Subgroups — #17026316

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541561	C₂	1	2	yes	maximal
2	#14541178	C₂	1	1	yes	maximal, characteristic
3	#14528393	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528364	C₂	1	2	yes	maximal, complemented
5	#14528188	C₂	1	1	yes	maximal, characteristic, complemented
6	#14528170	C₂	1	1	yes	maximal, characteristic
7	#14527990	C₂	1	2	yes	maximal
8	#14527942	C₂	1	2	yes	maximal, complemented
9	#14527681	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#63230	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763519	C₂²	1	2	yes	—
13	#11763214	C₂²	1	1	yes	characteristic
14	#11763177	—	2	2	no	—
15	#11763161	C₂²	1	2	yes	—
16	#11763159	C₂²	1	2	yes	—
17	#11763143	—	2	2	no	—
18	#11763116	—	2	2	no	—
19	#11763104	C₂²	1	2	yes	—
20	#11763104	C₂²	1	2	yes	—
21	#11763047	—	2	1	no	—
22	#11763011	—	2	2	no	—
23	#11763011	C₂²	1	2	yes	—
24	#11762940	C₂²	1	2	yes	—
25	#11762907	—	2	2	no	—
26	C₄×#63230	C₂²	1	1	yes	characteristic
27	C₈×D₆₈	C₂²	1	1	yes	characteristic
28	#107×D₁₇	—	2	1	no	—
29	#11762562	C₂²	1	1	yes	characteristic
30	#11762559	C₂²	1	1	yes	characteristic
31	#11762557	C₂²	1	2	yes	—
32	#11762555	C₂²	1	2	yes	—
33	Q₈⋊#3622	C₂²	1	1	yes	characteristic
34	Q₈⋊#3622	C₂²	1	2	yes	—
35	#11762536	C₂²	1	1	yes	characteristic
36	C₈×#3622	C₂²	1	1	yes	characteristic
37	C₁₇×#557	C₂²	1	2	yes	—
38	C₁₇×#500	C₂²	1	1	yes	characteristic
39	C₁₇×#499	—	2	1	no	—
40	C₁₇×#479	—	2	2	no	—
41	C₁₇×#475	C₂²	1	2	yes	—
42	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
43	C₄×Q₁₆×C₁₇	—	2	1	no	—
44	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₁₇	D₄	1	1	yes	characteristic
46	C₁₇×C₄⋊Q₈	—	2	2	no	—
47	C₁₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₁₇×#125	—	2	2	no	—
49	C₁₇×#125	—	4	2	no	—
50	C₄×Q₈×C₁₇	—	2	2	no	—
51	C₄×Q₈×C₁₇	—	2	2	no	—
52	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
53	C₄×Q₈×C₁₇	C₂³	1	1	yes	characteristic
54	#10587632	—	2	2	no	—
55	C₂×#63230	—	2	2	no	—
56	C₂×#63230	D₄	1	1	yes	characteristic
57	C₂×C₈×D₁₇	—	2	1	no	—
58	#10587582	—	4	2	no	—
59	C₄⋊C₄×D₁₇	—	2	2	no	—
60	C₄×D₆₈	—	2	2	no	—
61	C₄×D₆₈	C₂³	1	1	yes	characteristic
62	#10587556	—	2	2	no	—
63	#10587555	C₂³	1	2	yes	—
64	C₁₇×#107	—	2	1	no	—
65	C₁₇×#107	—	2	1	no	—
66	C₁₇×#107	D₄	1	1	yes	characteristic
67	C₁₇×#106	—	2	2	no	—
68	C₁₇×C₄⋊C₈	—	2	1	no	—
69	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
70	C₁₇×#103	—	2	2	no	—
71	C₁₇×#103	D₄	1	2	yes	—
72	C₄×C₈×C₁₇	—	2	1	no	—
73	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	#10587506	—	2	2	no	—
75	#10587506	—	2	2	no	—
76	#10587506	—	2	2	no	—
77	#10587506	—	2	2	no	—
78	#10587506	—	2	2	no	—
79	#10587506	D₄	1	1	yes	characteristic
