yes	maximal, complemented
4	C₂×#11762586	C₂	1	6	yes	maximal, complemented
5	C₁₇×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587634	C₄	1	1	yes	characteristic
7	#41×Dic₁₇	C₄	1	1	yes	characteristic
8	#11763226	C₄	1	8	yes	complemented
9	C₂×#10587483	C₂²	1	6	yes	—
10	#11762767	C₄	1	3	yes	—
11	#11762764	C₂²	1	3	yes	—
12	M₁₆×Dic₁₇	C₄	1	3	yes	—
13	C₂×#10587486	C₂²	1	3	yes	—
14	#11762586	C₂²	1	4	yes	—
15	#11762586	C₂²	1	12	yes	—
16	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
17	C₁₇×#504	C₄	1	8	yes	complemented
18	C₂×C₁₇×M₃₂	C₂²	1	6	yes	—
19	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₁₇×#131	C₂×C₄	1	4	yes	—
21	C₁₇×#131	C₂×C₄	1	4	yes	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
24	#10587634	—	2	4	no	—
25	Q₈×Dic₁₇	—	2	1	no	—
26	D₄×Dic₁₇	—	2	3	no	—
27	#10587612	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
29	C₂×#63223	C₂×C₄	1	3	yes	—
30	C₂²×#3622	C₂×C₄	1	3	yes	—
31	C₁₇×M₃₂	C₂×C₄	1	12	yes	complemented
32	C₁₇×M₃₂	C₂³	1	4	yes	—
33	C₂×C₁₆×C₁₇	C₂×C₄	1	12	yes	complemented
34	#10587486	C₂×C₄	1	3	yes	—
35	#10587486	C₂×C₄	1	3	yes	—
36	#10587486	C₂³	1	1	yes	characteristic
37	#10587486	C₂³	1	3	yes	—
38	C₈×Dic₁₇	C₂×C₄	1	6	yes	—
39	#10587483	C₂×C₄	1	12	yes	complemented
40	#10587483	C₂³	1	4	yes	—
41	C₂×#63214	C₂×C₄	1	12	yes	complemented
42	#41×C₁₇	C₄²	1	4	yes	—
43	#41×C₁₇	M₁₆	1	4	yes	—
44	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
45	C₂×D₄×C₁₇	M₁₆	1	3	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₁₇	—	2	3	no	—
48	M₁₆×C₁₇	C₄²	1	12	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
52	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
53	C₂²⋊Dic₁₇	—	2	3	no	—
54	C₄⋊Dic₁₇	—	2	3	no	—
55	C₄×Dic₁₇	—	2	3	no	—
56	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
57	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
58	#63223	—	2	6	no	—
59	C₂×#3622	—	2	6	no	—
60	C₂×#3622	C₂²×C₄	1	3	yes	—
61	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
62	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
63	#63214	C₂²×C₄	1	8	yes	—
64	#33329	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂×M₁₆	1	3	yes	—
66	Q₈×C₁₇	#97	1	4	yes	—
67	D₄×C₁₇	#97	1	12	yes	—
68	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
69	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
70	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
71	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₁₇	—	2	3	no	—
73	C₂×Dic₁₇	—	2	1	no	—
74	C₂×Dic₁₇	—	8	3	no	—
75	C₈×C₁₇	C₂×C₄²	1	6	yes	—
76	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
78	#3622	—	2	1	no	—
79	#3622	—	2	3	no	—
80	#3407	—	17	1	no	—
81	#3407	D₁₇	1	1	yes	characteristic
82	#2875	—	17	1	no	—
83	#2110	—	17	6	no	—
84	C₂×#347	—	17	6	no	—
85	C₂²×C₁₇	—	4	3	no	—
86	C₂²×C₁₇	C₂×#97	1	6	yes	—
87	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
88	C₄×C₁₇	C₂×#97	1	6	yes	—
89	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
90	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
91	Dic₁₇	—	4	3	no	—
92	Dic₁₇	—	4	1	no	—
93	C₂×#131	—	17	1	no	—
94	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
95	C₄×#41	—	17	1	no	—
96	#504	—	17	8	no	—
97	#504	Dic₁₇	1	8	yes	complemented
