Subgroups — #17978663

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545201	C₂	1	2	yes	maximal, complemented
2	#14545201	C₂	1	2	yes	maximal, complemented
3	#14545200	C₂	1	2	yes	maximal, complemented
4	#14545199	C₂	1	2	yes	maximal, complemented
5	#14545120	C₂	1	2	yes	maximal
6	#14545117	C₂	1	1	yes	maximal, characteristic
7	#14544994	C₂	1	2	yes	maximal
8	#14544985	C₂	1	1	yes	maximal, characteristic
9	#14537715	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587530	C₂²	1	2	yes	—
11	C₂×#10587529	C₂²	1	1	yes	characteristic
12	C₂×#10587529	C₂²	1	1	yes	characteristic
13	#11763730	C₂²	1	1	yes	characteristic
14	#11763730	C₂²	1	2	yes	—
15	#11763730	C₂²	1	1	yes	characteristic
16	#11763730	C₂²	1	2	yes	—
17	C₄⋊#63216	C₂²	1	2	yes	—
18	C₄⋊#63216	C₂²	1	2	yes	—
19	C₄⋊#63216	C₂²	1	2	yes	—
20	#11763698	C₂²	1	2	yes	—
21	#11763698	C₂²	1	2	yes	—
22	#11763697	C₂²	1	2	yes	—
23	#11763697	C₂²	1	2	yes	—
24	#11763695	C₂²	1	2	yes	—
25	C₄×#63216	C₂²	1	2	yes	—
26	#11763363	C₄	1	1	yes	characteristic
27	#11762939	C₄	1	1	yes	characteristic
28	#11762937	C₄	1	2	yes	—
29	Q₈×#3622	C₂²	1	1	yes	characteristic
30	#11762933	C₂²	1	2	yes	—
31	#11762933	C₂²	1	1	yes	characteristic
32	#11762834	C₄	1	2	yes	—
33	#11762831	C₄	1	1	yes	characteristic
34	#11762766	C₂²	1	2	yes	—
35	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
36	C₁₇×#446	C₄	1	1	yes	characteristic
37	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	#10587632	C₈	1	1	yes	characteristic
39	#10587630	C₈	1	1	yes	characteristic
40	C₂×#63218	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
43	#10587583	C₈	1	1	yes	characteristic
44	#10587582	C₈	1	2	yes	—
45	#10587575	C₈	1	2	yes	—
46	#10587574	C₈	1	1	yes	characteristic
47	#10587558	C₂×C₄	1	2	yes	—
48	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
50	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
51	C₂×#63216	C₂³	1	2	yes	—
52	C₂×#63216	C₂³	1	2	yes	—
53	C₂×#63216	C₂³	1	2	yes	—
54	C₂×#63216	C₂³	1	2	yes	—
55	#10587530	—	2	4	no	—
56	#10587529	—	2	2	no	—
57	#10587529	Q₈	1	4	yes	complemented
58	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₁₇×#97	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
62	#10587486	C₂×C₄	1	2	yes	—
63	#10587485	C₂×C₄	1	2	yes	—
64	#10587485	C₂×C₄	1	2	yes	—
65	#10587485	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₇	C₂³	1	2	yes	—
69	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#3622	C₂×C₄	1	2	yes	—
71	C₄×#3622	C₂³	1	1	yes	characteristic
72	C₄×#3622	C₂³	1	2	yes	—
73	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
78	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
84	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
86	#63227	C₂×C₈	1	2	yes	—
87	#63227	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
90	#63225	C₂×C₈	1	2	yes	—
91	#63225	C₂×C₈	1	2	yes	—
92	#63225	C₂×C₈	1	2	yes	—
93	#63225	C₂×C₈	1	2	yes	—
94	#63225	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
