Subgroups — #16950805

Order 4352.

370 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541886	C₂	1	1	yes	maximal, characteristic
2	#14530550	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530549	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530397	C₂	1	1	yes	maximal, characteristic
5	#14528585	C₂	1	1	yes	maximal, characteristic
6	#14526872	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587587	C₂²	1	1	yes	characteristic
9	#11763227	—	2	2	no	—
10	#11763225	—	2	2	no	—
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	M₃₂×D₁₇	—	2	2	no	—
14	#11762791	C₄	1	1	yes	characteristic
15	#11762782	C₄	1	1	yes	characteristic
16	#11762764	C₂²	1	1	yes	characteristic
17	#11762630	—	2	2	no	—
18	#11762629	—	2	2	no	—
19	#11762588	—	2	2	no	—
20	#11762581	C₄	1	1	yes	characteristic
21	#11762579	C₄	1	1	yes	characteristic
22	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#405	C₂²	1	1	yes	characteristic
25	C₂×#63252	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
29	#10587612	C₂×C₄	1	1	yes	characteristic
30	C₂×#63223	C₂×C₄	1	1	yes	characteristic
31	C₂×#63223	C₂×C₄	1	1	yes	characteristic
32	#10587587	—	2	1	no	—
33	#10587587	—	2	1	no	—
34	#10587587	—	2	1	no	—
35	#10587587	—	2	1	no	—
36	C₂×#63218	Q₈	1	2	yes	—
37	C₂×C₈×D₁₇	D₄	1	2	yes	—
38	C₁₇×M₃₂	—	2	1	no	—
39	C₁₇×M₃₂	D₄	1	2	yes	—
40	C₂×C₁₆×C₁₇	—	2	1	no	—
41	C₂×C₁₆×C₁₇	—	2	1	no	—
42	C₂×C₁₆×C₁₇	—	2	1	no	—
43	C₂×C₁₆×C₁₇	—	2	1	no	—
44	C₁₇×#97	—	2	1	no	—
45	C₄×C₈×C₁₇	—	2	1	no	—
46	#10587495	C₈	1	2	yes	—
47	#10587494	C₈	1	2	yes	—
48	#10587486	—	2	1	no	—
49	C₈×Dic₁₇	—	2	1	no	—
50	#10587483	—	2	1	no	—
51	#10587483	—	2	1	no	—
52	#10587483	—	2	1	no	—
53	#10587483	D₄	1	2	yes	—
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	—	2	1	no	—
56	#10587477	—	2	2	no	—
57	#10587469	—	2	4	no	—
58	C₁₆×D₁₇	—	2	4	no	—
59	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₇	M₁₆	1	2	yes	—
61	#63252	—	2	1	no	—
62	#63252	—	2	1	no	—
63	#63252	—	2	1	no	—
64	#63252	—	2	1	no	—
65	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
67	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
68	M₁₆×C₁₇	—	2	1	no	—
69	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₄²×C₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄×Dic₁₇	—	2	1	no	—
83	#63223	—	2	1	no	—
84	#63223	—	2	1	no	—
85	#63223	—	2	1	no	—
86	#63223	C₂×C₈	1	2	yes	—
87	C₂×#3622	—	2	1	no	—
88	C₂×#3622	C₄⋊C₄	1	2	yes	—
89	#63218	—	2	2	no	—
90	#63218	—	2	2	no	—
91	C₈×D₁₇	—	2	2	no	—
92	C₈×D₁₇	—	2	2	no	—
93	C₁₆×C₁₇	—	2	2	no	—
94	C₁₆×C₁₇	—	4	1	no	—
95	#63214	—	2	2	no	—
96	#63214	—	4	1	no	—
97	#12916	—	17	1	no	maximal, Sylow
98	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×D₁₇	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₇	—	4	1	no	—
101	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
106	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
107	C₂²⋊D₁₇	—	2	2	no	—
108	C₂²⋊D₁₇	—	2	2	no	—
109	C₂×Dic₁₇	—	4	1	no	—
110	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
111	D₆₈	—	2	1	no	—
112	D₆₈	—	2	1	no	—
113	C₄×D₁₇	—	2	2	no	—
114	C₄×D₁₇	—	2	2	no	—
115	Dic₃₄	—	2	1	no	—
116	Dic₃₄	—	2	1	no	—
117	C₈×C₁₇	—	2	2	no	—
118	C₈×C₁₇	#120	1	1	yes	characteristic
119	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
121	C₈×C₁₇	#123	1	1	yes	characteristic
122	#3622	—	2	2	no	—
123	#3622	—	2	2	no	—
124	#3407	—	17	1	no	—
125	#2153	—	17	1	no	—
126	#2153	—	17	1	no	—
127	#2113	—	17	1	no	—
128	#1757	—	17	1	no	—
129	#1379	—	17	1	no	—
130	#1379	D₁₇	1	1	yes	characteristic, complemented
131	C₂²×C₁₇	—	4	1	no	—
132	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₁₇	#435	1	1	yes	characteristic
134	C₂²×C₁₇	#435	1	1	yes	characteristic
135	C₂×D₁₇	—	2	2	no	—
136	C₂×D₁₇	—	4	2	no	—
137	C₄×C₁₇	—	8	1	no	—
138	C₄×C₁₇	#385	1	1	yes	characteristic
139	C₄×C₁₇	#424	1	1	yes	characteristic
140	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
141	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
142	Dic₁₇	—	2	2	no	—
143	Dic₁₇	—	8	1	no	—
144	C₂×#131	—	17	1	no	—
145	#504	—	34	2	no	—
146	#504	—	34	2	no	—
147	C₂×M₃₂	—	17	1	no	—
148	C₂×M₃₂	—	17	1	no	—
149	C₂×M₃₂	—	34	2	no	—
150	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
