Subgroups — #17189252

Order 4352.

610 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	C₂×C₄×D₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#14530979	C₂	1	2	yes	maximal, complemented
3	#14530928	C₂	1	2	yes	maximal, complemented
4	#14530826	C₂	1	2	yes	maximal, complemented
5	#14530712	C₂	1	2	yes	maximal
6	C₄×#10587499	C₂	1	4	yes	maximal, complemented
7	#14530583	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₆₈	—	2	1	no	—
11	C₂×#10587499	C₂²	1	4	yes	—
12	C₄×D₁₃₆	—	2	1	no	—
13	C₄×D₁₃₆	—	2	1	no	—
14	C₄×D₁₃₆	C₂²	1	2	yes	—
15	C₄×D₁₃₆	C₂²	1	2	yes	—
16	C₂×#10587489	C₂²	1	1	yes	characteristic
17	C₂×#10587492	—	2	1	no	—
18	#11762636	C₂²	1	4	yes	complemented
19	C₄⋊#63214	C₂²	1	2	yes	complemented
20	#11762613	C₂²	1	4	yes	complemented
21	#11762609	C₂²	1	4	yes	—
22	C₂²⋊#63214	C₂²	1	2	yes	—
23	C₄×#63214	C₂²	1	2	yes	complemented
24	C₁₇×#533	—	2	1	no	—
25	C₁₇×#500	C₂²	1	1	yes	characteristic, complemented
26	C₁₇×#494	C₂²	1	1	yes	characteristic, complemented
27	C₁₇×#487	—	2	1	no	—
28	C₁₇×#467	—	2	1	no	—
29	C₁₇×#467	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₁₇	—	2	1	no	—
31	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
33	C₂²×D₆₈	—	2	1	no	—
34	C₂²×C₄×D₁₇	—	4	1	no	—
35	C₂×Q₁₆×C₁₇	—	2	1	no	—
36	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×C₄⋊Q₈	—	2	1	no	—
40	C₁₇×C₄⋊Q₈	—	2	1	no	—
41	C₁₇×#125	—	4	1	no	—
42	C₁₇×C₂²⋊Q₈	—	2	2	no	—
43	C₁₇×C₂²⋊Q₈	—	4	1	no	—
44	C₄×Q₈×C₁₇	—	2	2	no	—
45	C₄×Q₈×C₁₇	—	4	1	no	—
46	C₁₇×#117	—	4	1	no	—
47	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
48	C₂×#63227	—	4	1	no	—
49	C₂×C₄⋊Dic₁₇	—	2	1	no	—
50	C₂×D₁₃₆	—	2	4	no	—
51	C₂×D₁₃₆	—	2	1	no	—
52	C₂×D₁₃₆	C₂³	1	2	yes	—
53	C₄×D₆₈	—	2	2	no	—
54	C₄×D₆₈	—	2	2	no	—
55	C₄×D₆₈	—	4	1	no	—
56	C₄×D₆₈	—	4	1	no	—
57	C₁₇×#107	—	2	1	no	—
58	C₁₇×#107	C₂³	1	2	yes	—
59	C₁₇×#103	—	2	2	no	—
60	C₁₇×#103	—	2	2	no	—
61	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
65	#10587499	—	2	8	no	—
66	#10587492	—	2	2	no	—
67	#10587492	—	2	2	no	—
68	#10587489	—	2	1	no	—
69	#10587489	C₂³	1	2	yes	—
70	C₂×#63214	C₂³	1	4	yes	—
71	C₂³×D₁₇	—	4	1	no	—
72	C₂×Q₈×C₁₇	—	2	2	no	—
73	C₂×Q₈×C₁₇	—	4	1	no	—
74	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₇	—	4	1	no	—
78	C₂×D₆₈	—	2	4	no	—
79	C₂×D₆₈	—	2	2	no	—
80	C₂×D₆₈	—	4	2	no	—
81	C₂×D₆₈	—	4	1	no	—
82	C₂×C₄×D₁₇	—	4	2	no	—
83	C₂×C₄×D₁₇	—	4	2	no	—
84	C₂×C₄×D₁₇	—	8	4	no	—
85	Q₁₆×C₁₇	—	2	4	no	—
86	Q₁₆×C₁₇	—	4	2	no	—
87	C₂×C₈×C₁₇	—	2	2	no	—
88	C₂×C₈×C₁₇	—	4	1	no	—
89	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₇	—	2	2	no	—
94	C₄⋊C₄×C₁₇	—	4	2	no	—
95	C₄⋊C₄×C₁₇	—	4	2	no	—
96	C₄⋊C₄×C₁₇	—	4	1	no	—
97	C₄⋊C₄×C₁₇	—	4	1	no	—
98	C₄⋊C₄×C₁₇	—	4	1	no	—
99	C₂²⋊C₄×C₁₇	—	4	2	no	—
100	C₄²×C₁₇	—	4	2	no	—
101	C₄²×C₁₇	D₈	1	1	yes	characteristic
102	C₄²×C₁₇	D₈	1	1	yes	characteristic
103	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	#63227	—	4	2	no	—
106	#63227	—	4	2	no	—
107	C₄⋊Dic₁₇	—	2	2	no	—
