Subgroups — #17004178

Order 4352.

646 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537897	C₂	1	2	yes	maximal, complemented
2	C₄×#10587590	C₂	1	1	yes	maximal, characteristic
3	#14527688	C₂	1	4	yes	maximal, complemented
4	#14527682	C₂	1	2	yes	maximal, complemented
5	#14527670	C₂	1	2	yes	maximal, complemented
6	C₂×#11762562	C₂	1	1	yes	maximal, characteristic, complemented
7	#14527172	C₂	1	2	yes	maximal, complemented
8	C₁₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10587590	C₄	1	1	yes	characteristic
10	C₄×#63252	C₂²	1	2	yes	—
11	#11762939	C₂²	1	4	yes	—
12	C₄×Dic₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
14	C₄×#63219	C₄	1	2	yes	—
15	C₄×#63219	C₂²	1	2	yes	—
16	C₄×D₁₃₆	C₄	1	1	yes	characteristic
17	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
18	#11762888	C₂²	1	4	yes	—
19	#11762877	—	2	4	no	—
20	#11762865	—	2	2	no	—
21	#11762847	C₄	1	1	yes	characteristic
22	#11762831	—	2	2	no	—
23	C₂×C₄⋊#3622	C₂²	1	2	yes	—
24	#11762806	C₄	1	2	yes	—
25	#11762562	C₂²	1	2	yes	—
26	#11762562	C₂²	1	2	yes	—
27	#11762541	C₂²	1	4	yes	—
28	#11762538	C₂²	1	4	yes	—
29	C₁₇×#431	C₂²	1	1	yes	characteristic
30	C₁₇×#431	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₁₇×#322	C₂²	1	2	yes	—
33	C₈²×C₁₇	C₂²	1	2	yes	—
34	C₂×#63252	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
38	C₄×C₂²⋊D₁₇	—	2	2	no	—
39	C₄×C₂²⋊D₁₇	—	2	2	no	—
40	#10587612	C₂×C₄	1	2	yes	—
41	C₂²×#3622	—	2	2	no	—
42	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
43	#10587590	—	2	4	no	—
44	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#63219	C₂×C₄	1	2	yes	—
46	C₂×#63218	—	2	4	no	—
47	C₄×D₆₈	C₂×C₄	1	2	yes	—
48	C₄×D₆₈	C₂³	1	2	yes	—
49	#10587558	—	2	2	no	—
50	C₄²×D₁₇	—	2	2	no	—
51	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
52	C₄×Dic₃₄	C₂³	1	2	yes	—
53	C₁₇×C₄⋊C₈	—	2	1	no	—
54	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
56	C₁₇×C₂²⋊C₈	—	2	1	no	—
57	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
58	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₇	C₂³	1	2	yes	—
61	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₇	C₂³	1	2	yes	—
64	#10587492	C₂×C₄	1	2	yes	—
65	#10587492	C₂×C₄	1	2	yes	—
66	#10587491	—	2	4	no	—
67	#10587489	C₂×C₄	1	1	yes	characteristic
68	#10587489	C₂×C₄	1	1	yes	characteristic
69	#10587488	C₂×C₄	1	2	yes	—
70	#10587487	C₂×C₄	1	2	yes	—
71	#10587487	C₂×C₄	1	2	yes	—
72	#10587486	—	2	4	no	—
73	#10587485	—	2	4	no	—
74	C₄⋊#3622	—	2	2	no	—
75	C₄⋊#3622	C₂³	1	4	yes	—
76	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₇	—	2	2	no	—
80	#63252	—	2	4	no	—
81	#63252	—	2	4	no	—
82	C₂×D₆₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₇	—	2	2	no	—
84	C₂×C₄×D₁₇	—	2	2	no	—
85	C₂×C₄×D₁₇	—	2	2	no	—
86	C₂×Dic₃₄	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₇	—	2	2	no	—
88	C₂×C₈×C₁₇	—	2	2	no	—
89	C₂×C₈×C₁₇	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
96	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₇	#41	1	1	yes	characteristic
99	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₇	—	2	2	no	—
101	#63227	—	2	2	no	—
102	#63227	—	2	2	no	—
103	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
105	#63225	—	2	2	no	—
106	#63225	—	2	2	no	—
107	C₄×Dic₁₇	—	2	2	no	—
108	C₄×Dic₁₇	—	2	2	no	—
109	C₄×Dic₁₇	—	2	2	no	—
110	C₂×#3622	—	2	4	no	—
111	C₂×#3622	—	8	2	no	—
