Subgroups — #16951194

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537685	C₂	1	1	yes	maximal, characteristic
2	#14530413	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762593	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530399	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526874	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526869	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763607	C₄	1	1	yes	characteristic
9	C₂×#10587587	C₂²	1	1	yes	characteristic
10	C₂×#10587483	C₂²	1	1	yes	characteristic
11	C₂²×#63214	C₂²	1	1	yes	characteristic
12	#11762929	—	2	2	no	—
13	#11762924	—	2	2	no	—
14	#11762857	C₄	1	1	yes	characteristic
15	#11762826	C₄	1	1	yes	characteristic
16	#11762825	C₄	1	1	yes	characteristic
17	#11762764	C₂²	1	1	yes	characteristic
18	#11762594	—	2	2	no	—
19	#11762593	—	2	2	no	—
20	#11762588	—	2	2	no	—
21	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
22	C₁₇×#504	—	2	2	no	—
23	C₁₇×#504	—	2	2	no	—
24	C₂×C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#63252	C₂×C₄	1	1	yes	characteristic
29	C₁₇×#131	—	2	1	no	—
30	C₁₇×#131	—	2	1	no	—
31	C₁₇×#131	D₄	1	2	yes	—
32	C₁₇×#131	D₄	1	2	yes	—
33	C₂×M₁₆×C₁₇	—	2	1	no	—
34	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	—	2	1	no	—
36	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
37	#10587637	C₈	1	2	yes	—
38	#10587636	—	2	2	no	—
39	#10587635	C₈	1	2	yes	—
40	#10587628	—	2	1	no	—
41	#10587618	—	2	1	no	—
42	#10587612	C₂×C₄	1	1	yes	characteristic
43	C₂×#63223	C₂×C₄	1	1	yes	characteristic
44	C₂×#63223	C₂×C₄	1	1	yes	characteristic
45	#10587587	—	2	1	no	—
46	#10587587	—	2	1	no	—
47	#10587587	—	2	1	no	—
48	#10587587	—	2	1	no	—
49	C₂×#63218	—	2	1	no	—
50	C₂×C₈×D₁₇	—	2	1	no	—
51	C₁₇×M₃₂	—	2	4	no	—
52	C₁₇×M₃₂	—	2	1	no	—
53	C₁₇×M₃₂	—	2	1	no	—
54	C₁₇×M₃₂	—	2	1	no	—
55	C₁₇×M₃₂	D₄	1	2	yes	—
56	C₂×C₁₆×C₁₇	—	2	4	no	—
57	C₂×C₁₆×C₁₇	—	2	1	no	—
58	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
59	#10587508	—	2	2	no	—
60	#10587496	—	2	2	no	—
61	#10587493	—	2	2	no	—
62	#10587486	—	2	1	no	—
63	C₈×Dic₁₇	—	2	1	no	—
64	#10587483	—	2	1	no	—
65	#10587483	—	2	1	no	—
66	C₂×#63214	—	2	1	no	—
67	C₂×#63214	—	2	1	no	—
68	C₂×#63214	—	2	1	no	—
69	C₂×#63214	—	2	1	no	—
70	#41×C₁₇	—	2	1	no	—
71	#41×C₁₇	—	2	1	no	—
72	#41×C₁₇	C₂²⋊C₄	1	2	yes	—
73	#41×C₁₇	C₂×C₈	1	2	yes	—
74	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
75	C₂×D₄×C₁₇	—	2	1	no	—
76	C₂×D₄×C₁₇	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	—	2	1	no	—
78	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₇	—	2	1	no	—
80	#63252	—	2	1	no	—
81	#63252	—	2	1	no	—
82	#63252	—	2	1	no	—
83	#63252	—	2	1	no	—
84	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₇	—	2	1	no	—
86	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
87	M₁₆×C₁₇	—	2	1	no	—
88	M₁₆×C₁₇	—	2	2	no	—
89	M₁₆×C₁₇	—	2	2	no	—
90	M₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₁₇	—	2	1	no	—
92	C₂×C₈×C₁₇	—	2	2	no	—
93	C₂×C₈×C₁₇	—	2	2	no	—
94	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
100	C₂²⋊Dic₁₇	—	2	1	no	—
101	#63231	—	2	2	no	—
102	#63230	—	2	2	no	—
103	#63229	—	2	2	no	—
104	#63228	—	2	2	no	—
105	C₄⋊Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	#63223	—	2	1	no	—
108	#63223	—	2	1	no	—
109	#63223	—	2	1	no	—
110	#63223	C₂×C₈	1	2	yes	—
111	C₂×#3622	—	2	1	no	—
112	C₂×#3622	—	2	1	no	—
113	#63218	—	4	1	no	—
114	#63218	—	4	1	no	—
