Subgroups — #17025878

Order 4352.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14541207	C₂	1	1	yes	maximal, characteristic
3	#14541080	C₂	1	2	yes	maximal, complemented
4	#14528394	C₂	1	1	yes	maximal, characteristic, complemented
5	#14528360	C₂	1	2	yes	maximal, complemented
6	#14528188	C₂	1	1	yes	maximal, characteristic, complemented
7	#14528173	C₂	1	1	yes	maximal, characteristic, complemented
8	#14528014	C₂	1	2	yes	maximal, complemented
9	#14527951	C₂	1	2	yes	maximal, complemented
10	#14527670	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763524	C₂²	1	2	yes	—
13	Q₈×D₆₈	C₂²	1	2	yes	—
14	#11763214	—	2	1	no	—
15	#11763214	C₂²	1	1	yes	characteristic
16	#11763214	C₂²	1	1	yes	characteristic
17	#11763160	C₂²	1	2	yes	—
18	#11763160	C₂²	1	2	yes	—
19	#11763153	—	2	2	no	—
20	#11763134	—	2	2	no	—
21	#11763104	C₂²	1	2	yes	—
22	#11763100	C₂²	1	2	yes	—
23	#11763090	—	2	2	no	—
24	#11763090	C₂²	1	2	yes	—
25	#11763054	—	2	2	no	—
26	#11763045	—	2	1	no	—
27	#11763014	C₂²	1	2	yes	—
28	#11762946	—	2	2	no	—
29	C₄×D₁₃₆	—	2	1	no	—
30	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
31	#11762888	C₂²	1	2	yes	—
32	#11762847	—	2	1	no	—
33	#11762816	—	2	2	no	—
34	#11762562	C₂²	1	1	yes	characteristic
35	#11762557	C₂²	1	2	yes	—
36	#11762554	C₂²	1	2	yes	—
37	#11762538	C₂²	1	2	yes	—
38	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
41	C₁₇×#334	C₂²	1	1	yes	characteristic
42	C₁₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₁₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₁₇	—	2	2	no	—
46	C₂×SD₁₆×C₁₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₇	—	2	1	no	—
48	C₄×Q₈×C₁₇	C₂³	1	1	yes	characteristic
49	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
50	#10587624	—	2	2	no	—
51	C₂×#63229	—	2	2	no	—
52	#10587616	—	2	2	no	—
53	#10587612	—	2	2	no	—
54	C₂×D₁₃₆	D₄	1	1	yes	characteristic
55	C₂×#63218	—	2	2	no	—
56	#10587583	—	2	2	no	—
57	#10587583	—	4	2	no	—
58	#10587578	—	2	2	no	—
59	#10587571	—	4	2	no	—
60	C₄×D₆₈	—	2	2	no	—
61	C₄×D₆₈	C₂³	1	2	yes	—
62	#10587555	—	2	2	no	—
63	#10587555	C₂³	1	2	yes	—
64	#10587555	C₂³	1	2	yes	—
65	C₁₇×#106	D₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	—	2	1	no	—
67	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
68	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₁₇×#103	D₄	1	1	yes	characteristic
70	C₁₇×#102	D₄	1	1	yes	characteristic
71	C₁₇×C₂²⋊C₈	—	2	1	no	—
72	C₄×C₈×C₁₇	—	2	1	no	—
73	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	#10587506	—	2	2	no	—
77	#10587492	—	2	2	no	—
78	#10587492	—	2	2	no	—
79	#10587492	D₄	1	2	yes	—