112	Dic₆₈	—	2	2	no	—
113	D₁₃₆	—	2	2	no	—
114	#63219	—	2	4	no	—
115	#63218	—	8	4	no	—
116	#17898	—	17	1	no	maximal, Sylow
117	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	4	1	no	—
122	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₁₇	#131	1	2	yes	—
127	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	4	4	no	—
131	C₂×Dic₁₇	—	2	2	no	—
132	C₂×Dic₁₇	—	2	2	no	—
133	C₂×Dic₁₇	—	2	2	no	—
134	D₆₈	—	2	2	no	—
135	D₆₈	—	2	2	no	—
136	C₄×D₁₇	—	4	2	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	4	4	no	—
139	C₄×D₁₇	—	8	2	no	—
140	Dic₃₄	—	2	2	no	—
141	Dic₃₄	—	2	2	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	—	2	2	no	—
145	C₈×C₁₇	—	2	2	no	—
146	C₈×C₁₇	—	4	2	no	—
147	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
148	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
149	#3622	—	4	4	no	—
150	#2989	—	17	2	no	—
151	C₄×#135	—	17	1	no	—
152	#1592	—	17	4	no	—
153	#1591	—	17	2	no	—
154	#1589	—	17	2	no	—
155	C₂×#335	—	17	1	no	—
156	#1477	—	17	2	no	—
157	#1450	D₁₇	1	1	yes	characteristic, complemented
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	#436	1	2	yes	—
161	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×D₁₇	—	4	2	no	—
163	C₂×D₁₇	—	4	2	no	—
164	C₂×D₁₇	—	4	2	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	#436	1	2	yes	—
169	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
170	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
171	C₄×C₁₇	#576	1	1	yes	characteristic
172	C₄×C₁₇	#578	1	1	yes	characteristic
173	Dic₁₇	—	4	2	no	—
174	Dic₁₇	—	4	2	no	—
175	Dic₁₇	—	4	2	no	—
176	C₂×#135	—	17	1	no	—
177	C₄×#41	—	17	2	no	—
178	#446	—	17	4	no	—
179	C₄×Q₁₆	—	17	1	no	—
180	C₄×Q₁₆	—	17	1	no	—
181	C₄×SD₁₆	—	17	2	no	—
182	C₄×SD₁₆	—	17	2	no	—
183	C₄×D₈	—	17	1	no	—
184	C₄×D₈	—	17	1	no	—
185	#436	—	17	4	no	—
186	#435	—	34	4	no	—
187	#432	—	34	2	no	—
188	#431	C₂×D₁₇	1	1	yes	characteristic
189	#431	C₂×D₁₇	1	1	yes	characteristic
190	#427	—	17	1	no	—
191	#423	—	34	2	no	—
192	C₂×C₄⋊C₈	—	17	2	no	—
193	#416	—	17	2	no	—
194	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
195	#335	—	17	2	no	—
196	#335	—	17	2	no	—
197	Q₈⋊C₈	—	17	4	no	—
198	D₄⋊C₈	—	17	4	no	—
199	#322	C₂×D₁₇	1	2	yes	—
200	C₈²	C₂×D₁₇	1	2	yes	—
201	C₂×C₁₇	—	2	2	no	—
202	C₂×C₁₇	C₂×#436	1	1	yes	characteristic
203	C₂×C₁₇	#2951	1	1	yes	characteristic
204	C₂×C₁₇	#2957	1	1	yes	characteristic
205	D₁₇	—	8	2	no	—
206	C₂×#41	—	17	2	no	—
207	#135	—	34	4	no	—
208	C₂×Q₁₆	—	17	1	no	—
209	C₂×SD₁₆	—	17	2	no	—
210	C₂×D₈	—	17	1	no	—
211	C₂×M₁₆	—	34	4	no	—
212	C₂²×C₈	—	34	2	no	—
213	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
215	C₄×Q₈	—	17	2	no	—
216	C₄×Q₈	—	17	2	no	—
217	C₄×D₄	—	17	2	no	—
218	C₄×D₄	—	17	2	no	—
219	C₄×D₄	—	34	2	no	—
220	C₄×D₄	—	34	2	no	—
221	#117	—	17	2	no	—
222	#117	—	34	2	no	—
223	C₂×C₄²	—	34	2	no	—
224	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
225	#107	—	17	1	no	—
226	#107	—	17	1	no	—
227	#106	—	17	2	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	17	4	no	—
230	C₄⋊C₈	—	34	4	no	—
231	C₄⋊C₈	—	34	2	no	—
232	C₄⋊C₈	D₆₈	1	1	yes	characteristic
233	C₄⋊C₈	D₆₈	1	1	yes	characteristic
234	#103	—	17	2	no	—
235	#103	—	17	2	no	—
236	#102	—	17	2	no	—
237	#102	—	17	2	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	34	4	no	—
240	C₂²⋊C₈	D₆₈	1	2	yes	—
241	#97	—	34	4	no	—