151	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
152	#413	—	17	1	no	—
153	#410	—	17	1	no	—
154	#405	—	17	1	no	—
155	#405	C₂×D₁₇	1	1	yes	characteristic
156	#364	—	34	2	no	—
157	#364	—	34	2	no	—
158	C₂²⋊C₁₆	—	34	2	no	—
159	#344	—	17	1	no	—
160	#344	—	17	1	no	—
161	C₂×C₁₇	—	2	1	no	—
162	C₂×C₁₇	—	2	1	no	—
163	C₂×C₁₇	—	2	1	no	—
164	C₂×C₁₇	#1850	1	1	yes	characteristic
165	D₁₇	—	4	2	no	—
166	C₂×#41	—	17	1	no	—
167	#131	—	34	1	no	—
168	#131	—	34	1	no	—
169	#131	—	34	1	no	—
170	#131	—	34	1	no	—
171	C₂×M₁₆	—	17	1	no	—
172	C₂×M₁₆	—	17	2	no	—
173	C₂×M₁₆	—	17	1	no	—
174	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
175	C₂²×C₈	—	17	2	no	—
176	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
177	#117	—	17	1	no	—
178	#117	C₄×D₁₇	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	17	2	no	—
181	M₃₂	—	34	4	no	—
182	M₃₂	—	34	1	no	—
183	M₃₂	—	34	1	no	—
184	M₃₂	—	34	1	no	—
185	M₃₂	D₆₈	1	2	yes	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	34	4	no	—
191	C₂×C₁₆	—	34	1	no	—
192	C₂×C₁₆	—	34	1	no	—
193	#101	—	17	2	no	—
194	#100	—	17	2	no	—
195	#99	—	34	2	no	—
196	#97	—	2	1	no	—
197	#97	—	34	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	34	1	no	—
200	C₁₇	#12916	1	1	yes	characteristic, complemented, Sylow
201	#41	—	34	1	no	—
202	#41	—	34	1	no	—
203	#41	—	34	1	no	—
204	#41	—	34	1	no	—
205	C₂×Q₈	—	17	1	no	—
206	C₂×D₄	—	17	2	no	—
207	C₂×D₄	—	17	1	no	—
208	C₂²×C₄	—	17	2	no	—
209	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	17	2	no	—
212	M₁₆	—	34	1	no	—
213	M₁₆	—	34	1	no	—
214	M₁₆	—	34	1	no	—
215	M₁₆	—	34	2	no	—
216	M₁₆	—	34	2	no	—
217	M₁₆	C₈×D₁₇	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	17	2	no	—
220	C₂×C₈	—	34	1	no	—
221	C₂×C₈	—	34	2	no	—
222	C₂×C₈	—	34	2	no	—
223	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
224	C₂×C₈	#63252	1	1	yes	characteristic
225	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
226	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
227	C₂×C₈	#63254	1	1	yes	characteristic
228	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	34	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	34	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	34	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	34	2	no	—
238	C₁₆	—	68	1	no	—
239	C₂³	—	17	2	no	—
240	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
241	Q₈	—	34	1	no	—
242	Q₈	—	34	1	no	—
243	D₄	—	34	2	no	—
244	D₄	—	34	2	no	—
245	D₄	—	34	1	no	—
246	D₄	—	34	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	17	2	no	—
249	C₂×C₄	—	34	2	no	—
250	C₂×C₄	—	34	2	no	—
251	C₂×C₄	—	68	1	no	—
252	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
253	C₂×C₄	#10587567	1	1	yes	characteristic
254	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
256	C₂×C₄	#10587587	1	1	yes	characteristic
257	C₂×C₄	#10587593	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	34	2	no	—
260	C₈	—	34	2	no	—
261	C₈	#10587568	1	1	yes	characteristic
262	C₈	#10587569	1	1	yes	characteristic
263	C₈	#10587582	1	1	yes	characteristic
264	C₈	#10587583	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	34	2	no	—
267	C₂²	—	68	2	no	—
268	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
269	C₂²	#11762882	1	1	yes	characteristic
270	C₂²	#11762883	1	1	yes	characteristic
271	C₄	—	8	1	no	—
272	C₄	—	34	2	no	—
273	C₄	—	136	1	no	—
274	C₄	#11762683	1	1	yes	characteristic
275	C₄	#11762834	1	1	yes	characteristic
276	C₄	#11762872	1	1	yes	characteristic
277	C₄	C₈×D₆₈	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	68	2	no	—
282	C₂	#14528963	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587568
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1379	C₂
socle	C₂×C₁₇	#1850
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1379	D₁₇
hypocenter	C₁₇	#12916
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
2. #14530550	C₂	semidirect
3. #14530549	C₂	semidirect
6. #14526872	C₂	semidirect
7. C₁₇×#1379	C₂	semidirect
130. #1379	D₁₇	semidirect
200. C₁₇	#12916	semidirect

Maximal subgroups

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

subgroup	index
#14541886	2
#14530550	2
#14530549	2
#14530397	2
#14528585	2
#14526872	2
C₁₇×#1379	2
#12916	17