Subgroups — #17187572

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 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, complemented
2	#14530626	C₂	1	4	yes	maximal, complemented
3	#14530621	C₂	1	4	yes	maximal, complemented
4	#14530550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762593	C₂	1	2	yes	maximal, complemented
6	#14530381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#2955	C₂	1	1	yes	maximal, characteristic
8	#11763226	—	2	4	no	—
9	#11763226	C₂²	1	8	yes	complemented
10	C₂×#10587483	—	2	4	no	—
11	C₂×#10587483	C₂²	1	2	yes	—
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	C₂²×#63214	C₂²	1	1	yes	characteristic
14	#11762630	C₂²	1	4	yes	complemented
15	#11762593	C₂²	1	4	yes	complemented
16	#11762593	C₂²	1	4	yes	complemented
17	#11762585	C₂²	1	2	yes	complemented
18	C₄×#63214	C₂²	1	2	yes	complemented
19	C₁₇×#575	C₄	1	1	yes	characteristic
20	C₂×C₁₇×#131	C₂²	1	2	yes	—
21	C₁₇×#444	C₄	1	4	yes	—
22	C₁₇×#443	C₂²	1	4	yes	—
23	C₁₇×#430	C₄	1	1	yes	characteristic
24	C₁₇×#421	C₄	1	2	yes	—
25	C₁₇×#405	C₂²	1	1	yes	characteristic
26	C₂×#41×C₁₇	C₂×C₄	1	2	yes	—
27	C₁₇×#137	C₈	1	4	yes	—
28	C₁₇×AGL(1,ℤ/8)	C₈	1	4	yes	—
29	C₁₇×#135	C₂×C₄	1	4	yes	—
30	C₁₇×#131	—	2	2	no	—
31	C₁₇×#131	—	2	2	no	—
32	C₁₇×#131	C₂×C₄	1	4	yes	—
33	C₁₇×#131	C₂³	1	4	yes	—
34	C₂×M₁₆×C₁₇	—	2	2	no	—
35	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	—	2	2	no	—
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
40	C₁₇×#108	C₂×C₄	1	4	yes	—
41	C₁₇×#104	C₂×C₄	1	8	yes	—
42	C₁₇×#97	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₇	C₂³	1	2	yes	—
44	#10587483	—	2	8	no	—
45	#10587483	—	2	2	no	—
46	#10587483	D₄	1	4	yes	—
47	#10587483	C₂³	1	4	yes	—
48	C₂×#63214	—	2	8	no	—
49	C₂×#63214	—	2	2	no	—
50	C₂×#63214	—	2	2	no	—
51	C₂×#63214	C₂³	1	1	yes	characteristic
52	C₂×#63214	C₂³	1	1	yes	characteristic
53	C₂×#63214	C₂³	1	2	yes	—
54	#41×C₁₇	—	2	2	no	—
55	#41×C₁₇	—	2	2	no	—
56	#41×C₁₇	C₂×C₈	1	4	yes	—
57	#41×C₁₇	C₂²×C₄	1	4	yes	—
58	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₁₇	—	2	2	no	—
60	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₇	—	2	2	no	—
62	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
64	SD₁₆×C₁₇	C₂×C₈	1	8	yes	—
65	D₈×C₁₇	C₂×C₈	1	4	yes	—
66	M₁₆×C₁₇	—	2	2	no	—
67	M₁₆×C₁₇	—	2	4	no	—
68	M₁₆×C₁₇	—	2	4	no	—
69	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
70	M₁₆×C₁₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	—	2	4	no	—
73	C₂×C₈×C₁₇	—	2	4	no	—
74	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₇	#41	1	2	yes	—
78	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
82	#63214	—	2	4	no	—
83	#63214	—	2	1	no	—
84	#63214	—	2	1	no	—
85	#63214	C₂×D₄	1	4	yes	—
86	#33442	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	—	2	2	no	—
88	C₂³×C₁₇	#131	1	1	yes	characteristic
89	Q₈×C₁₇	—	2	2	no	—
90	Q₈×C₁₇	C₂²×C₈	1	4	yes	—
91	D₄×C₁₇	—	2	4	no	—
92	D₄×C₁₇	—	2	4	no	—
93	D₄×C₁₇	—	2	2	no	—
94	D₄×C₁₇	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₁₇	—	2	2	no	—
96	C₂×C₄×C₁₇	—	2	4	no	—
97	C₂×C₄×C₁₇	—	2	4	no	—
98	C₂×C₄×C₁₇	—	2	2	no	—
99	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₁₇	—	2	4	no	—
104	C₈×C₁₇	C₄×D₄	1	2	yes	—
105	C₈×C₁₇	C₂²×C₈	1	4	yes	—
106	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
108	#3407	—	17	2	no	—
109	#2955	D₁₇	1	1	yes	characteristic
110	#2171	—	17	4	no	—
111	#2170	—	17	4	no	—
112	#2153	—	17	1	no	—
113	C₂×#350	—	17	2	no	—
114	#2108	—	17	1	no	—
115	C₂²×C₁₇	—	2	4	no	—
116	C₂²×C₁₇	—	4	1	no	—
117	C₂²×C₁₇	—	8	2	no	—
118	C₂²×C₁₇	C₈×D₄	1	2	yes	—
119	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
120	C₄×C₁₇	—	2	4	no	—
121	C₄×C₁₇	—	4	2	no	—
122	C₄×C₁₇	C₈×D₄	1	2	yes	—
123	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
125	#575	Dic₁₇	1	1	yes	characteristic
