Subgroups — #17189074

Order 4352.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539315	C₂	1	1	yes	maximal, characteristic, complemented
2	#14530960	C₂	1	4	yes	maximal, complemented
3	#14530955	C₂	1	4	yes	maximal, complemented
4	#14530558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762633	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530550	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530520	C₂	1	1	yes	maximal, characteristic
8	C₂×#11762619	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#2957	C₂	1	1	yes	maximal, characteristic
10	#11763437	—	2	1	no	—
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	#11763218	—	2	2	no	—
14	#11763211	C₂²	1	2	yes	complemented
15	#11763207	C₂²	1	2	yes	complemented
16	#11762851	—	2	1	no	—
17	#11762847	C₂²	1	1	yes	characteristic
18	C₂×#10587488	C₂²	1	1	yes	characteristic
19	#11762633	C₂²	1	2	yes	complemented
20	#11762633	C₂²	1	4	yes	complemented
21	#11762633	C₂²	1	2	yes	complemented
22	#11762630	C₂²	1	4	yes	complemented
23	#11762619	C₂²	1	2	yes	—
24	#11762619	C₂²	1	4	yes	—
25	#11762619	C₂²	1	2	yes	—
26	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
27	C₁₇×#568	—	2	1	no	—
28	C₁₇×#444	—	2	2	no	—
29	C₁₇×#443	C₂²	1	4	yes	—
30	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
31	C₁₇×#421	—	2	1	no	—
32	C₁₇×#405	C₂²	1	1	yes	characteristic
33	C₂×#41×C₁₇	—	2	1	no	—
34	C₁₇×#135	—	2	1	no	—
35	C₁₇×#135	—	2	1	no	—
36	C₁₇×#135	Q₈	1	2	yes	complemented
37	C₁₇×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₁₇	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₁₇	—	2	1	no	—
40	C₂×D₈×C₁₇	Q₈	1	1	yes	characteristic, complemented
41	C₁₇×#131	—	2	4	no	—
42	C₁₇×#131	—	4	2	no	—
43	C₂×M₁₆×C₁₇	—	2	1	no	—
44	C₂×M₁₆×C₁₇	—	4	2	no	—
45	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
47	C₁₇×#117	D₄	1	1	yes	characteristic
48	#10587612	—	2	1	no	—
49	C₂×C₄⋊Dic₁₇	—	2	1	no	—
50	#10587576	—	4	1	no	—
51	#10587574	—	4	1	no	—
52	#10587564	—	4	2	no	—
53	#10587556	—	2	2	no	—
54	#10587555	—	2	2	no	—
55	C₁₇×#108	—	2	1	no	—
56	C₁₇×#108	C₂³	1	2	yes	—
57	C₁₇×#104	—	2	4	no	—
58	C₁₇×#97	D₄	1	2	yes	—
59	C₄×C₈×C₁₇	C₂³	1	2	yes	—
60	#10587489	—	2	2	no	—
61	#10587489	—	2	1	no	—
62	#10587488	—	2	2	no	—
63	#10587488	—	2	1	no	—
64	#10587488	C₂³	1	1	yes	characteristic
65	#10587488	C₂³	1	2	yes	—
66	#10587488	C₂³	1	1	yes	characteristic
67	#10587483	—	2	1	no	—
68	#10587483	—	2	1	no	—
69	#10587483	C₂³	1	2	yes	—
70	#10587483	C₂³	1	2	yes	—
71	C₂×#63214	—	2	1	no	—
72	C₂×#63214	—	2	1	no	—
73	#41×C₁₇	—	2	2	no	—
74	#41×C₁₇	—	2	2	no	—
75	#41×C₁₇	—	4	1	no	—
76	#41×C₁₇	—	4	1	no	—
77	C₂×Q₈×C₁₇	—	2	1	no	—
78	C₂×D₄×C₁₇	—	2	1	no	—
79	C₂×D₄×C₁₇	—	4	1	no	—
80	C₂²×C₄×C₁₇	—	4	1	no	—
81	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₁₇	—	4	1	no	—
83	C₂×Dic₃₄	—	4	2	no	—
84	Q₁₆×C₁₇	—	2	1	no	—
85	Q₁₆×C₁₇	C₂×Q₈	1	2	yes	—
86	SD₁₆×C₁₇	—	2	2	no	—
87	SD₁₆×C₁₇	—	2	2	no	—
88	D₈×C₁₇	—	2	1	no	—
89	D₈×C₁₇	C₂×Q₈	1	2	yes	—
90	M₁₆×C₁₇	—	2	2	no	—
91	M₁₆×C₁₇	—	4	1	no	—
92	M₁₆×C₁₇	—	4	4	no	—
93	M₁₆×C₁₇	SD₁₆	1	2	yes	—
94	M₁₆×C₁₇	SD₁₆	1	2	yes	—
95	C₂×C₈×C₁₇	—	4	1	no	—
96	C₂×C₈×C₁₇	—	4	4	no	—
97	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₁₇	—	2	1	no	—
105	C₂²⋊C₄×C₁₇	—	2	1	no	—
106	C₄²×C₁₇	C₂×D₄	1	2	yes	—