115	C₈×D₁₇	—	4	1	no	—
116	C₈×D₁₇	—	4	1	no	—
117	C₁₆×C₁₇	—	2	2	no	—
118	C₁₆×C₁₇	—	2	2	no	—
119	#63214	—	4	1	no	—
120	#63214	—	4	1	no	—
121	#12979	—	17	1	no	maximal, Sylow
122	C₂³×C₁₇	—	2	1	no	—
123	C₂³×C₁₇	#131	1	1	yes	characteristic
124	C₂²×D₁₇	—	2	1	no	—
125	Q₈×C₁₇	—	2	1	no	—
126	Q₈×C₁₇	C₂²⋊C₈	1	2	yes	—
127	D₄×C₁₇	—	2	2	no	—
128	D₄×C₁₇	—	2	2	no	—
129	D₄×C₁₇	—	2	1	no	—
130	D₄×C₁₇	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₁₇	—	2	1	no	—
132	C₂×C₄×C₁₇	—	2	2	no	—
133	C₂×C₄×C₁₇	—	2	2	no	—
134	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
140	C₂²⋊D₁₇	—	4	1	no	—
141	C₂²⋊D₁₇	—	4	1	no	—
142	C₂×Dic₁₇	—	2	1	no	—
143	C₂×Dic₁₇	—	4	1	no	—
144	D₆₈	—	2	1	no	—
145	D₆₈	—	2	1	no	—
146	C₄×D₁₇	—	4	1	no	—
147	C₄×D₁₇	—	4	1	no	—
148	Dic₃₄	—	2	1	no	—
149	Dic₃₄	—	2	1	no	—
150	C₈×C₁₇	—	2	2	no	—
151	C₈×C₁₇	#120	1	1	yes	characteristic
152	C₈×C₁₇	#121	1	1	yes	characteristic
153	C₈×C₁₇	#121	1	1	yes	characteristic
154	C₈×C₁₇	#123	1	1	yes	characteristic
155	#3622	—	2	2	no	—
156	#3622	—	4	1	no	—
157	#3407	D₁₇	1	1	yes	characteristic, complemented
158	#2955	—	17	1	no	—
159	#2117	—	17	1	no	—
160	C₂×#350	—	17	1	no	—
161	#2113	—	17	1	no	—
162	#1379	—	17	1	no	—
163	#1378	—	17	1	no	—
164	C₂²×C₁₇	—	2	2	no	—
165	C₂²×C₁₇	—	4	1	no	—
166	C₂²×C₁₇	—	8	1	no	—
167	C₂²×C₁₇	#408	1	1	yes	characteristic
168	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
169	C₂²×C₁₇	#435	1	1	yes	characteristic
170	C₂×D₁₇	—	4	2	no	—
171	C₂×D₁₇	—	4	1	no	—
172	C₄×C₁₇	—	2	2	no	—
173	C₄×C₁₇	#386	1	1	yes	characteristic
174	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
176	C₄×C₁₇	#435	1	1	yes	characteristic
177	Dic₁₇	—	4	1	no	—
178	Dic₁₇	—	8	1	no	—
179	#575	—	17	1	no	—
180	C₂×#131	—	17	1	no	—
181	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
182	#504	—	2	2	no	—
183	#504	—	2	2	no	—
184	C₂×M₃₂	—	2	2	no	—
185	C₂×M₃₂	—	17	1	no	—
186	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
187	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
188	C₂²×C₁₆	—	17	1	no	—
189	#444	—	34	2	no	—
190	#443	—	34	2	no	—
191	#430	—	17	1	no	—
192	#421	—	17	1	no	—
193	#421	—	17	1	no	—
194	#405	—	17	1	no	—
195	#350	—	34	2	no	—
196	#350	—	34	2	no	—
197	C₂²⋊C₁₆	—	34	2	no	—
198	C₂×C₁₇	—	2	1	no	—
199	C₂×C₁₇	—	2	1	no	—
200	C₂×C₁₇	—	2	1	no	—
201	C₂×C₁₇	—	4	2	no	—
202	C₂×C₁₇	#1869	1	1	yes	characteristic
203	D₁₇	—	8	1	no	—
204	C₂×#41	—	17	1	no	—
205	C₂×#41	C₄×D₁₇	1	1	yes	characteristic
206	#137	—	17	2	no	—
207	#137	—	34	1	no	—
208	AGL(1,ℤ/8)	—	17	2	no	—
209	AGL(1,ℤ/8)	—	34	1	no	—
210	#135	—	34	2	no	—
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	34	1	no	—
214	#131	—	34	1	no	—
215	#131	—	34	1	no	—
216	#131	—	34	1	no	—
217	#131	D₆₈	1	2	yes	—
218	#131	C₂²⋊D₁₇	1	2	yes	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	17	1	no	—
221	C₂×M₁₆	—	17	1	no	—
222	C₂×M₁₆	—	34	1	no	—
223	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	34	1	no	—
226	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
227	#117	—	17	1	no	—
228	M₃₂	—	2	4	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	2	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	34	1	no	—
233	M₃₂	—	34	1	no	—
234	M₃₂	C₂²⋊D₁₇	1	2	yes	—
235	C₂×C₁₆	—	2	4	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	34	1	no	—
238	C₂×C₁₆	—	34	1	no	—
239	C₂×C₁₆	—	34	1	no	—
240	C₂×C₁₆	—	34	1	no	—
241	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
242	#108	—	34	2	no	—
243	#104	—	34	2	no	—
