Subgroups — #17043188

Order 4352.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537719	C₂	1	2	yes	maximal
2	#14529762	C₂	1	2	yes	maximal
3	#14529538	C₂	1	4	yes	maximal
4	#14528956	C₂	1	1	yes	maximal, characteristic
5	#14528670	C₂	1	4	yes	maximal
6	C₄×#10587486	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
8	#11762940	—	2	4	no	—
9	Q₈×#3622	—	2	4	no	—
10	C₂×#10587490	C₂²	1	4	yes	—
11	#11762836	C₂²	1	4	yes	—
12	#11762833	C₂²	1	2	yes	—
13	#11762765	C₂²	1	4	yes	—
14	C₂×#10587486	C₂²	1	2	yes	—
15	C₂×#10587486	C₂²	1	2	yes	—
16	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
17	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
18	#11762568	C₂²	1	8	yes	—
19	C₁₇×#533	C₂²	1	2	yes	—
20	C₁₇×#500	C₄	1	2	yes	complemented
21	C₁₇×#499	C₄	1	4	yes	complemented
22	C₁₇×#498	C₄	1	2	yes	complemented
23	C₁₇×#427	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
26	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₁₇×#125	C₂×C₄	1	4	yes	—
28	C₁₇×C₂²⋊Q₈	—	2	4	no	—
29	C₄×Q₈×C₁₇	—	2	4	no	—
30	C₁₇×#117	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₁₇	—	2	2	no	—
33	C₂×C₄×Dic₁₇	C₂³	1	2	yes	—
34	C₂×#63223	C₂³	1	4	yes	—
35	C₂²×#3622	C₂³	1	1	yes	characteristic
36	C₂²×#3622	C₂³	1	1	yes	characteristic
37	C₁₇×#107	C₂×C₄	1	8	yes	—
38	C₁₇×#106	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	#10587490	—	2	8	no	—
43	#10587486	—	2	4	no	—
44	#10587486	Q₈	1	8	yes	complemented
45	C₄⋊#3622	—	2	8	no	—
46	C₄⋊#3622	—	2	4	no	—
47	#10587474	—	2	4	no	—
48	C₄×#3622	—	2	4	no	—
49	C₄×#3622	D₄	1	4	yes	—
50	C₂×Q₈×C₁₇	—	2	4	no	—
51	C₂²×C₄×C₁₇	#41	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₁₇	#41	1	4	yes	—
58	C₄⋊C₄×C₁₇	—	2	8	no	—
59	C₄⋊C₄×C₁₇	—	2	4	no	—
60	C₄⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
61	C₂²⋊C₄×C₁₇	—	2	4	no	—
62	C₄²×C₁₇	—	2	4	no	—
63	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂×D₄	1	2	yes	—
66	C₄×Dic₁₇	—	2	2	no	—
67	C₄×Dic₁₇	—	4	4	no	—
68	C₄×Dic₁₇	C₂×Q₈	1	4	yes	—
69	#63223	—	2	8	no	—
70	C₂×#3622	—	2	2	no	—
71	C₂×#3622	—	2	4	no	—
72	C₂×#3622	C₂×D₄	1	2	yes	—
73	C₂×#3622	C₂×D₄	1	2	yes	—
74	C₂×#3622	C₂×Q₈	1	4	yes	—
75	#21734	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
77	Q₈×C₁₇	—	4	8	no	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	—	2	8	no	—
80	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
86	C₂×Dic₁₇	—	2	4	no	—
87	C₂×Dic₁₇	—	4	4	no	—
88	C₈×C₁₇	C₄×Q₈	1	4	yes	—
89	C₈×C₁₇	C₄×Q₈	1	4	yes	—
90	#3622	—	2	2	no	—
91	#3622	—	4	4	no	—
92	#3622	C₄⋊Q₈	1	4	yes	—
93	#3160	D₁₇	1	1	yes	characteristic
94	#2960	—	17	2	no	—
95	#1994	—	17	2	no	—
96	#1946	—	17	4	no	—
97	#1848	—	17	1	no	—
98	#1773	—	17	4	no	—
99	C₄×#97	—	17	1	no	—
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₁₇	#533	1	2	yes	—
103	C₄×C₁₇	—	2	2	no	—
104	C₄×C₁₇	—	4	8	no	—
105	C₄×C₁₇	#444	1	4	yes	—
106	C₄×C₁₇	C₂×C₄×Q₈	1	2	yes	—
107	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₄×C₁₇	#532	1	1	yes	characteristic
109	Dic₁₇	—	4	4	no	—
110	#533	C₂×D₁₇	1	2	yes	—
111	#500	Dic₁₇	1	2	yes	complemented
112	#499	Dic₁₇	1	4	yes	complemented
113	#498	Dic₁₇	1	2	yes	complemented
114	#446	—	34	4	no	—
115	C₈×Q₈	—	34	4	no	—
116	C₂×#108	—	17	4	no	—
117	#427	C₂×D₁₇	1	4	yes	—
118	#424	—	17	4	no	—
119	#423	—	17	2	no	—