108	C₄⋊Dic₁₇	—	4	1	no	—
109	D₁₃₆	—	2	4	no	—
110	D₁₃₆	—	2	2	no	—
111	D₁₃₆	#41	1	4	yes	—
112	#63214	—	2	4	no	—
113	#33661	—	17	1	no	maximal, Sylow
114	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₁₇	—	4	4	no	—
116	C₂²×D₁₇	—	4	2	no	—
117	C₂²×D₁₇	—	8	2	no	—
118	C₂²×D₁₇	—	8	2	no	—
119	Q₈×C₁₇	—	4	2	no	—
120	Q₈×C₁₇	—	4	2	no	—
121	Q₈×C₁₇	—	8	2	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	—	4	2	no	—
126	C₂×C₄×C₁₇	—	4	2	no	—
127	C₂×C₄×C₁₇	SD₃₂	1	4	yes	complemented
128	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	—	4	2	no	—
135	C₂×Dic₁₇	—	8	2	no	—
136	D₆₈	—	2	4	no	—
137	D₆₈	—	4	4	no	—
138	D₆₈	—	4	2	no	—
139	C₄×D₁₇	—	8	4	no	—
140	C₄×D₁₇	—	8	4	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	C₂×#41	1	2	yes	—
145	#3170	D₁₇	1	1	yes	characteristic, complemented
146	C₂×C₄×D₈	—	17	1	no	—
147	#2235	—	17	2	no	—
148	#2226	—	17	2	no	—
149	#2208	—	17	2	no	—
150	#2187	—	17	2	no	—
151	C₄×SD₃₂	—	17	4	no	—
152	#2162	—	17	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
156	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₁₇	—	4	4	no	—
158	C₂×D₁₇	—	8	8	no	—
159	C₂×D₁₇	—	8	2	no	—
160	C₂×D₁₇	—	8	2	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	8	2	no	—
165	C₄×C₁₇	—	8	2	no	—
166	C₄×C₁₇	C₂×SD₃₂	1	2	yes	—
167	C₄×C₁₇	C₂×SD₃₂	1	2	yes	—
168	C₄×C₁₇	#525	1	1	yes	characteristic
169	C₄×C₁₇	#575	1	1	yes	characteristic
170	Dic₁₇	—	8	2	no	—
171	C₂²×D₈	—	17	1	no	—
172	#533	—	2	1	no	—
173	C₂×C₄×D₄	—	34	1	no	—
174	C₂×SD₃₂	—	17	4	no	—
175	#500	C₂×D₁₇	1	1	yes	characteristic, complemented
176	#494	C₂×D₁₇	1	1	yes	characteristic, complemented
177	#487	—	2	1	no	—
178	#467	—	2	1	no	—
179	#467	C₂×D₁₇	1	2	yes	—
180	C₄×Q₁₆	—	2	1	no	—
181	C₄×Q₁₆	C₂×D₁₇	1	2	yes	complemented
182	C₄×D₈	—	17	2	no	—
183	C₄×D₈	—	17	2	no	—
184	C₄×D₈	—	34	1	no	—
185	C₄×D₈	—	34	1	no	—
186	C₂×#107	—	17	1	no	—
187	C₂×#102	—	34	1	no	—
188	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
189	#367	—	17	4	no	—
190	C₄⋊C₁₆	—	17	2	no	—
191	#358	—	17	4	no	—
192	#357	—	17	4	no	—
193	C₂²⋊C₁₆	—	17	2	no	—
194	C₄×C₁₆	—	17	2	no	—
195	C₂×C₁₇	—	2	2	no	—
196	C₂×C₁₇	#3100	1	1	yes	characteristic
197	C₂×C₁₇	C₂²×SD₃₂	1	1	yes	characteristic
198	C₂×C₁₇	#3414	1	1	yes	characteristic
199	D₁₇	—	8	4	no	—
200	C₂²×D₄	—	34	1	no	—
201	C₂³×C₄	—	68	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
204	C₂×D₈	—	17	2	no	—
205	C₂×D₈	—	34	4	no	—
206	C₂×D₈	—	34	1	no	—
207	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×D₄	—	34	2	no	—
217	C₄×D₄	—	34	2	no	—
218	C₄×D₄	—	68	1	no	—
219	C₄×D₄	—	68	1	no	—
220	#117	—	4	1	no	—
221	C₂×C₄⋊C₄	—	34	1	no	—
222	C₂×C₂²⋊C₄	—	68	1	no	—
223	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
224	SD₃₂	—	34	8	no	—
225	C₂×C₁₆	—	17	4	no	—
226	#107	—	2	1	no	—
227	#107	—	17	2	no	—
228	#107	—	34	1	no	—
229	#107	C₂²×D₁₇	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#102	—	34	2	no	—
233	#102	—	34	2	no	—
