Subgroups — #16933549

Order 4352.

454 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	#14529733	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762562	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762558	C₂	1	2	yes	maximal
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	—	2	1	no	—
9	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
10	#11762571	—	2	4	no	—
11	#11762563	—	2	1	no	—
12	#11762562	—	2	2	no	—
13	#11762558	—	2	4	no	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₇×#500	—	2	2	no	—
16	C₁₇×#500	C₄	1	4	yes	complemented
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₁₇×#107	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	—	2	2	no	—
24	C₁₇×C₄⋊Q₈	—	2	2	no	—
25	C₁₇×C₄⋊Q₈	—	2	2	no	—
26	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₇	—	2	1	no	—
32	C₂²×#3622	—	2	1	no	—
33	C₂²×#3622	—	2	2	no	—
34	C₁₇×#107	—	2	4	no	—
35	C₁₇×#107	—	2	4	no	—
36	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₇	D₄	1	2	yes	—
38	#10587502	—	2	1	no	—
39	C₈×Dic₁₇	—	4	2	no	—
40	C₄⋊#3622	—	2	2	no	—
41	C₄⋊#3622	—	2	2	no	—
42	C₄⋊#3622	—	2	1	no	—
43	C₄⋊#3622	D₄	1	2	yes	—
44	C₂×Q₈×C₁₇	—	2	4	no	—
45	C₂×Q₈×C₁₇	—	4	2	no	—
46	C₂×Q₈×C₁₇	—	8	4	no	—
47	C₂²×C₄×C₁₇	—	4	2	no	—
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
51	C₂²×Dic₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	—	2	1	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
58	C₄⋊C₄×C₁₇	—	2	4	no	—
59	C₄⋊C₄×C₁₇	—	2	4	no	—
60	C₄⋊C₄×C₁₇	—	4	2	no	—
61	C₄⋊C₄×C₁₇	—	4	2	no	—
62	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₇	#41	1	1	yes	characteristic
66	C₄⋊Dic₁₇	—	2	2	no	—
67	C₄⋊Dic₁₇	Q₁₆	1	4	yes	—
68	C₄×Dic₁₇	—	4	1	no	—
69	C₄×Dic₁₇	—	4	1	no	—
70	C₂×#3622	—	2	2	no	—
71	C₂×#3622	—	4	1	no	—
72	C₂×#3622	—	4	1	no	—
73	C₂×#3622	—	4	2	no	—
74	C₂×#3622	—	4	4	no	—
75	#9730	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	#121	1	1	yes	characteristic
77	Q₈×C₁₇	—	8	4	no	—
78	Q₈×C₁₇	—	8	4	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	4	4	no	—
81	C₂×C₄×C₁₇	—	8	4	no	—
82	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#135	1	2	yes	—
92	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
94	C₂×Dic₁₇	—	2	4	no	—
95	C₂×Dic₁₇	—	4	2	no	—
96	C₂×Dic₁₇	—	4	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	#103	1	4	yes	—
99	#3622	—	4	2	no	—
100	#3622	—	8	2	no	—
101	C₂×#500	D₁₇	1	1	yes	characteristic
102	#1987	—	17	2	no	—
103	C₈×C₄⋊C₄	—	17	1	no	—
104	C₂×#335	—	17	1	no	—
105	C₂×#333	—	17	2	no	—
106	C₂²×C₁₇	#390	1	1	yes	characteristic
107	C₂²×C₁₇	#461	1	1	yes	characteristic
108	C₂²×C₁₇	#462	1	1	yes	characteristic
109	C₂²×C₁₇	#467	1	2	yes	—
110	C₂²×C₁₇	#472	1	2	yes	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	8	4	no	—
114	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
115	C₄×C₁₇	#419	1	1	yes	characteristic
116	C₄×C₁₇	C₄×Q₁₆	1	2	yes	—
117	C₄×C₁₇	#461	1	1	yes	characteristic
118	C₄×C₁₇	#462	1	1	yes	characteristic
119	C₄×C₁₇	#494	1	1	yes	characteristic
120	C₄×C₁₇	#495	1	1	yes	characteristic
121	Dic₁₇	—	4	4	no	—
122	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
123	#500	—	2	2	no	—
124	#500	Dic₁₇	1	4	yes	complemented
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	2	1	no	—
127	C₂×C₄⋊C₈	—	17	1	no	—
128	C₂×C₄⋊C₈	—	17	2	no	—
129	C₂×C₄×C₈	—	34	1	no	—
130	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
131	C₄×C₄⋊C₄	—	17	1	no	—
