Subgroups — #17187745

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541920	C₂	1	1	yes	maximal, characteristic
2	#14537702	C₂	1	2	yes	maximal, complemented
3	#14530671	C₂	1	4	yes	maximal, complemented
4	#14530666	C₂	1	4	yes	maximal, complemented
5	#14530579	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762617	C₂	1	2	yes	maximal, complemented
7	C₁₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587590	C₂²	1	2	yes	—
9	#11763575	—	2	2	no	—
10	#11763252	C₄	1	4	yes	complemented
11	#11763247	C₄	1	4	yes	complemented
12	#11763241	C₂²	1	4	yes	—
13	#11762932	—	2	4	no	—
14	#11762927	C₂²	1	8	yes	complemented
15	C₂×#10587490	C₂²	1	2	yes	—
16	#11762827	—	2	2	no	—
17	#11762639	C₂²	1	4	yes	complemented
18	#11762617	C₂²	1	4	yes	complemented
19	#11762617	C₂²	1	4	yes	complemented
20	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#405	C₂²	1	1	yes	characteristic
23	C₁₇×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₁₇	C₂²	1	2	yes	complemented
25	C₂×#63252	—	2	2	no	—
26	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#117	D₄	1	1	yes	characteristic
29	C₂×#63223	—	2	2	no	—
30	#10587593	—	4	4	no	—
31	M₁₆×D₁₇	—	4	4	no	—
32	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
33	#10587590	—	2	2	no	—
34	#10587590	—	2	2	no	—
35	#10587590	C₂×C₄	1	4	yes	—
36	#10587590	C₂³	1	4	yes	—
37	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
38	C₂×#63219	—	2	2	no	—
39	#10587587	—	2	8	no	—
40	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₁₇	—	2	2	no	—
42	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
45	C₁₇×#97	D₄	1	2	yes	—
46	C₄×C₈×C₁₇	C₂³	1	2	yes	—
47	#10587490	—	2	2	no	—
48	#10587490	C₂³	1	4	yes	—
49	#10587476	—	2	8	no	—
50	Dic₁₃₆	C₂×C₄	1	4	yes	—
51	#10587471	C₂×C₄	1	8	yes	—
52	D₂₇₂	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₇	—	4	2	no	—
55	#63252	—	2	4	no	—
56	#63252	—	2	4	no	—
57	#63252	—	4	2	no	—
58	#63252	—	4	2	no	—
59	C₂×D₆₈	—	2	2	no	—
60	C₂×C₄×D₁₇	—	4	2	no	—
61	C₂×Dic₃₄	—	2	2	no	—
62	M₁₆×C₁₇	D₈	1	4	yes	—
63	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₇	#41	1	2	yes	—
67	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₂²⋊C₄×C₁₇	—	2	1	no	—
70	C₄²×C₁₇	C₂×D₄	1	2	yes	—
71	#63223	—	2	4	no	—
72	#63223	—	4	2	no	—
73	C₂×#3622	—	4	2	no	—
74	Dic₆₈	—	2	2	no	—
75	Dic₆₈	C₂²×C₄	1	4	yes	—
76	D₁₃₆	—	2	2	no	—
77	D₁₃₆	C₂²×C₄	1	4	yes	—
78	#63219	—	2	4	no	—
79	#63219	—	2	4	no	—
80	#63218	—	4	8	no	—
81	C₈×D₁₇	—	4	8	no	—
82	C₁₆×C₁₇	—	2	1	no	—
83	C₁₆×C₁₇	—	2	1	no	—
84	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
85	#33464	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	#135	1	1	yes	characteristic
87	C₂²×D₁₇	—	4	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²⋊D₁₇	—	4	4	no	—
94	C₂²⋊D₁₇	—	4	4	no	—
95	C₂×Dic₁₇	—	4	2	no	—
96	D₆₈	—	2	4	no	—
97	D₆₈	—	4	2	no	—
98	C₄×D₁₇	—	4	4	no	—
99	C₄×D₁₇	—	4	4	no	—
100	Dic₃₄	—	2	4	no	—
101	Dic₃₄	—	4	2	no	—
102	C₈×C₁₇	C₄×D₄	1	2	yes	—
103	C₈×C₁₇	C₂×D₈	1	4	yes	—
104	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
105	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
106	#3622	—	4	4	no	—
107	#3414	—	17	1	no	—
108	#2957	—	17	2	no	—
109	#2179	—	17	4	no	—
110	#2178	—	17	4	no	—
111	#2161	—	17	1	no	—
112	C₂×#359	—	17	2	no	—
113	#2108	D₁₇	1	1	yes	characteristic, complemented
114	C₂²×C₁₇	—	4	1	no	—
115	C₂²×C₁₇	C₄×D₈	1	2	yes	—
116	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
117	C₂×D₁₇	—	4	4	no	—
118	C₂×D₁₇	—	16	2	no	—
119	C₄×C₁₇	—	4	2	no	—
120	C₄×C₁₇	C₄×D₈	1	2	yes	—
121	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
123	Dic₁₇	—	4	4	no	—
124	C₂×#135	—	17	2	no	—
125	#568	—	34	2	no	—
126	#510	—	17	4	no	—
127	#509	—	17	4	no	—
128	#508	—	17	4	no	—
129	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