107	C₂²⋊Dic₁₇	—	4	1	no	—
108	C₄⋊Dic₁₇	—	2	1	no	—
109	C₄⋊Dic₁₇	—	2	2	no	—
110	C₄⋊Dic₁₇	—	2	1	no	—
111	C₄⋊Dic₁₇	—	4	2	no	—
112	C₄⋊Dic₁₇	—	4	1	no	—
113	#63225	—	4	2	no	—
114	#63225	—	4	2	no	—
115	C₄×Dic₁₇	—	4	1	no	—
116	#63214	—	2	2	no	—
117	#63214	—	2	2	no	—
118	#33638	—	17	1	no	maximal, Sylow
119	C₂³×C₁₇	—	4	1	no	—
120	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈×C₁₇	—	2	2	no	—
122	Q₈×C₁₇	—	4	1	no	—
123	D₄×C₁₇	—	2	2	no	—
124	D₄×C₁₇	—	4	2	no	—
125	D₄×C₁₇	—	4	2	no	—
126	D₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	2	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	—	4	1	no	—
138	C₂×Dic₁₇	—	4	1	no	—
139	C₂×Dic₁₇	—	4	2	no	—
140	C₂×Dic₁₇	—	16	1	no	—
141	Dic₃₄	—	16	2	no	—
142	C₈×C₁₇	—	4	2	no	—
143	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
146	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
148	#3163	—	17	1	no	—
149	#2957	D₁₇	1	1	yes	characteristic
150	#2231	—	17	4	no	—
151	#2230	—	17	4	no	—
152	#2155	—	17	1	no	—
153	C₂×#365	—	17	1	no	—
154	#2153	—	17	1	no	—
155	#2145	—	17	1	no	—
156	C₂×#360	—	17	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	—	4	2	no	—
159	C₂²×C₁₇	—	16	1	no	—
160	C₂²×C₁₇	#476	1	1	yes	characteristic
161	C₂²×C₁₇	#476	1	1	yes	characteristic
162	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₁₇	—	4	2	no	—
164	C₄×C₁₇	—	4	2	no	—
165	C₄×C₁₇	#476	1	1	yes	characteristic
166	C₄×C₁₇	#476	1	1	yes	characteristic
167	C₄×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
169	Dic₁₇	—	8	1	no	—
170	Dic₁₇	—	8	2	no	—
171	Dic₁₇	—	8	1	no	—
172	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
173	#568	—	2	1	no	—
174	#544	—	34	1	no	—
175	C₂×M₃₂	—	17	1	no	—
176	C₂×M₃₂	—	17	1	no	—
177	#501	—	34	2	no	—
178	#499	—	17	2	no	—
179	#498	—	17	2	no	—
180	#444	—	2	2	no	—
181	#443	C₂×D₁₇	1	4	yes	—
182	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
183	#428	—	34	1	no	—
184	#427	—	17	1	no	—
185	C₂×#106	—	17	1	no	—
186	#421	—	2	1	no	—
187	#405	C₂×D₁₇	1	1	yes	characteristic
188	#365	—	17	4	no	—
189	#365	—	17	2	no	—
190	#365	—	17	2	no	—
191	#364	—	17	4	no	—
192	#360	—	17	4	no	—
193	#360	—	17	2	no	—
194	#360	—	17	2	no	—
195	C₂×C₁₇	—	2	1	no	—
196	C₂×C₁₇	—	2	1	no	—
197	C₂×C₁₇	—	2	1	no	—
198	C₂×C₁₇	—	8	2	no	—
199	C₂×C₁₇	C₂×#476	1	1	yes	characteristic
200	C₂×#41	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	Dic₃₄	1	2	yes	complemented
204	#135	C₂²×D₁₇	1	2	yes	—
205	C₂×Q₁₆	Dic₃₄	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	2	1	no	—
207	C₂×D₈	Dic₃₄	1	1	yes	characteristic, complemented
208	#131	—	2	4	no	—
209	#131	—	4	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	4	2	no	—
212	C₂×M₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	34	2	no	—
215	C₄⋊Q₈	—	68	1	no	—
216	#125	—	34	2	no	—
217	#125	—	68	1	no	—
218	C₂²⋊Q₈	—	68	2	no	—
219	#117	—	34	1	no	—
220	#117	C₂²⋊D₁₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	34	1	no	—
222	M₃₂	—	17	2	no	—
223	M₃₂	—	17	2	no	—
224	M₃₂	—	34	1	no	—
225	M₃₂	—	34	1	no	—
226	C₂×C₁₆	—	34	1	no	—
227	C₂×C₁₆	—	34	1	no	—
228	#108	—	2	1	no	—
229	#108	C₂²×D₁₇	1	2	yes	—
230	#107	—	34	1	no	—
231	#107	—	34	2	no	—
232	#106	—	17	2	no	—
233	#106	—	17	1	no	—
234	#106	—	17	1	no	—
235	#106	—	34	1	no	—
236	#106	—	34	2	no	—
237	#104	—	2	4	no	—