126	C₂×#131	C₂×D₁₇	1	2	yes	—
127	#504	—	17	8	no	—
128	#504	—	34	4	no	—
129	C₂×M₃₂	—	17	1	no	—
130	C₂×M₃₂	—	17	2	no	—
131	C₂×M₃₂	—	34	4	no	—
132	C₂²×C₁₆	—	17	1	no	—
133	#444	Dic₁₇	1	4	yes	—
134	#443	C₂×D₁₇	1	4	yes	—
135	#430	Dic₁₇	1	1	yes	characteristic
136	#421	Dic₁₇	1	2	yes	—
137	#405	C₂×D₁₇	1	1	yes	characteristic
138	#364	—	17	4	no	—
139	#350	—	17	4	no	—
140	#350	—	17	4	no	—
141	#346	—	17	2	no	—
142	C₄×C₁₆	—	17	2	no	—
143	C₂×C₁₇	—	2	1	no	—
144	C₂×C₁₇	—	2	2	no	—
145	C₂×C₁₇	—	4	4	no	—
146	C₂×C₁₇	C₂×C₈×D₄	1	1	yes	characteristic
147	C₂×#41	C₂×Dic₁₇	1	2	yes	—
148	#137	#3622	1	4	yes	—
149	AGL(1,ℤ/8)	#3622	1	4	yes	—
150	#135	C₂×Dic₁₇	1	4	yes	—
151	#131	—	2	2	no	—
152	#131	—	2	2	no	—
153	#131	C₂×Dic₁₇	1	4	yes	—
154	#131	C₂²×D₁₇	1	4	yes	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
157	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
160	#117	C₂×Dic₁₇	1	1	yes	characteristic
161	M₃₂	—	17	4	no	—
162	M₃₂	—	17	4	no	—
163	M₃₂	—	34	8	no	—
164	M₃₂	—	34	2	no	—
165	C₂×C₁₆	—	17	2	no	—
166	C₂×C₁₆	—	17	1	no	—
167	C₂×C₁₆	—	17	1	no	—
168	C₂×C₁₆	—	34	8	no	—
169	C₂×C₁₆	—	34	2	no	—
170	C₂×C₁₆	—	34	2	no	—
171	#108	C₂×Dic₁₇	1	4	yes	—
172	#104	C₂×Dic₁₇	1	8	yes	—
173	#97	C₂×Dic₁₇	1	2	yes	—
174	C₄×C₈	C₂²×D₁₇	1	2	yes	—
175	C₁₇	#33442	1	1	yes	characteristic, complemented, Sylow
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×#3622	1	4	yes	—
179	#41	C₂²×Dic₁₇	1	4	yes	—
180	C₂×Q₈	C₂×#3622	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×#3622	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
185	Q₁₆	C₂×#3622	1	4	yes	—
186	SD₁₆	C₂×#3622	1	8	yes	—
187	D₈	C₂×#3622	1	4	yes	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	C₂×#3622	1	4	yes	—
192	M₁₆	C₂²×Dic₁₇	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	D₄×D₁₇	1	2	yes	—
197	C₂×C₈	#63254	1	2	yes	—
198	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
199	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
200	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂²×Dic₁₇	1	2	yes	—
204	C₁₆	—	17	4	no	—
205	C₁₆	—	34	4	no	—
206	C₁₆	—	34	1	no	—
207	C₁₆	—	34	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#10587634	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×#3622	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×#3622	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₂²×#3622	1	2	yes	—
221	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
222	C₂×C₄	#10587634	1	1	yes	characteristic
223	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂²×#3622	1	4	yes	—
226	C₈	D₄×Dic₁₇	1	2	yes	—
227	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
228	C₈	C₂×#63254	1	1	yes	characteristic
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	D₄×#3622	1	2	yes	—
233	C₂²	C₂×#10587634	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	D₄×#3622	1	2	yes	—
237	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
238	C₄	C₂³×#3622	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	C₂	C₂×D₄×#3622	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2955	C₂
socle	C₂×C₁₇	C₂×C₈×D₄
prime radical	C₁₇×#575	C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2955	D₁₇
hypocenter	C₁₇	#33442
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. #14541887	C₂	semidirect
2. #14530626	C₂	semidirect
3. #14530621	C₂	semidirect
4. #14530550	C₂	semidirect
5. C₂×#11762593	C₂	semidirect
6. #14530381	C₂	semidirect
9. #11763226	C₂²	semidirect
14. #11762630	C₂²	semidirect
15. #11762593	C₂²	semidirect
16. #11762593	C₂²	semidirect
17. #11762585	C₂²	semidirect
18. C₄×#63214	C₂²	semidirect
175. C₁₇	#33442	semidirect

Maximal subgroups

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

subgroup	index
#14541887	2
#14530626	2
#14530621	2
#14530550	2
C₂×#11762593	2
#14530381	2
C₁₇×#2955	2
#33442	17