99	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
100	C₂×#3622	C₂²×C₄	1	2	yes	—
101	C₂×#3622	C₂⁴	1	1	yes	characteristic
102	#63218	—	2	4	no	—
103	C₈×D₁₇	—	2	2	no	—
104	C₈×D₁₇	C₂×Q₈	1	2	yes	—
105	C₈×D₁₇	#41	1	2	yes	—
106	#63216	—	2	2	no	—
107	#63216	—	2	2	no	—
108	#63216	—	2	2	no	—
109	#63216	C₂×Q₈	1	4	yes	—
110	#33475	—	17	1	no	maximal, Sylow
111	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
112	Q₈×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
120	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
121	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
123	C₄×D₁₇	—	4	1	no	—
124	C₄×D₁₇	—	4	1	no	—
125	C₄×D₁₇	C₄×Q₈	1	2	yes	—
126	C₄×D₁₇	#131	1	2	yes	—
127	Dic₃₄	—	4	2	no	—
128	Dic₃₄	—	4	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	2	no	—
131	C₈×C₁₇	C₄×Q₈	1	2	yes	—
132	#3622	—	2	1	no	—
133	#3622	—	2	2	no	—
134	#3622	C₂²×Q₈	1	1	yes	characteristic
135	#3622	C₂×#41	1	1	yes	characteristic
136	#2960	—	17	1	no	—
137	#2183	—	17	2	no	—
138	#2183	—	17	2	no	—
139	#2183	—	17	2	no	—
140	#2183	—	17	2	no	—
141	#2151	—	17	2	no	—
142	#2150	—	17	1	no	—
143	#2108	—	17	2	no	—
144	C₂×#346	—	17	1	no	—
145	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	—	2	1	no	—
148	C₂×D₁₇	C₈×Q₈	1	2	yes	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	2	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	2	no	—
153	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	2	2	no	—
157	Dic₁₇	—	2	2	no	—
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	C₈×Q₈	1	2	yes	—
160	#533	—	17	1	no	—
161	C₂×M₃₂	—	17	2	no	—
162	C₂²×C₁₆	—	17	1	no	—
163	C₂²×C₁₆	—	17	1	no	—
164	#446	—	17	2	no	—
165	#446	—	17	1	no	—
166	#446	#595	1	1	yes	characteristic
167	C₈×Q₈	—	17	1	no	—
168	C₈×Q₈	—	17	1	no	—
169	C₈×Q₈	—	17	2	no	—
170	#424	—	17	2	no	—
171	#423	—	17	1	no	—
172	#405	—	17	2	no	—
173	C₂×C₄×C₈	—	17	1	no	—
174	C₄⋊C₁₆	—	17	2	no	—
175	C₄⋊C₁₆	—	17	2	no	—
176	C₄⋊C₁₆	—	17	2	no	—
177	C₄⋊C₁₆	—	17	1	no	—
178	C₄⋊C₁₆	—	17	1	no	—
179	C₄⋊C₁₆	—	17	2	no	—
180	C₄⋊C₁₆	—	17	2	no	—
181	#346	—	17	2	no	—
182	#346	—	17	2	no	—
183	#346	—	17	2	no	—
184	#346	—	17	2	no	—
185	C₄×C₁₆	—	17	2	no	—
186	C₄×C₁₆	—	17	2	no	—
187	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
188	C₂×C₁₇	#3407	1	1	yes	characteristic
189	C₂×C₁₇	#3407	1	1	yes	characteristic
190	D₁₇	—	2	2	no	—
191	C₂×M₁₆	—	17	2	no	—
192	C₂²×C₈	—	17	1	no	—
193	C₂²×C₈	—	17	1	no	—
194	C₄⋊Q₈	—	17	1	no	—
195	C₄⋊Q₈	—	17	1	no	—
196	#125	—	17	2	no	—
197	C₂²⋊Q₈	—	17	1	no	—
198	C₂²⋊Q₈	—	17	2	no	—
199	C₂²⋊Q₈	—	17	1	no	—
200	C₄×Q₈	—	17	2	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×Q₈	C₂×#595	1	1	yes	characteristic
203	#117	—	17	2	no	—
204	C₂×C₄²	—	17	1	no	—
205	M₃₂	—	34	4	no	—
206	C₂×C₁₆	—	17	2	no	—
207	C₂×C₁₆	—	17	4	no	—
208	C₂×C₁₆	—	17	2	no	—
209	C₂×C₁₆	—	17	2	no	—
210	C₂×C₁₆	—	17	2	no	—
211	C₂×C₁₆	—	34	2	no	—
212	C₄⋊C₈	—	17	2	no	—
213	C₄⋊C₈	—	17	2	no	—
214	C₄⋊C₈	—	17	2	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	2	no	—
