Subgroups — #17187741

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541918	C₂	1	1	yes	maximal, characteristic
2	#14537702	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530668	C₂	1	4	yes	maximal, complemented
4	#14530663	C₂	1	4	yes	maximal, complemented
5	#14530578	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762616	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11762615	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587590	C₂²	1	1	yes	characteristic
11	#11763575	—	2	1	no	—
12	#11763250	C₄	1	2	yes	complemented
13	#11763249	C₄	1	2	yes	complemented
14	#11763245	C₄	1	2	yes	complemented
15	#11763244	C₄	1	2	yes	complemented
16	#11763239	C₂²	1	4	yes	—
17	C₂×#10587483	C₂²	1	1	yes	characteristic
18	C₂²×#63214	C₂²	1	1	yes	characteristic
19	#11762932	—	2	2	no	—
20	#11762927	C₂²	1	4	yes	complemented
21	C₂×#10587490	C₂²	1	1	yes	characteristic
22	#11762827	—	2	1	no	—
23	#11762638	C₂²	1	4	yes	complemented
24	#11762616	C₂²	1	2	yes	complemented
25	#11762616	C₂²	1	2	yes	complemented
26	#11762615	C₂²	1	2	yes	complemented
27	#11762615	C₂²	1	2	yes	complemented
28	#11762585	C₂²	1	2	yes	complemented
29	C₄×#63214	C₂²	1	2	yes	complemented
30	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
31	C₁₇×#568	—	2	1	no	—
32	C₁₇×#444	—	2	2	no	—
33	C₁₇×#443	C₂²	1	4	yes	complemented
34	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
35	C₁₇×#421	—	2	1	no	—
36	C₁₇×#405	C₂²	1	1	yes	characteristic
37	C₂×#41×C₁₇	—	2	1	no	—
38	C₂×#63252	—	2	1	no	—
39	C₁₇×#135	—	2	1	no	—
40	C₁₇×#135	—	2	1	no	—
41	C₁₇×#135	C₂×C₄	1	2	yes	—
42	C₁₇×#135	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×SD₁₆×C₁₇	—	2	1	no	—
45	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#131	—	2	4	no	—
47	C₁₇×#131	—	4	2	no	—
48	C₂×M₁₆×C₁₇	—	2	1	no	—
49	C₂×M₁₆×C₁₇	—	4	2	no	—
50	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₁₇×#117	D₄	1	1	yes	characteristic
53	C₂×#63223	—	2	1	no	—
54	#10587593	—	4	2	no	—
55	M₁₆×D₁₇	—	4	2	no	—
56	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
57	#10587590	—	2	1	no	—
58	#10587590	—	2	1	no	—
59	#10587590	C₂×C₄	1	2	yes	—
60	#10587590	C₂³	1	2	yes	—
61	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
62	C₂×#63219	—	2	1	no	—
63	#10587587	—	2	4	no	—
64	C₁₇×#108	—	2	1	no	—
65	C₁₇×#108	C₂³	1	2	yes	—
66	C₁₇×#104	—	2	4	no	—
67	C₁₇×#97	D₄	1	2	yes	—
68	C₄×C₈×C₁₇	C₂³	1	2	yes	—
69	#10587500	C₂×C₄	1	4	yes	—
70	#10587499	C₂×C₄	1	4	yes	—
71	#10587498	C₂×C₄	1	4	yes	—
72	#10587497	C₂×C₄	1	4	yes	—
73	#10587490	—	2	1	no	—
74	#10587490	C₂³	1	2	yes	—
75	#10587483	C₂×C₄	1	2	yes	—
76	#10587483	C₂×C₄	1	2	yes	—
77	C₂×#63214	—	2	1	no	—
78	C₂×#63214	—	2	1	no	—
79	C₂×#63214	C₂³	1	1	yes	characteristic
80	C₂×#63214	C₂³	1	1	yes	characteristic
81	C₂×#63214	C₂³	1	2	yes	—
82	#10587476	—	2	4	no	—
83	#41×C₁₇	—	2	2	no	—
84	#41×C₁₇	—	2	2	no	—
85	#41×C₁₇	—	4	1	no	—
86	#41×C₁₇	—	4	1	no	—
87	C₂×Q₈×C₁₇	—	2	1	no	—
88	C₂×D₄×C₁₇	—	2	1	no	—
89	C₂×D₄×C₁₇	—	4	1	no	—
90	C₂²×C₄×C₁₇	—	4	1	no	—
