Subgroups — #16927563

Order 4352.

350 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	#14528915	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic
3	#14527348	C₂	1	2	yes	maximal
4	#14526644	C₂	1	2	yes	maximal
5	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	#11762675	C₂²	1	2	yes	—
8	#11762572	—	2	1	no	—
9	#11762572	—	2	1	no	—
10	#11762562	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762546	—	2	4	no	—
13	C₁₇×#533	C₂²	1	1	yes	characteristic
14	C₁₇×#446	—	2	1	no	—
15	C₁₇×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₁₇	—	2	1	no	—
17	C₈×Q₈×C₁₇	C₄	1	2	yes	complemented
18	C₁₇×#424	—	2	1	no	—
19	C₁₇×#423	C₂²	1	1	yes	characteristic
20	C₁₇×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₁₇×#125	—	2	1	no	—
28	C₁₇×C₂²⋊Q₈	—	2	1	no	—
29	C₁₇×C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈×C₁₇	—	2	1	no	—
31	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₁₇×#117	—	2	1	no	—
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₇	—	2	2	no	—
35	C₁₇×C₄⋊C₈	—	2	2	no	—
36	C₁₇×C₄⋊C₈	—	2	2	no	—
37	C₁₇×C₄⋊C₈	—	2	1	no	—
38	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₇×#97	—	2	2	no	—
40	C₄×C₈×C₁₇	—	2	2	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	#10587502	—	2	2	no	—
46	#10587502	—	2	2	no	—
47	C₂²⋊#3622	—	2	2	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄×#3622	—	2	2	no	—
50	C₂×Q₈×C₁₇	—	2	1	no	—
51	C₂×Q₈×C₁₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
55	C₂²×Dic₁₇	—	4	2	no	—
56	M₁₆×C₁₇	—	4	2	no	—
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	—	2	2	no	—
66	C₄⋊C₄×C₁₇	—	2	1	no	—
67	C₄⋊C₄×C₁₇	—	2	1	no	—
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
70	C₂²⋊C₄×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₁₇	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₇	—	4	2	no	—
77	C₄⋊Dic₁₇	—	4	2	no	—
78	C₂×#3622	—	4	2	no	—
79	#8681	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	#121	1	1	yes	characteristic
81	Q₈×C₁₇	—	4	2	no	—
82	Q₈×C₁₇	—	4	2	no	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	—	2	2	no	—
85	C₂×C₄×C₁₇	—	4	1	no	—
86	C₂×C₄×C₁₇	—	4	1	no	—
87	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#135	1	2	yes	—
94	C₂×C₄×C₁₇	#137	1	2	yes	—
95	C₂×Dic₁₇	—	8	2	no	—
96	C₂×Dic₁₇	—	8	2	no	—
97	C₂×Dic₁₇	—	8	2	no	—
98	C₈×C₁₇	—	2	1	no	—
99	C₈×C₁₇	—	2	1	no	—
100	C₈×C₁₇	—	2	1	no	—
101	C₈×C₁₇	—	2	1	no	—
102	C₈×C₁₇	—	4	2	no	—
103	#3622	—	8	2	no	—
104	#2960	D₁₇	1	1	yes	characteristic
105	#1835	—	17	1	no	—
106	#1564	—	17	1	no	—
107	#1518	—	17	2	no	—
108	#1286	—	17	2	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	—	2	1	no	—
111	C₂²×C₁₇	#380	1	1	yes	characteristic
112	C₂²×C₁₇	#464	1	2	yes	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	4	2	no	—
118	C₄×C₁₇	—	4	2	no	—
119	C₄×C₁₇	#416	1	1	yes	characteristic
120	C₄×C₁₇	#419	1	1	yes	characteristic
121	C₄×C₁₇	#479	1	2	yes	—
122	Dic₁₇	—	16	2	no	—
123	#533	C₂×D₁₇	1	1	yes	characteristic
124	#446	—	2	1	no	—
125	#446	Dic₁₇	1	2	yes	complemented
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	Dic₁₇	1	2	yes	complemented
128	#431	—	17	2	no	—
129	#424	—	2	1	no	—
130	#423	C₂×D₁₇	1	1	yes	characteristic
131	#405	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
133	#382	—	17	2	no	—
134	#340	—	34	1	no	—
135	#340	—	34	1	no	—
136	#335	—	34	1	no	—
137	#334	—	34	1	no	—
138	#328	—	34	4	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	#1805	1	1	yes	characteristic
141	C₂×C₁₇	#1812	1	1	yes	characteristic
142	C₂×C₁₇	#1814	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	no	—
144	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄⋊Q₈	Dic₃₄	1	1	yes	characteristic
147	C₄⋊Q₈	Dic₃₄	1	1	yes	characteristic
148	#125	—	2	1	no	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	D₆₈	1	2	yes	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
153	#117	—	2	1	no	—
154	C₂×C₄⋊C₄	—	34	2	no	—
155	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	34	2	no	—
160	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
161	C₂²⋊C₈	—	34	2	no	—
162	#97	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	34	2	no	—
165	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
166	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
169	#95	—	34	2	no	—
170	#95	—	34	2	no	—
171	C₁₇	#8681	1	1	yes	characteristic, complemented, Sylow
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
174	C₂²×C₄	—	68	2	no	—
175	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
176	C₂²×C₄	#63252	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	68	2	no	—
193	C₄⋊C₄	—	68	2	no	—
194	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
198	C₄²	#63252	1	1	yes	characteristic
199	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
201	C₂³	#10587616	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	136	2	no	—
209	C₂×C₄	—	136	2	no	—
210	C₂×C₄	—	136	2	no	—
211	C₂×C₄	#10587610	1	1	yes	characteristic
212	C₂×C₄	#10587610	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
214	C₂×C₄	#10587612	1	1	yes	characteristic
215	C₂×C₄	#10587616	1	1	yes	characteristic
216	C₂×C₄	#10587628	1	2	yes	—
217	C₂×C₄	#10587636	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	136	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#11762669	1	1	yes	characteristic
228	C₂²	#11763031	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	272	2	no	—
236	C₄	#11762805	1	1	yes	characteristic
237	C₄	#11762818	1	1	yes	characteristic
238	C₄	#11763116	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#14528799	1	1	yes	characteristic
241	C₂	#14528830	1	1	yes	characteristic
242	C₂	#14528838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#8681
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
15. C₁₇×#446	C₄	semidirect
17. C₈×Q₈×C₁₇	C₄	semidirect
125. #446	Dic₁₇	semidirect
127. C₈×Q₈	Dic₁₇	semidirect
171. C₁₇	#8681	semidirect

Maximal subgroups

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

subgroup	index
#14528915	2
#14527532	2
#14527348	2
#14526644	2
C₁₇×#2960	2
#8681	17