80	#10587491	—	2	2	no	—
81	#10587489	—	2	2	no	—
82	#10587489	—	2	1	no	—
83	#10587489	—	2	1	no	—
84	#10587489	—	2	1	no	—
85	#10587489	—	2	1	no	—
86	#10587489	D₄	1	1	yes	characteristic
87	#10587488	—	2	2	no	—
88	#10587487	—	2	2	no	—
89	#10587487	—	2	2	no	—
90	#10587487	—	4	2	no	—
91	#10587481	—	2	2	no	—
92	#10587478	—	2	2	no	—
93	C₄⋊#3622	C₂³	1	2	yes	—
94	C₂×Q₈×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₇	—	2	1	no	—
97	C₂×C₂²⋊D₁₇	—	4	2	no	—
98	Q₈×D₁₇	—	4	4	no	—
99	Q₈×D₁₇	—	8	2	no	—
100	C₂×D₆₈	C₂×D₄	1	2	yes	—
101	C₂×C₄×D₁₇	—	2	2	no	—
102	C₂×C₄×D₁₇	—	4	2	no	—
103	C₂×Dic₃₄	—	2	2	no	—
104	C₂×Dic₃₄	—	2	2	no	—
105	C₂×Dic₃₄	—	4	2	no	—
106	SD₁₆×C₁₇	—	2	2	no	—
107	C₂×C₈×C₁₇	—	2	1	no	—
108	C₂×C₈×C₁₇	—	2	1	no	—
109	C₂×C₈×C₁₇	—	2	1	no	—
110	C₂×C₈×C₁₇	—	4	2	no	—
111	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₁₇	—	2	1	no	—
116	C₄⋊C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₁₇	—	2	1	no	—
119	C₄²×C₁₇	—	2	1	no	—
120	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊Dic₁₇	—	4	2	no	—
122	#63229	—	8	2	no	—
123	#63227	—	2	2	no	—
124	#63227	—	4	2	no	—
125	#63227	—	4	2	no	—
126	C₄⋊Dic₁₇	—	2	2	no	—
127	C₄⋊Dic₁₇	—	2	2	no	—
128	C₄⋊Dic₁₇	—	2	2	no	—
129	C₄⋊Dic₁₇	—	2	2	no	—
130	C₄⋊Dic₁₇	—	4	2	no	—
131	C₄⋊Dic₁₇	C₂×D₄	1	2	yes	—
132	C₄×Dic₁₇	—	4	2	no	—
133	C₂×#3622	—	2	2	no	—
134	D₁₃₆	—	2	2	no	—
135	#63218	—	8	2	no	—
136	#19905	—	17	1	no	maximal, Sylow
137	C₂³×C₁₇	—	2	1	no	—
138	C₂²×D₁₇	—	2	2	no	—
139	Q₈×C₁₇	—	2	1	no	—
140	Q₈×C₁₇	#135	1	2	yes	—
141	D₄×C₁₇	—	2	1	no	—
142	D₄×C₁₇	—	2	1	no	—
143	C₂×C₄×C₁₇	—	2	1	no	—
144	C₂×C₄×C₁₇	—	2	1	no	—
145	C₂×C₄×C₁₇	—	2	1	no	—
146	C₂×C₄×C₁₇	—	4	1	no	—
147	C₂×C₄×C₁₇	—	4	1	no	—
148	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²⋊D₁₇	—	16	2	no	—
152	C₂×Dic₁₇	—	2	2	no	—
153	C₂×Dic₁₇	—	4	2	no	—
154	C₂×Dic₁₇	—	4	2	no	—
155	D₆₈	—	2	2	no	—
156	D₆₈	—	2	2	no	—
157	C₄×D₁₇	—	4	2	no	—
158	C₄×D₁₇	—	4	2	no	—
159	C₄×D₁₇	—	8	4	no	—
160	Dic₃₄	—	4	2	no	—
161	Dic₃₄	—	4	2	no	—
162	Dic₃₄	—	8	4	no	—
163	Dic₃₄	—	8	2	no	—
164	C₈×C₁₇	—	2	1	no	—
165	C₈×C₁₇	—	2	1	no	—
166	C₈×C₁₇	—	2	1	no	—
167	C₈×C₁₇	—	4	1	no	—
168	C₈×C₁₇	—	4	1	no	—
169	C₈×C₁₇	—	4	1	no	—
170	C₈×C₁₇	#137	1	2	yes	—
171	#3622	—	4	2	no	—
172	Q₈×D₈	—	17	1	no	—
173	#3341	—	17	1	no	—
174	#3329	—	17	2	no	—
175	#1710	—	17	1	no	—
176	#1703	—	17	2	no	—
177	#1676	—	17	1	no	—
178	#1673	—	17	1	no	—
179	#1646	—	17	2	no	—
180	#1637	—	17	2	no	—
181	#1589	—	17	1	no	—
182	C₈×SD₁₆	D₁₇	1	1	yes	characteristic, complemented
183	C₂²×C₁₇	—	4	1	no	—
184	C₂²×C₁₇	—	4	1	no	—
185	C₂²×C₁₇	—	4	1	no	—
186	C₂²×C₁₇	D₄²	1	1	yes	characteristic
187	C₂×D₁₇	—	4	2	no	—