238	#97	C₂²⋊D₁₇	1	2	yes	—
239	C₄×C₈	C₂²×D₁₇	1	2	yes	—
240	C₁₇	#33638	1	1	yes	characteristic, complemented, Sylow
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	68	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	68	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Dic₃₄	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Dic₃₄	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	#63229	1	2	yes	—
262	M₁₆	#63229	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
266	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
267	C₂×C₈	#63252	1	1	yes	characteristic
268	C₂×C₈	#63252	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
271	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	34	2	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	68	2	no	—
277	C₄⋊C₄	—	68	2	no	—
278	C₄⋊C₄	—	68	2	no	—
279	C₄⋊C₄	—	68	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	68	1	no	—
282	C₄²	—	68	1	no	—
283	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
284	C₁₆	—	34	2	no	—
285	C₁₆	—	34	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#10587635	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	272	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	272	1	no	—
300	C₂×C₄	—	68	1	no	—
301	C₂×C₄	—	68	1	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	#10587610	1	1	yes	characteristic
304	C₂×C₄	#10587610	1	1	yes	characteristic
305	C₂×C₄	C₂×#63229	1	1	yes	characteristic
306	C₂×C₄	C₂×#63229	1	1	yes	characteristic
307	C₂×C₄	#10587635	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	#10587610	1	1	yes	characteristic
311	C₈	#10587610	1	1	yes	characteristic
312	C₈	C₂×#63229	1	4	yes	—
313	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
314	C₈	C₂×#63252	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#11763099	1	1	yes	characteristic
319	C₂²	#11763099	1	1	yes	characteristic
320	C₂²	C₂×#10587635	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	136	2	no	—
324	C₄	—	136	1	no	—
325	C₄	—	136	1	no	—
326	C₄	#11763099	1	1	yes	characteristic
327	C₄	#11763099	1	1	yes	characteristic
328	C₄	C₂×#10587610	1	1	yes	characteristic
329	C₄	C₂²×#63229	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#11763099	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11763099
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2957	C₂
socle	C₂×C₁₇	C₂×#476
prime radical	C₂×C₁₇×#135	C₂²
hypercenter	#2957	D₁₇
hypocenter	C₁₇	#33638
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
1. #14539315	C₂	semidirect
2. #14530960	C₂	semidirect
3. #14530955	C₂	semidirect
4. #14530558	C₂	semidirect
5. C₂×#11762633	C₂	semidirect
6. #14530550	C₂	semidirect
14. #11763211	C₂²	semidirect
15. #11763207	C₂²	semidirect
19. #11762633	C₂²	semidirect
20. #11762633	C₂²	semidirect
21. #11762633	C₂²	semidirect
22. #11762630	C₂²	semidirect
36. C₁₇×#135	Q₈	semidirect
38. C₂×Q₁₆×C₁₇	Q₈	semidirect
40. C₂×D₈×C₁₇	Q₈	semidirect
203. #135	Dic₃₄	semidirect
205. C₂×Q₁₆	Dic₃₄	semidirect
207. C₂×D₈	Dic₃₄	semidirect
240. C₁₇	#33638	semidirect

Maximal subgroups

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

subgroup	index
#14539315	2
#14530960	2
#14530955	2
#14530558	2
C₂×#11762633	2
#14530550	2
#14530520	2
C₂×#11762619	2
C₁₇×#2957	2
#33638	17