132	#340	—	34	4	no	—
133	#336	—	34	1	no	—
134	#335	—	34	2	no	—
135	#333	—	34	4	no	—
136	C₂×C₁₇	#1675	1	2	yes	—
137	C₂×C₁₇	#1676	1	2	yes	—
138	C₂×C₁₇	#1969	1	1	yes	characteristic
139	C₂×C₁₇	#1978	1	1	yes	characteristic
140	C₂×C₁₇	#1981	1	1	yes	characteristic
141	C₂²×Q₈	—	2	2	no	—
142	C₂²×C₈	—	34	1	no	—
143	C₂²×C₈	—	34	2	no	—
144	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	17	2	no	—
153	C₂×C₄²	—	34	1	no	—
154	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
155	#107	—	2	4	no	—
156	#107	—	2	4	no	—
157	C₄⋊C₈	—	17	2	no	—
158	C₄⋊C₈	—	34	2	no	—
159	C₄⋊C₈	—	34	1	no	—
160	C₄⋊C₈	—	34	2	no	—
161	C₄×C₈	—	68	2	no	—
162	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
163	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
164	#95	—	34	1	no	—
165	C₁₇	#9730	1	1	yes	characteristic, complemented, Sylow
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	8	4	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	34	2	no	—
171	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
172	C₂²×C₄	#63254	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	34	2	no	—
177	C₂×C₈	—	68	1	no	—
178	C₂×C₈	—	68	1	no	—
179	C₂×C₈	—	68	4	no	—
180	C₂×C₈	—	68	2	no	—
181	C₂×C₈	#63230	1	2	yes	—
182	C₂×C₈	#63231	1	2	yes	—
183	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	17	4	no	—
190	C₄⋊C₄	—	34	2	no	—
191	C₄²	—	68	1	no	—
192	C₄²	—	68	1	no	—
193	C₄²	D₄×D₁₇	1	1	yes	characteristic
194	C₄²	#63254	1	1	yes	characteristic
195	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
197	C₂³	#10587624	1	1	yes	characteristic
198	Q₈	—	8	4	no	—
199	Q₈	—	8	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	4	no	—
203	C₂×C₄	—	34	4	no	—
204	C₂×C₄	—	68	2	no	—
205	C₂×C₄	—	68	2	no	—
206	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
207	C₂×C₄	#10587605	1	2	yes	—
208	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
209	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
210	C₂×C₄	#10587622	1	1	yes	characteristic
211	C₂×C₄	#10587624	1	1	yes	characteristic
212	C₂×C₄	#10587626	1	1	yes	characteristic
213	C₂×C₄	C₂×#63230	1	1	yes	characteristic
214	C₂×C₄	#10587628	1	1	yes	characteristic
215	C₂×C₄	#10587628	1	1	yes	characteristic
216	C₂×C₄	C₂×#63231	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	68	2	no	—
220	C₈	—	136	2	no	—
221	C₈	#10587506	1	4	yes	—
222	C₂²	#11762712	1	1	yes	characteristic
223	C₂²	#11763011	1	1	yes	characteristic
224	C₂²	#11763016	1	1	yes	characteristic
225	C₂²	#11763047	1	2	yes	—
226	C₂²	#11763076	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	4	no	—
230	C₄	—	68	4	no	—
231	C₄	C₂×#10587506	1	1	yes	characteristic
232	C₄	#11762818	1	1	yes	characteristic
233	C₄	Q₁₆×Dic₁₇	1	2	yes	—
234	C₄	#11763011	1	1	yes	characteristic
235	C₄	#11763016	1	1	yes	characteristic
236	C₄	#11763187	1	1	yes	characteristic
237	C₄	#11763194	1	1	yes	characteristic
238	C₂	#14528180	1	2	yes	—
239	C₂	#14528185	1	2	yes	—
240	C₂	#14529646	1	1	yes	characteristic
241	C₂	#14529695	1	1	yes	characteristic
242	C₂	#14529708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#9730
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. C₁₇×#500	C₄	semidirect
124. #500	Dic₁₇	semidirect
165. C₁₇	#9730	semidirect

Maximal subgroups

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

subgroup	index
#14529733	2
C₈×C₄⋊Dic₁₇	2
C₂×#11762562	2
C₂×#11762558	2
C₂×C₁₇×#500	2
#9730	17