80	#10587491	—	2	1	no	—
81	#10587489	—	2	1	no	—
82	#10587489	—	2	1	no	—
83	#10587481	—	4	2	no	—
84	#10587480	D₄	1	1	yes	characteristic
85	#10587479	D₄	1	1	yes	characteristic
86	#10587478	—	2	2	no	—
87	#10587478	—	2	2	no	—
88	C₄⋊#3622	—	2	2	no	—
89	C₄⋊#3622	C₂³	1	2	yes	—
90	C₄⋊#3622	C₂³	1	1	yes	characteristic
91	#10587474	—	2	2	no	—
92	C₄×#3622	C₂³	1	1	yes	characteristic
93	C₄×#3622	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₇	—	2	2	no	—
95	C₂×Q₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
97	C₂×D₆₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₁₇	—	2	1	no	—
99	C₂×C₄×D₁₇	—	4	2	no	—
100	C₂×Dic₃₄	—	2	2	no	—
101	Q₁₆×C₁₇	—	2	2	no	—
102	C₂×C₈×C₁₇	—	2	1	no	—
103	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₄⋊C₄×C₁₇	—	2	1	no	—
106	C₄⋊C₄×C₁₇	—	2	2	no	—
107	C₄⋊C₄×C₁₇	—	2	2	no	—
108	C₄⋊C₄×C₁₇	—	2	2	no	—
109	C₄⋊C₄×C₁₇	—	4	2	no	—
110	C₄⋊C₄×C₁₇	—	4	2	no	—
111	C₄⋊C₄×C₁₇	—	4	2	no	—
112	C₄⋊C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
114	C₄²×C₁₇	—	2	1	no	—
115	C₄²×C₁₇	—	2	2	no	—
116	C₄²×C₁₇	—	4	2	no	—
117	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
118	#63230	—	2	2	no	—
119	#63230	—	4	4	no	—
120	#63227	—	2	1	no	—
121	#63227	—	4	2	no	—
122	C₄⋊Dic₁₇	—	2	2	no	—
123	C₄⋊Dic₁₇	—	2	2	no	—
124	C₄⋊Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	#63225	—	4	2	no	—
126	#63225	—	4	2	no	—
127	C₂×#3622	—	2	2	no	—
128	C₂×#3622	—	2	1	no	—
129	C₂×#3622	—	2	1	no	—
130	C₂×#3622	C₂×D₄	1	1	yes	characteristic
131	C₂×#3622	C₂×D₄	1	1	yes	characteristic
132	C₈×D₁₇	—	4	2	no	—
133	#19938	—	17	1	no	maximal, Sylow
134	C₂²×D₁₇	—	2	1	no	—
135	Q₈×C₁₇	—	2	1	no	—
136	Q₈×C₁₇	—	2	2	no	—
137	Q₈×C₁₇	—	2	1	no	—
138	Q₈×C₁₇	—	2	2	no	—
139	Q₈×C₁₇	—	4	2	no	—
140	Q₈×C₁₇	—	4	2	no	—
141	C₂×C₄×C₁₇	—	2	1	no	—
142	C₂×C₄×C₁₇	—	2	2	no	—
143	C₂×C₄×C₁₇	—	2	1	no	—
144	C₂×C₄×C₁₇	—	2	2	no	—
145	C₂×C₄×C₁₇	—	4	2	no	—
146	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₁₇	—	2	1	no	—
150	C₂×Dic₁₇	—	4	2	no	—
151	D₆₈	—	2	1	no	—
152	D₆₈	#135	1	2	yes	—
153	C₄×D₁₇	—	4	1	no	—
154	C₄×D₁₇	—	4	1	no	—
155	C₄×D₁₇	—	16	2	no	—
156	Dic₃₄	—	8	2	no	—
157	C₈×C₁₇	—	2	1	no	—
158	C₈×C₁₇	—	4	1	no	—
159	C₈×C₁₇	#137	1	2	yes	—
160	#3622	—	2	1	no	—
161	#3622	—	2	1	no	—
162	#3622	—	4	1	no	—
163	#3622	—	4	2	no	—
164	#3622	—	4	1	no	—
165	#3383	D₁₇	1	1	yes	characteristic, complemented
166	#3372	—	17	2	no	—
167	#3338	—	17	1	no	—
168	#1710	—	17	1	no	—
169	#1704	—	17	2	no	—
170	#1676	—	17	1	no	—
171	#1673	—	17	1	no	—
172	#1643	—	17	2	no	—
173	#1636	—	17	2	no	—
174	#1591	—	17	1	no	—
175	C₈×SD₁₆	—	17	1	no	—
176	C₂²×C₁₇	D₄²	1	1	yes	characteristic
177	C₂×D₁₇	—	2	2	no	—
178	C₂×D₁₇	—	4	1	no	—