91	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₁₇	—	4	1	no	—
93	#63252	—	2	2	no	—
94	#63252	—	2	2	no	—
95	#63252	—	4	1	no	—
96	#63252	—	4	1	no	—
97	C₂×D₆₈	—	2	1	no	—
98	C₂×C₄×D₁₇	—	4	1	no	—
99	C₂×Dic₃₄	—	2	1	no	—
100	Q₁₆×C₁₇	—	2	1	no	—
101	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
102	SD₁₆×C₁₇	—	2	2	no	—
103	SD₁₆×C₁₇	—	2	2	no	—
104	D₈×C₁₇	—	2	1	no	—
105	D₈×C₁₇	C₂²×C₄	1	2	yes	—
106	M₁₆×C₁₇	—	2	2	no	—
107	M₁₆×C₁₇	—	4	1	no	—
108	M₁₆×C₁₇	—	4	4	no	—
109	M₁₆×C₁₇	D₈	1	2	yes	—
110	M₁₆×C₁₇	D₈	1	2	yes	—
111	C₂×C₈×C₁₇	—	4	1	no	—
112	C₂×C₈×C₁₇	—	4	4	no	—
113	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
118	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
119	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₁₇	—	2	1	no	—
121	C₂²⋊C₄×C₁₇	—	2	1	no	—
122	C₄²×C₁₇	C₂×D₄	1	2	yes	—
123	#63223	—	2	2	no	—
124	#63223	—	4	1	no	—
125	C₂×#3622	—	4	1	no	—
126	Dic₆₈	—	2	1	no	—
127	Dic₆₈	C₂²×C₄	1	2	yes	—
128	D₁₃₆	—	2	1	no	—
129	D₁₃₆	C₂²×C₄	1	2	yes	—
130	#63219	—	2	2	no	—
131	#63219	—	2	2	no	—
132	#63218	—	4	4	no	—
133	C₈×D₁₇	—	4	4	no	—
134	#63214	—	2	1	no	—
135	#63214	—	2	1	no	—
136	#63214	C₂²×C₄	1	4	yes	—
137	#33464	—	17	1	no	maximal, Sylow
138	C₂³×C₁₇	—	4	1	no	—
139	C₂³×C₁₇	#135	1	1	yes	characteristic
140	C₂²×D₁₇	—	4	1	no	—
141	Q₈×C₁₇	—	2	2	no	—
142	Q₈×C₁₇	—	4	1	no	—
143	D₄×C₁₇	—	2	2	no	—
144	D₄×C₁₇	—	4	2	no	—
145	D₄×C₁₇	—	4	2	no	—
146	D₄×C₁₇	—	4	1	no	—
147	C₂×C₄×C₁₇	—	2	2	no	—
148	C₂×C₄×C₁₇	—	4	1	no	—
149	C₂×C₄×C₁₇	—	4	2	no	—
150	C₂×C₄×C₁₇	—	4	2	no	—
151	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
156	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₁₇	—	4	2	no	—
158	C₂²⋊D₁₇	—	4	2	no	—
159	C₂×Dic₁₇	—	4	1	no	—
160	D₆₈	—	2	2	no	—
161	D₆₈	—	4	1	no	—
162	C₄×D₁₇	—	4	2	no	—
163	C₄×D₁₇	—	4	2	no	—
164	Dic₃₄	—	2	2	no	—
165	Dic₃₄	—	4	1	no	—
166	C₈×C₁₇	—	4	2	no	—
167	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
168	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
169	C₈×C₁₇	C₂×D₈	1	4	yes	—
170	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
171	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
172	#3622	—	4	2	no	—
173	#3414	—	17	1	no	—
174	#2957	—	17	1	no	—
175	#2957	D₁₇	1	1	yes	characteristic, complemented
176	#2179	—	17	4	no	—
177	#2178	—	17	4	no	—
178	#2161	—	17	1	no	—
179	C₂×#359	—	17	1	no	—
180	C₂×#359	—	17	1	no	—
181	#2108	—	17	1	no	—
182	C₂²×C₁₇	—	4	1	no	—
183	C₂²×C₁₇	—	4	2	no	—
184	C₂²×C₁₇	—	16	1	no	—
185	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
186	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
187	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
188	C₂×D₁₇	—	4	2	no	—
189	C₂×D₁₇	—	16	1	no	—
190	C₄×C₁₇	—	4	2	no	—
191	C₄×C₁₇	—	4	2	no	—
192	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
193	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
194	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
196	Dic₁₇	—	4	2	no	—
197	C₂×#135	—	17	1	no	—
198	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