130	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
131	#444	—	34	4	no	—
132	#443	—	17	8	no	—
133	C₂×#108	—	17	2	no	—
134	#421	—	34	2	no	—
135	#405	C₂×D₁₇	1	1	yes	characteristic
136	#368	—	17	4	no	—
137	#359	—	17	4	no	—
138	#359	—	17	4	no	—
139	#346	C₂×D₁₇	1	2	yes	complemented
140	C₄×C₁₆	C₂×D₁₇	1	2	yes	complemented
141	C₂×C₁₇	—	2	1	no	—
142	C₂×C₁₇	—	2	2	no	—
143	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
144	D₁₇	—	8	4	no	—
145	C₂×#41	—	34	2	no	—
146	#135	—	17	4	no	—
147	#135	—	17	4	no	—
148	#135	—	34	2	no	—
149	#135	—	34	2	no	—
150	C₂×Q₁₆	—	17	2	no	—
151	C₂×SD₁₆	—	34	2	no	—
152	C₂×D₈	—	17	2	no	—
153	#131	—	34	8	no	—
154	#131	—	68	4	no	—
155	C₂×M₁₆	—	34	2	no	—
156	C₂×M₁₆	—	68	4	no	—
157	C₂×M₁₆	D₆₈	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
159	#117	D₆₈	1	1	yes	characteristic
160	Q₃₂	—	17	4	no	—
161	SD₃₂	—	17	8	no	—
162	D₁₆	—	17	4	no	—
163	M₃₂	C₄×D₁₇	1	4	yes	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
168	#108	—	17	4	no	—
169	#108	—	34	2	no	—
170	#104	—	34	8	no	—
171	#97	D₆₈	1	2	yes	—
172	C₄×C₈	C₂²×D₁₇	1	2	yes	—
173	C₁₇	#33464	1	1	yes	characteristic, complemented, Sylow
174	#41	—	34	4	no	—
175	#41	—	34	4	no	—
176	#41	—	68	2	no	—
177	#41	—	68	2	no	—
178	C₂×Q₈	—	34	2	no	—
179	C₂×D₄	—	34	2	no	—
180	C₂×D₄	—	68	2	no	—
181	C₂²×C₄	—	68	2	no	—
182	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
183	Q₁₆	—	17	4	no	—
184	Q₁₆	—	34	2	no	—
185	SD₁₆	—	34	4	no	—
186	SD₁₆	—	34	4	no	—
187	D₈	—	17	4	no	—
188	D₈	—	34	2	no	—
189	M₁₆	—	34	4	no	—
190	M₁₆	—	68	2	no	—
191	M₁₆	—	68	8	no	—
192	M₁₆	D₁₃₆	1	4	yes	—
193	C₂×C₈	—	68	2	no	—
194	C₂×C₈	—	68	8	no	—
195	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
196	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
197	C₂×C₈	C₂×D₆₈	1	2	yes	—
198	C₂×C₈	#63252	1	2	yes	—
199	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₆₈	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
206	C₂³	—	68	2	no	—
207	C₂³	#10587590	1	1	yes	characteristic
208	Q₈	—	34	4	no	—
209	Q₈	—	68	2	no	—
210	D₄	—	34	4	no	—
211	D₄	—	68	4	no	—
212	D₄	—	68	4	no	—
213	D₄	—	68	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	68	2	no	—
216	C₂×C₄	—	68	4	no	—
217	C₂×C₄	—	68	4	no	—
218	C₂×C₄	C₄×D₆₈	1	2	yes	—
219	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
220	C₂×C₄	#10587590	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
222	C₈	—	68	4	no	—
223	C₈	C₄×D₆₈	1	2	yes	—
224	C₈	C₂×D₁₃₆	1	4	yes	—
225	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
226	C₈	C₂×#63252	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	272	2	no	—
229	C₂²	—	68	4	no	—
230	C₂²	C₄×D₁₃₆	1	2	yes	—
231	C₂²	C₂×#10587590	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	68	4	no	—
234	C₄	C₄×D₁₃₆	1	2	yes	—
235	C₄	C₂×C₄×D₆₈	1	1	yes	characteristic
236	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	136	4	no	—
240	C₂	C₂×C₄×D₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2108	C₂
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	#14541920	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2108	D₁₇
hypocenter	C₁₇	#33464
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #14537702	C₂	semidirect
3. #14530671	C₂	semidirect
4. #14530666	C₂	semidirect
5. #14530579	C₂	semidirect
6. C₂×#11762617	C₂	semidirect
7. C₁₇×#2108	C₂	semidirect
10. #11763252	C₄	semidirect
11. #11763247	C₄	semidirect
14. #11762927	C₂²	semidirect
17. #11762639	C₂²	semidirect
18. #11762617	C₂²	semidirect
19. #11762617	C₂²	semidirect
23. C₁₇×#346	C₂²	semidirect
24. C₄×C₁₆×C₁₇	C₂²	semidirect
113. #2108	D₁₇	semidirect
139. #346	C₂×D₁₇	semidirect
140. C₄×C₁₆	C₂×D₁₇	semidirect
173. C₁₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#14541920	2
#14537702	2
#14530671	2
#14530666	2
#14530579	2
C₂×#11762617	2
C₁₇×#2108	2
#33464	17