217	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
218	C₄⋊C₈	C₂×#595	1	2	yes	—
219	#97	—	17	2	no	—
220	#97	C₂×#595	1	2	yes	—
221	C₄×C₈	—	17	2	no	—
222	C₄×C₈	—	17	1	no	—
223	C₄×C₈	—	17	1	no	—
224	C₄×C₈	—	17	2	no	—
225	C₄×C₈	—	17	1	no	—
226	C₄×C₈	C₂×#595	1	1	yes	characteristic
227	C₁₇	#33475	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	17	2	no	—
229	C₂×Q₈	—	17	1	no	—
230	C₂×Q₈	#63242	1	1	yes	characteristic
231	C₂²×C₄	—	17	1	no	—
232	C₂²×C₄	—	17	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	M₁₆	—	34	4	no	—
235	C₂×C₈	—	17	1	no	—
236	C₂×C₈	—	17	2	no	—
237	C₂×C₈	—	17	2	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	—	17	2	no	—
240	C₂×C₈	—	34	2	no	—
241	C₂×C₈	C₂²×#595	1	1	yes	characteristic
242	C₂×C₈	C₂²×#595	1	2	yes	—
243	C₂×C₈	C₂²×#595	1	1	yes	characteristic
244	C₄⋊C₄	—	17	2	no	—
245	C₄⋊C₄	—	17	2	no	—
246	C₄⋊C₄	—	17	2	no	—
247	C₄⋊C₄	—	17	2	no	—
248	C₄⋊C₄	—	17	2	no	—
249	C₄⋊C₄	—	17	2	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	#63242	1	2	yes	—
252	C₄⋊C₄	#63242	1	1	yes	characteristic
253	C₂²⋊C₄	—	17	2	no	—
254	C₂²⋊C₄	—	17	2	no	—
255	C₄²	—	17	1	no	—
256	C₄²	—	17	1	no	—
257	C₄²	—	17	1	no	—
258	C₄²	—	17	2	no	—
259	C₄²	C₂²×#595	1	1	yes	characteristic
260	C₄²	C₂²×#595	1	2	yes	—
261	C₁₆	—	17	4	no	—
262	C₁₆	—	34	2	no	—
263	C₁₆	—	34	2	no	—
264	C₁₆	—	34	2	no	—
265	C₂³	—	17	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	68	2	no	—
268	Q₈	—	68	1	no	—
269	C₂×C₄	—	17	1	no	—
270	C₂×C₄	—	17	2	no	—
271	C₂×C₄	—	17	2	no	—
272	C₂×C₄	—	17	1	no	—
273	C₂×C₄	—	17	1	no	—
274	C₂×C₄	—	17	2	no	—
275	C₂×C₄	—	17	2	no	—
276	C₂×C₄	—	68	1	no	—
277	C₂×C₄	—	68	1	no	—
278	C₂×C₄	C₂×#63242	1	1	yes	characteristic
279	C₂×C₄	C₂×#63242	1	1	yes	characteristic
280	C₂×C₄	C₂×#63242	1	2	yes	—
281	C₂×C₄	C₂×#63242	1	2	yes	—
282	C₂×C₄	C₂³×#595	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	2	no	—
285	C₈	—	17	1	no	—
286	C₈	—	17	1	no	—
287	C₈	—	34	1	no	—
288	C₈	—	34	2	no	—
289	C₈	Q₈×#595	1	2	yes	—
290	C₂²	—	17	2	no	—
291	C₂²	—	34	1	no	—
292	C₂²	—	34	1	no	—
293	C₂²	C₂²×#63242	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	17	2	no	—
299	C₄	—	34	1	no	—
300	C₄	—	34	2	no	—
301	C₄	—	34	1	no	—
302	C₄	—	34	2	no	—
303	C₄	C₂×Q₈×#595	1	1	yes	characteristic
304	C₄	D₁₇⋊#131	1	1	yes	characteristic
305	C₂	—	34	2	no	—
306	C₂	Q₈×#63242	1	1	yes	characteristic
307	C₂	#14545752	1	1	yes	characteristic
308	C₂	#14545753	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#446	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#446	#595
hypocenter	C₁₇	#33475
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #14545201	C₂	semidirect
2. #14545201	C₂	semidirect
3. #14545200	C₂	semidirect
4. #14545199	C₂	semidirect
57. #10587529	Q₈	semidirect
227. C₁₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#14545201	2
#14545200	2
#14545199	2
#14545120	2
#14545117	2
#14544994	2
#14544985	2
#14537715	2
#33475	17