120	#405	—	17	4	no	—
121	C₂×#97	—	17	2	no	—
122	C₂×#97	—	17	2	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	C₄³	—	17	1	no	—
126	#339	—	17	8	no	—
127	C₂×C₁₇	—	2	2	no	—
128	C₂×C₁₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#444	1	2	yes	—
130	C₂×M₁₆	—	17	4	no	—
131	C₂²×C₈	—	17	1	no	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
134	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
135	#125	C₂×Dic₁₇	1	4	yes	—
136	C₂²⋊Q₈	—	2	4	no	—
137	C₄×Q₈	—	2	4	no	—
138	#117	C₂²×D₁₇	1	4	yes	—
139	C₂×C₄²	—	17	2	no	—
140	C₂×C₄²	—	34	2	no	—
141	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
142	#108	—	34	8	no	—
143	#107	C₂×Dic₁₇	1	8	yes	—
144	#106	C₂×Dic₁₇	1	8	yes	—
145	C₄⋊C₈	—	34	8	no	—
146	C₄⋊C₈	—	34	4	no	—
147	#97	—	17	8	no	—
148	#97	—	34	4	no	—
149	#97	—	34	4	no	—
150	C₄×C₈	—	17	4	no	—
151	C₄×C₈	—	34	4	no	—
152	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
154	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
155	C₁₇	#21734	1	1	yes	characteristic, complemented, Sylow
156	C₂×Q₈	—	2	4	no	—
157	C₂²×C₄	—	34	2	no	—
158	C₂²×C₄	#63254	1	2	yes	—
159	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
160	M₁₆	—	34	8	no	—
161	C₂×C₈	—	17	2	no	—
162	C₂×C₈	—	17	4	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	34	2	no	—
165	C₂×C₈	—	34	4	no	—
166	C₂×C₈	Q₈×D₁₇	1	4	yes	—
167	C₂×C₈	#63256	1	4	yes	—
168	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
169	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	C₂²×Dic₁₇	1	8	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	17	4	no	—
176	C₄²	—	34	2	no	—
177	C₄²	—	68	4	no	—
178	C₄²	D₄×D₁₇	1	2	yes	—
179	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
180	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
181	C₂³	C₂×#63254	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	34	4	no	—
186	C₂×C₄	—	68	4	no	—
187	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
188	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
189	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₈×D₁₇	1	2	yes	—
191	C₂×C₄	C₂×#63256	1	2	yes	—
192	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
193	C₈	—	17	4	no	—
194	C₈	—	34	2	no	—
195	C₈	—	68	4	no	—
196	C₈	Q₈×Dic₁₇	1	4	yes	—
197	C₈	Q₈×Dic₁₇	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
200	C₂²	#11763364	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	8	no	—
203	C₄	—	68	4	no	—
204	C₄	#11762931	1	4	yes	—
205	C₄	C₂×Q₈×Dic₁₇	1	2	yes	—
206	C₄	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
207	C₄	#11763360	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄⋊Q₈×Dic₁₇	1	1	yes	characteristic
210	C₂	C₂×#11762931	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₂×#41
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#21734
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
20. C₁₇×#500	C₄	semidirect
21. C₁₇×#499	C₄	semidirect
22. C₁₇×#498	C₄	semidirect
44. #10587486	Q₈	semidirect
111. #500	Dic₁₇	semidirect
112. #499	Dic₁₇	semidirect
113. #498	Dic₁₇	semidirect
155. C₁₇	#21734	semidirect

Maximal subgroups

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

subgroup	index
#14537719	2
#14529762	2
#14529538	2
#14528956	2
#14528670	2
C₄×#10587486	2
C₁₇×#3160	2
#21734	17