179	C₂×D₁₇	—	4	1	no	—
180	C₄×C₁₇	—	2	1	no	—
181	C₄×C₁₇	—	4	2	no	—
182	C₄×C₁₇	—	4	1	no	—
183	C₄×C₁₇	—	4	2	no	—
184	C₄×C₁₇	—	4	1	no	—
185	C₄×C₁₇	—	8	2	no	—
186	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
187	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
188	C₄×C₁₇	#578	1	1	yes	characteristic
189	C₄×C₁₇	#578	1	1	yes	characteristic
190	Dic₁₇	—	4	1	no	—
191	Dic₁₇	—	8	2	no	—
192	#557	C₂×D₁₇	1	2	yes	—
193	#554	—	17	2	no	—
194	#500	—	17	1	no	—
195	#500	C₂×D₁₇	1	1	yes	characteristic
196	#499	—	2	1	no	—
197	#491	—	34	2	no	—
198	#487	—	17	2	no	—
199	#487	—	17	2	no	—
200	#484	—	34	2	no	—
201	#479	—	2	2	no	—
202	#479	—	34	2	no	—
203	#477	—	17	2	no	—
204	#477	—	17	2	no	—
205	#475	C₂×D₁₇	1	2	yes	—
206	#467	—	34	1	no	—
207	#461	—	17	2	no	—
208	#461	—	34	2	no	—
209	#446	—	17	2	no	—
210	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
211	#440	—	34	2	no	—
212	C₄×Q₁₆	—	2	1	no	—
213	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
214	C₄×SD₁₆	—	17	1	no	—
215	C₈×D₄	—	17	1	no	—
216	C₂×#107	—	34	1	no	—
217	#335	—	17	1	no	—
218	#334	—	17	1	no	—
219	#333	—	17	2	no	—
220	#332	—	17	2	no	—
221	Q₈⋊C₈	—	17	1	no	—
222	Q₈⋊C₈	—	17	2	no	—
223	D₄⋊C₈	—	17	1	no	—
224	C₈²	—	17	1	no	—
225	C₂×C₁₇	#3276	1	1	yes	characteristic
226	C₂×C₁₇	#3285	1	1	yes	characteristic
227	C₂×C₁₇	#3290	1	1	yes	characteristic
228	D₁₇	—	4	2	no	—
229	C₂×Q₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
230	C₂×SD₁₆	—	17	1	no	—
231	C₂×SD₁₆	—	34	2	no	—
232	C₂²×C₈	—	34	1	no	—
233	C₄⋊Q₈	—	2	2	no	—
234	C₄⋊Q₈	—	17	2	no	—
235	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
236	#125	—	2	2	no	—
237	#125	—	4	2	no	—
238	#125	—	34	2	no	—
239	C₂²⋊Q₈	—	34	2	no	—
240	C₂²⋊Q₈	—	68	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
245	C₄×D₄	—	17	1	no	—
246	C₄×D₄	—	34	2	no	—
247	C₂×C₄⋊C₄	—	34	2	no	—
248	#107	—	2	1	no	—
249	#107	—	2	1	no	—
250	#107	—	34	1	no	—
251	#107	—	34	1	no	—
252	#107	—	34	2	no	—
253	#107	—	34	2	no	—
254	#107	C₂²⋊D₁₇	1	1	yes	characteristic
255	#106	—	2	2	no	—
256	#106	—	17	1	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	17	1	no	—
259	C₄⋊C₈	—	17	2	no	—
260	C₄⋊C₈	—	34	2	no	—
261	C₄⋊C₈	C₂²×D₁₇	1	1	yes	characteristic
262	#103	—	2	2	no	—
263	#103	—	17	1	no	—
264	#103	—	34	2	no	—
265	#103	—	34	2	no	—
266	#103	—	34	2	no	—
267	#103	—	34	2	no	—
268	#103	—	34	2	no	—
269	#103	—	68	2	no	—
270	#103	C₂²⋊D₁₇	1	2	yes	—
271	#102	—	17	1	no	—
272	C₂²⋊C₈	—	34	1	no	—
273	#97	—	34	2	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	—	17	1	no	—
276	C₄×C₈	—	17	1	no	—
277	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
278	C₁₇	#19938	1	1	yes	characteristic, complemented, Sylow