234	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₁₇	#33661	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	—	68	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	34	4	no	—
243	C₂×D₄	—	34	2	no	—
244	C₂×D₄	—	68	2	no	—
245	C₂×D₄	—	68	1	no	—
246	C₂²×C₄	—	68	1	no	—
247	C₂²×C₄	—	68	2	no	—
248	C₂²×C₄	—	68	2	no	—
249	C₂²×C₄	—	136	4	no	—
250	C₂²×C₄	#63228	1	1	yes	characteristic
251	C₂²×C₄	#63228	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	4	2	no	—
255	D₈	—	17	4	no	—
256	D₈	—	34	4	no	—
257	D₈	—	34	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	34	2	no	—
271	C₄⋊C₄	—	68	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	68	2	no	—
274	C₂²⋊C₄	—	68	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	#63228	1	1	yes	characteristic
277	C₄²	#63228	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
280	C₁₆	—	34	4	no	—
281	C₂³	—	68	4	no	—
282	C₂³	—	68	2	no	—
283	C₂³	—	136	2	no	—
284	C₂³	—	136	2	no	—
285	C₂³	C₂×#63228	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	34	4	no	—
290	D₄	—	68	4	no	—
291	D₄	—	68	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	68	2	no	—
298	C₂×C₄	—	136	2	no	—
299	C₂×C₄	—	136	4	no	—
300	C₂×C₄	—	136	4	no	—
301	C₂×C₄	#10587499	1	4	yes	complemented
302	C₂×C₄	C₂×#63228	1	1	yes	characteristic
303	C₂×C₄	C₂×#63228	1	1	yes	characteristic
304	C₂×C₄	C₂×#63228	1	1	yes	characteristic
305	C₂×C₄	C₂×#63228	1	1	yes	characteristic
306	C₂×C₄	C₂×#63228	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₁₇	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	68	4	no	—
315	C₂²	—	136	8	no	—
316	C₂²	—	136	2	no	—
317	C₂²	—	136	2	no	—
318	C₂²	C₂×#10587499	1	2	yes	—
319	C₂²	C₂²×#63228	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	136	2	no	—
326	C₄	C₂×#10587499	1	2	yes	—
327	C₄	C₂×#10587499	1	2	yes	—
328	C₄	#11763316	1	1	yes	characteristic
329	C₄	#11763607	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	136	4	no	—
332	C₂	#14538780	1	1	yes	characteristic
333	C₂	C₂²×#10587499	1	1	yes	characteristic
334	C₂	#14541918	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3170	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂²×D₁₃₆	C₂²
hypercenter	#3170	D₁₇
hypocenter	C₁₇	#33661
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #14530979	C₂	semidirect
3. #14530928	C₂	semidirect
4. #14530826	C₂	semidirect
6. C₄×#10587499	C₂	semidirect
7. #14530583	C₂	semidirect
8. C₁₇×#3170	C₂	semidirect
18. #11762636	C₂²	semidirect
19. C₄⋊#63214	C₂²	semidirect
20. #11762613	C₂²	semidirect
23. C₄×#63214	C₂²	semidirect
25. C₁₇×#500	C₂²	semidirect
26. C₁₇×#494	C₂²	semidirect
31. C₄×Q₁₆×C₁₇	C₂²	semidirect
127. C₂×C₄×C₁₇	SD₃₂	semidirect
145. #3170	D₁₇	semidirect
175. #500	C₂×D₁₇	semidirect
176. #494	C₂×D₁₇	semidirect
181. C₄×Q₁₆	C₂×D₁₇	semidirect
238. C₁₇	#33661	semidirect
301. C₂×C₄	#10587499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₃₆	2
#14530979	2
#14530928	2
#14530826	2
#14530712	2
C₄×#10587499	2
#14530583	2
C₁₇×#3170	2
#33661	17