242	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
243	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₇	1	2	yes	—
245	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₁₇	1	2	yes	—
248	C₁₇	#17898	1	1	yes	characteristic, complemented, Sylow
249	#41	—	34	4	no	—
250	#41	—	34	4	no	—
251	C₂×Q₈	—	17	2	no	—
252	C₂×D₄	—	17	2	no	—
253	C₂×D₄	—	34	2	no	—
254	C₂²×C₄	—	34	2	no	—
255	C₂²×C₄	—	34	2	no	—
256	C₂²×C₄	—	34	2	no	—
257	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
259	C₂²×C₄	#63252	1	1	yes	characteristic
260	Q₁₆	—	34	2	no	—
261	SD₁₆	—	34	4	no	—
262	D₈	—	34	2	no	—
263	M₁₆	—	136	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	34	4	no	—
267	C₂×C₈	—	136	2	no	—
268	C₂×C₈	#63218	1	4	yes	complemented
269	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
273	C₂×C₈	C₂×D₆₈	1	2	yes	—
274	C₂×C₈	C₂×D₆₈	1	2	yes	—
275	C₄⋊C₄	—	17	2	no	—
276	C₄⋊C₄	—	17	2	no	—
277	C₄⋊C₄	—	34	2	no	—
278	C₄⋊C₄	—	34	2	no	—
279	C₂²⋊C₄	—	34	2	no	—
280	C₂²⋊C₄	—	34	2	no	—
281	C₂²⋊C₄	—	34	2	no	—
282	C₄²	—	34	2	no	—
283	C₄²	—	34	2	no	—
284	C₄²	—	34	2	no	—
285	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
286	C₄²	C₂×D₆₈	1	1	yes	characteristic
287	C₄²	#63252	1	1	yes	characteristic
288	C₄²	C₂³×D₁₇	1	1	yes	characteristic
289	C₂³	—	34	2	no	—
290	C₂³	C₄×D₆₈	1	1	yes	characteristic
291	Q₈	—	34	2	no	—
292	Q₈	—	34	2	no	—
293	D₄	—	34	2	no	—
294	D₄	—	34	2	no	—
295	D₄	—	68	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	2	no	—
301	C₂×C₄	—	34	2	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	—	68	4	no	—
305	C₂×C₄	—	136	2	no	—
306	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
307	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
308	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
309	C₂×C₄	C₂×#63218	1	2	yes	—
310	C₂×C₄	#10587587	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
313	C₂×C₄	C₂×#63252	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	68	4	no	—
320	C₈	C₂×#63218	1	2	yes	—
321	C₈	C₂×#63218	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	68	2	no	—
325	C₂²	—	68	2	no	—
326	C₂²	—	68	2	no	—
327	C₂²	#11762888	1	2	yes	—
328	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	68	2	no	—
333	C₄	—	68	2	no	—
334	C₄	—	68	2	no	—
335	C₄	#11762888	1	2	yes	—
336	C₄	C₂²×#63218	1	1	yes	characteristic
337	C₄	C₂×#10587587	1	1	yes	characteristic
338	C₄	#11763614	1	1	yes	characteristic
339	C₄	#11763626	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	136	2	no	—
342	C₂	C₂×#11762888	1	1	yes	characteristic
343	C₂	#14537661	1	1	yes	characteristic
344	C₂	#14537702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#1450	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂×#10587590	C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1450	D₁₇
hypocenter	C₁₇	#17898
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #14537897	C₂	semidirect
3. #14527688	C₂	semidirect
4. #14527682	C₂	semidirect
5. #14527670	C₂	semidirect
6. C₂×#11762562	C₂	semidirect
7. #14527172	C₂	semidirect
8. C₁₇×#1450	C₂	semidirect
89. C₂×C₈×C₁₇	M₁₆	semidirect
157. #1450	D₁₇	semidirect
248. C₁₇	#17898	semidirect
268. C₂×C₈	#63218	semidirect

Maximal subgroups

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

subgroup	index
#14537897	2
C₄×#10587590	2
#14527688	2
#14527682	2
#14527670	2
C₂×#11762562	2
#14527172	2
C₁₇×#1450	2
#17898	17