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	34	2	no	—
281	C₂×Q₈	D₄×D₁₇	1	1	yes	characteristic
282	C₂×Q₈	C₂×C₂²⋊D₁₇	1	2	yes	—
283	C₂×D₄	—	17	1	no	—
284	C₂²×C₄	—	34	1	no	—
285	C₂²×C₄	—	68	2	no	—
286	Q₁₆	—	2	2	no	—
287	SD₁₆	—	34	2	no	—
288	SD₁₆	—	68	4	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	17	1	no	—
291	C₂×C₈	—	17	1	no	—
292	C₂×C₈	—	34	2	no	—
293	C₂×C₈	—	34	1	no	—
294	C₂×C₈	—	34	1	no	—
295	C₂×C₈	—	68	2	no	—
296	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	17	1	no	—
306	C₄⋊C₄	—	34	2	no	—
307	C₄⋊C₄	—	34	2	no	—
308	C₄⋊C₄	—	68	2	no	—
309	C₄⋊C₄	—	68	2	no	—
310	C₄⋊C₄	D₄×D₁₇	1	1	yes	characteristic
311	C₄⋊C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
312	C₂²⋊C₄	—	34	1	no	—
313	C₂²⋊C₄	—	68	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	1	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂³×D₁₇	1	1	yes	characteristic
318	C₂³	—	34	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	136	2	no	—
326	D₄	—	17	2	no	—
327	D₄	—	34	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	272	2	no	—
334	C₂×C₄	—	34	1	no	—
335	C₂×C₄	—	68	1	no	—
336	C₂×C₄	—	68	1	no	—
337	C₂×C₄	—	68	2	no	—
338	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
339	C₂×C₄	#10587691	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	34	1	no	—
344	C₈	—	34	1	no	—
345	C₈	—	68	1	no	—
346	C₈	—	68	2	no	—
347	C₈	—	68	1	no	—
348	C₈	#10587628	1	2	yes	—
349	C₂²	—	34	2	no	—
350	C₂²	—	68	1	no	—
351	C₂²	—	68	1	no	—
352	C₂²	D₄×C₂²⋊D₁₇	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	8	2	no	—
359	C₄	—	68	1	no	—
360	C₄	—	136	2	no	—
361	C₄	#135×D₁₇	1	1	yes	characteristic
362	C₄	C₂×#10587628	1	1	yes	characteristic
363	C₄	#11763623	1	1	yes	characteristic
364	C₄	#11763624	1	1	yes	characteristic
365	C₂	—	68	2	no	—
366	C₂	#14540430	1	1	yes	characteristic
367	C₂	#14540519	1	1	yes	characteristic
368	C₂	#14540570	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₁₇
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#3383	C₂
socle	C₂²×C₁₇	D₄²
prime radical	C₂×D₆₈	C₂×D₄
hypercenter	#3383	D₁₇
hypocenter	C₁₇	#19938
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3383

Complements

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

N	H ≅ G/N	type
3. #14528393	C₂	semidirect
4. #14528364	C₂	semidirect
5. #14528188	C₂	semidirect
8. #14527942	C₂	semidirect
9. #14527681	C₂	semidirect
11. C₁₇×#3383	C₂	semidirect
165. #3383	D₁₇	semidirect
278. C₁₇	#19938	semidirect

Maximal subgroups

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

subgroup	index
#14541561	2
#14541178	2
#14528393	2
#14528364	2
#14528188	2
#14528170	2
#14527990	2
#14527942	2
#14527681	2
C₈×#63230	2
C₁₇×#3383	2
#19938	17