199	#568	—	2	1	no	—
200	#568	—	34	1	no	—
201	#510	—	17	2	no	—
202	#510	—	17	2	no	—
203	#509	—	17	2	no	—
204	#509	—	17	2	no	—
205	#508	—	17	4	no	—
206	C₂×M₃₂	—	17	1	no	—
207	C₂²×C₁₆	—	17	1	no	—
208	#444	—	2	2	no	—
209	#444	—	34	2	no	—
210	#443	—	17	4	no	—
211	#443	C₂×D₁₇	1	4	yes	complemented
212	C₂×#108	—	17	1	no	—
213	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
214	#421	—	2	1	no	—
215	#421	—	34	1	no	—
216	#405	C₂×D₁₇	1	1	yes	characteristic
217	#368	—	17	4	no	—
218	#359	—	17	2	no	—
219	#359	—	17	2	no	—
220	#359	—	17	2	no	—
221	#359	—	17	2	no	—
222	#346	—	17	2	no	—
223	C₄×C₁₆	—	17	2	no	—
224	C₂×C₁₇	—	2	1	no	—
225	C₂×C₁₇	—	2	1	no	—
226	C₂×C₁₇	—	2	1	no	—
227	C₂×C₁₇	—	8	2	no	—
228	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
229	D₁₇	—	8	2	no	—
230	C₂×#41	—	2	1	no	—
231	C₂×#41	—	34	1	no	—
232	#135	—	2	1	no	—
233	#135	—	2	1	no	—
234	#135	—	17	2	no	—
235	#135	—	17	2	no	—
236	#135	—	34	1	no	—
237	#135	—	34	1	no	—
238	#135	C₄×D₁₇	1	2	yes	—
239	#135	C₂²×D₁₇	1	2	yes	—
240	C₂×Q₁₆	—	17	1	no	—
241	C₂×Q₁₆	C₄×D₁₇	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	34	1	no	—
244	C₂×D₈	—	17	1	no	—
245	C₂×D₈	C₄×D₁₇	1	1	yes	characteristic
246	#131	—	2	4	no	—
247	#131	—	4	2	no	—
248	#131	—	34	4	no	—
249	#131	—	68	2	no	—
250	C₂×M₁₆	—	2	1	no	—
251	C₂×M₁₆	—	4	2	no	—
252	C₂×M₁₆	—	34	1	no	—
253	C₂×M₁₆	—	68	2	no	—
254	C₂×M₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
255	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
256	#117	C₂²⋊D₁₇	1	1	yes	characteristic
257	Q₃₂	—	17	4	no	—
258	SD₃₂	—	17	4	no	—
259	SD₃₂	—	17	4	no	—
260	D₁₆	—	17	4	no	—
261	M₃₂	—	17	2	no	—
262	M₃₂	—	17	2	no	—
263	C₂×C₁₆	—	17	2	no	—
264	C₂×C₁₆	—	17	1	no	—
265	C₂×C₁₆	—	17	1	no	—
266	C₂×C₁₆	—	34	1	no	—
267	C₂×C₁₆	—	34	1	no	—
268	#108	—	2	1	no	—
269	#108	—	17	2	no	—
270	#108	—	34	1	no	—
271	#108	C₂²×D₁₇	1	2	yes	—
272	#104	—	2	4	no	—
273	#104	—	34	4	no	—
274	#97	C₂²⋊D₁₇	1	2	yes	—
275	C₄×C₈	C₂²×D₁₇	1	2	yes	—
276	C₁₇	#33464	1	1	yes	characteristic, complemented, Sylow
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	#41	—	34	2	no	—
282	#41	—	34	2	no	—
283	#41	—	68	1	no	—
284	#41	—	68	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	34	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	34	1	no	—
290	C₂×D₄	—	68	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	68	1	no	—
293	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	17	2	no	—
296	Q₁₆	—	34	1	no	—
297	Q₁₆	C₂×C₄×D₁₇	1	2	yes	—
298	SD₁₆	—	2	2	no	—
299	SD₁₆	—	2	2	no	—
300	SD₁₆	—	34	2	no	—
301	SD₁₆	—	34	2	no	—
302	D₈	—	2	1	no	—
303	D₈	—	17	2	no	—
304	D₈	—	34	1	no	—
305	D₈	C₂×C₄×D₁₇	1	2	yes	—
306	M₁₆	—	2	2	no	—
307	M₁₆	—	4	1	no	—
308	M₁₆	—	4	4	no	—
309	M₁₆	—	34	2	no	—
310	M₁₆	—	68	1	no	—
311	M₁₆	—	68	4	no	—
312	M₁₆	#63228	1	2	yes	—
313	M₁₆	#63228	1	2	yes	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	4	no	—
316	C₂×C₈	—	68	1	no	—
317	C₂×C₈	—	68	4	no	—
318	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