98	C₂×M₃₂	—	17	6	no	—
99	C₂×M₃₂	C₂×D₁₇	1	6	yes	—
100	#405	—	17	3	no	—
101	#405	—	17	3	no	—
102	C₄×M₁₆	—	17	3	no	—
103	C₂×#97	—	17	3	no	—
104	#347	—	17	4	no	—
105	#347	—	17	12	no	—
106	C₂×C₁₇	—	2	1	no	—
107	C₂×C₁₇	—	2	6	no	—
108	C₂×C₁₇	C₂²×#97	1	1	yes	characteristic
109	C₂×#41	C₄×D₁₇	1	1	yes	characteristic
110	#131	—	34	4	no	—
111	#131	C₄×D₁₇	1	4	yes	—
112	#131	C₂×Dic₁₇	1	4	yes	—
113	C₂×M₁₆	—	17	3	no	—
114	C₂×M₁₆	C₄×D₁₇	1	3	yes	—
115	C₂²×C₈	—	17	3	no	—
116	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
117	C₄×Q₈	—	34	1	no	—
118	C₄×D₄	—	34	3	no	—
119	#117	—	17	3	no	—
120	C₂×C₄²	—	17	3	no	—
121	M₃₂	—	17	12	no	—
122	M₃₂	—	17	4	no	—
123	M₃₂	C₂×Dic₁₇	1	12	yes	complemented
124	M₃₂	C₂²×D₁₇	1	4	yes	—
125	C₂×C₁₆	—	17	12	no	—
126	C₂×C₁₆	C₂×Dic₁₇	1	12	yes	complemented
127	#97	—	17	3	no	—
128	#97	—	17	3	no	—
129	#97	—	17	1	no	—
130	#97	—	17	3	no	—
131	C₄×C₈	—	17	6	no	—
132	C₁₇	#33329	1	1	yes	characteristic, complemented, Sylow
133	#41	#63218	1	4	yes	—
134	#41	C₄×Dic₁₇	1	4	yes	—
135	C₂×Q₈	#63218	1	1	yes	characteristic
136	C₂×D₄	#63218	1	3	yes	—
137	C₂²×C₄	—	34	3	no	—
138	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
139	M₁₆	—	34	6	no	—
140	M₁₆	C₄×Dic₁₇	1	12	yes	—
141	C₂×C₈	—	17	3	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	34	6	no	—
144	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
145	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
146	C₂×C₈	C₂²×Dic₁₇	1	6	yes	—
147	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
148	C₄⋊C₄	—	34	3	no	—
149	C₂²⋊C₄	—	34	3	no	—
150	C₄²	—	17	3	no	—
151	C₄²	—	17	1	no	—
152	C₄²	—	34	3	no	—
153	C₁₆	—	17	8	no	—
154	C₁₆	C₂²×Dic₁₇	1	8	yes	—
155	C₂³	C₂×#63218	1	3	yes	—
156	Q₈	#10587486	1	4	yes	—
157	D₄	#10587486	1	12	yes	—
158	C₂×C₄	—	34	1	no	—
159	C₂×C₄	—	34	3	no	—
160	C₂×C₄	—	136	3	no	—
161	C₂×C₄	C₂×#63218	1	3	yes	—
162	C₂×C₄	C₂×#63218	1	6	yes	—
163	C₂×C₄	C₂×C₄×Dic₁₇	1	6	yes	—
164	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
165	C₈	—	34	1	no	—
166	C₈	—	34	3	no	—
167	C₈	C₂×C₄×Dic₁₇	1	6	yes	—
168	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
169	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#10587486	1	6	yes	—
172	C₂²	C₂²×#63218	1	1	yes	characteristic
173	C₄	—	68	1	no	—
174	C₄	—	68	3	no	—
175	C₄	C₂×#10587486	1	6	yes	—
176	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
177	C₄	C₂²×#63218	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#10587486	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₁₇
commutator	C₄×C₁₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3407	C₂
socle	C₂×C₁₇	C₂²×#97
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#3407	D₁₇
hypocenter	C₁₇	#33329
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #14530388	C₂	semidirect
4. C₂×#11762586	C₂	semidirect
8. #11763226	C₄	semidirect
17. C₁₇×#504	C₄	semidirect
31. C₁₇×M₃₂	C₂×C₄	semidirect
33. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
39. #10587483	C₂×C₄	semidirect
41. C₂×#63214	C₂×C₄	semidirect
97. #504	Dic₁₇	semidirect
123. M₃₂	C₂×Dic₁₇	semidirect
126. C₂×C₁₆	C₂×Dic₁₇	semidirect
132. 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
#14537210	2
#14530388	2
C₂×#11762586	2
C₁₇×#3407	2
#33329	17