244	#104	—	34	2	no	—
245	#97	—	34	1	no	—
246	C₄×C₈	—	34	1	no	—
247	C₁₇	#12979	1	1	yes	characteristic, complemented, Sylow
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	34	1	no	—
251	#41	—	34	1	no	—
252	#41	—	34	1	no	—
253	#41	—	34	1	no	—
254	#41	C₈×D₁₇	1	2	yes	—
255	#41	#63227	1	2	yes	—
256	C₂×Q₈	—	17	1	no	—
257	C₂×Q₈	#63218	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	17	1	no	—
260	C₂×D₄	—	34	1	no	—
261	C₂×D₄	#63218	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
265	Q₁₆	—	34	2	no	—
266	SD₁₆	—	34	2	no	—
267	SD₁₆	—	34	2	no	—
268	D₈	—	34	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	17	2	no	—
273	M₁₆	—	34	1	no	—
274	M₁₆	—	34	1	no	—
275	M₁₆	—	34	1	no	—
276	M₁₆	—	68	1	no	—
277	M₁₆	—	68	1	no	—
278	M₁₆	#63227	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	34	1	no	—
283	C₂×C₈	—	34	1	no	—
284	C₂×C₈	—	68	1	no	—
285	C₂×C₈	—	68	1	no	—
286	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
287	C₂×C₈	#63252	1	1	yes	characteristic
288	C₂×C₈	#63252	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
292	C₄⋊C₄	—	34	1	no	—
293	C₂²⋊C₄	—	34	1	no	—
294	C₄²	—	34	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	68	1	no	—
298	C₁₆	—	68	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	34	1	no	—
301	C₂³	#10587587	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	34	1	no	—
304	Q₈	—	34	1	no	—
305	Q₈	#10587491	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	34	1	no	—
310	D₄	—	34	1	no	—
311	D₄	—	68	1	no	—
312	D₄	—	68	1	no	—
313	D₄	#10587491	1	2	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	68	1	no	—
319	C₂×C₄	—	68	1	no	—
320	C₂×C₄	—	68	1	no	—
321	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
322	C₂×C₄	C₂×#63218	1	1	yes	characteristic
323	C₂×C₄	#10587587	1	1	yes	characteristic
324	C₂×C₄	C₂×#63227	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	34	2	no	—
329	C₈	—	68	1	no	—
330	C₈	#10587615	1	1	yes	characteristic
331	C₈	#10587616	1	1	yes	characteristic
332	C₈	#10587616	1	1	yes	characteristic
333	C₈	#10587639	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	68	2	no	—
338	C₂²	—	68	1	no	—
339	C₂²	#11762775	1	1	yes	characteristic
340	C₂²	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
341	C₂²	#11762880	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	68	1	no	—
344	C₄	—	136	1	no	—
345	C₄	#11762688	1	1	yes	characteristic
346	C₄	C₂×#10587491	1	1	yes	characteristic
347	C₄	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
348	C₄	#11762880	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	136	1	no	—
354	C₂	#14529076	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587639
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#3407	C₂
socle	C₂×C₁₇	#1869
prime radical	#11763607	C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#3407	D₁₇
hypocenter	C₁₇	#12979
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

Complements

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

N	H ≅ G/N	type
2. #14530413	C₂	semidirect
3. C₂×#11762593	C₂	semidirect
4. #14530399	C₂	semidirect
5. #14526874	C₂	semidirect
6. #14526869	C₂	semidirect
7. C₁₇×#3407	C₂	semidirect
157. #3407	D₁₇	semidirect
247. C₁₇	#12979	semidirect

Maximal subgroups

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

subgroup	index
#14537685	2
#14530413	2
C₂×#11762593	2
#14530399	2
#14526874	2
#14526869	2
C₁₇×#3407	2
#12979	17