319	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
320	C₂×C₈	#63252	1	1	yes	characteristic
321	C₂×C₈	#63252	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
324	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	2	1	no	—
327	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
328	C₁₆	—	17	4	no	—
329	C₁₆	—	34	1	no	—
330	C₁₆	—	34	1	no	—
331	C₂³	—	4	1	no	—
332	C₂³	—	68	1	no	—
333	C₂³	#10587636	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	34	2	no	—
337	Q₈	—	68	1	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	34	2	no	—
343	D₄	—	68	2	no	—
344	D₄	—	68	2	no	—
345	D₄	—	68	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	68	1	no	—
351	C₂×C₄	—	68	2	no	—
352	C₂×C₄	—	68	2	no	—
353	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
354	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
355	C₂×C₄	C₂×#63228	1	1	yes	characteristic
356	C₂×C₄	C₂×#63228	1	1	yes	characteristic
357	C₂×C₄	#10587636	1	1	yes	characteristic
358	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	68	2	no	—
361	C₈	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
362	C₈	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
363	C₈	C₂×#63228	1	4	yes	—
364	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
365	C₈	C₂×#63252	1	1	yes	characteristic
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	16	1	no	—
369	C₂²	—	272	1	no	—
370	C₂²	—	68	2	no	—
371	C₂²	C₄×#63228	1	1	yes	characteristic
372	C₂²	C₄×#63228	1	1	yes	characteristic
373	C₂²	C₂×#10587636	1	1	yes	characteristic
374	C₄	—	4	2	no	—
375	C₄	—	4	2	no	—
376	C₄	—	68	2	no	—
377	C₄	C₄×#63228	1	1	yes	characteristic
378	C₄	C₄×#63228	1	1	yes	characteristic
379	C₄	C₂×C₄×C₂²⋊D₁₇	1	1	yes	characteristic
380	C₄	C₂²×#63228	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	136	2	no	—
386	C₂	C₂×C₄×#63228	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#63228
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2957	C₂
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	#14541918	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2957	D₁₇
hypocenter	C₁₇	#33464
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
2. #14537702	C₂	semidirect
3. #14530668	C₂	semidirect
4. #14530663	C₂	semidirect
5. #14530578	C₂	semidirect
6. C₂×#11762616	C₂	semidirect
7. C₂×#11762615	C₂	semidirect
8. #14530381	C₂	semidirect
9. C₁₇×#2957	C₂	semidirect
12. #11763250	C₄	semidirect
13. #11763249	C₄	semidirect
14. #11763245	C₄	semidirect
15. #11763244	C₄	semidirect
20. #11762927	C₂²	semidirect
23. #11762638	C₂²	semidirect
24. #11762616	C₂²	semidirect
25. #11762616	C₂²	semidirect
26. #11762615	C₂²	semidirect
27. #11762615	C₂²	semidirect
28. #11762585	C₂²	semidirect
29. C₄×#63214	C₂²	semidirect
33. C₁₇×#443	C₂²	semidirect
175. #2957	D₁₇	semidirect
211. #443	C₂×D₁₇	semidirect
276. C₁₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#14541918	2
#14537702	2
#14530668	2
#14530663	2
#14530578	2
C₂×#11762616	2
C₂×#11762615	2
#14530381	2
C₁₇×#2957	2
#33464	17