188	C₂×D₁₇	—	4	2	no	—
189	C₂×D₁₇	—	4	2	no	—
190	C₄×C₁₇	—	2	2	no	—
191	C₄×C₁₇	—	2	1	no	—
192	C₄×C₁₇	—	4	1	no	—
193	C₄×C₁₇	—	4	1	no	—
194	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
195	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
196	C₄×C₁₇	#576	1	1	yes	characteristic
197	C₄×C₁₇	#578	1	1	yes	characteristic
198	Dic₁₇	—	4	2	no	—
199	Dic₁₇	—	8	2	no	—
200	Dic₁₇	—	8	2	no	—
201	#555	—	17	2	no	—
202	D₄×Q₈	—	17	2	no	—
203	#500	—	17	1	no	—
204	#500	—	17	1	no	—
205	#500	—	34	1	no	—
206	#487	—	17	2	no	—
207	#487	—	17	2	no	—
208	#485	—	34	2	no	—
209	#482	—	34	2	no	—
210	#477	—	17	2	no	—
211	#476	—	17	2	no	—
212	#474	—	17	2	no	—
213	#474	—	34	2	no	—
214	#468	—	34	2	no	—
215	#466	—	34	1	no	—
216	#461	—	17	2	no	—
217	#448	—	34	2	no	—
218	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
219	C₄×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
220	C₄×D₈	—	17	1	no	—
221	C₄×D₈	—	34	1	no	—
222	#436	—	17	2	no	—
223	C₈×D₄	C₂×D₁₇	1	1	yes	characteristic
224	#427	—	34	1	no	—
225	#419	—	34	2	no	—
226	#335	—	17	1	no	—
227	#334	C₂×D₁₇	1	1	yes	characteristic
228	#333	—	17	2	no	—
229	#332	—	17	2	no	—
230	Q₈⋊C₈	C₂×D₁₇	1	1	yes	characteristic
231	D₄⋊C₈	—	17	2	no	—
232	D₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
233	C₈²	C₂×D₁₇	1	1	yes	characteristic
234	C₂×C₁₇	—	8	1	no	—
235	C₂×C₁₇	#3276	1	1	yes	characteristic
236	C₂×C₁₇	#3280	1	1	yes	characteristic
237	C₂×C₁₇	#3290	1	1	yes	characteristic
238	D₁₇	—	8	2	no	—
239	C₂²×Q₈	—	34	2	no	—
240	C₂×SD₁₆	—	34	2	no	—
241	C₂×SD₁₆	D₆₈	1	1	yes	characteristic
242	C₂×D₈	—	17	1	no	—
243	C₂×M₁₆	—	34	2	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₄⋊Q₈	—	17	2	no	—
246	C₄⋊Q₈	—	17	2	no	—
247	C₄⋊Q₈	—	34	2	no	—
248	#124	—	68	2	no	—
249	C₂²⋊Q₈	—	34	2	no	—
250	C₂²⋊Q₈	—	68	2	no	—
251	#121	—	34	2	no	—
252	#121	—	34	2	no	—
253	C₄×Q₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₄×D₄	—	17	2	no	—
255	C₄×D₄	—	34	2	no	—
256	C₄×D₄	C₂²×D₁₇	1	1	yes	characteristic
257	#117	—	34	2	no	—
258	#117	—	34	2	no	—
259	#107	—	17	1	no	—
260	#107	—	34	2	no	—
261	#107	—	34	1	no	—
262	#107	—	34	1	no	—
263	#107	—	34	1	no	—
264	#107	—	34	1	no	—
265	#107	—	34	2	no	—
266	#106	—	34	2	no	—
267	#106	D₆₈	1	1	yes	characteristic
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	17	2	no	—
270	C₄⋊C₈	C₂²×D₁₇	1	1	yes	characteristic
271	C₄⋊C₈	C₂²×D₁₇	1	1	yes	characteristic
272	#103	—	34	2	no	—
273	#103	—	34	2	no	—
274	#103	—	34	2	no	—
275	#103	—	34	2	no	—
276	#103	—	68	2	no	—
277	#103	D₆₈	1	1	yes	characteristic
278	#102	—	17	2	no	—
279	#102	—	34	2	no	—
280	#102	—	34	2	no	—
281	#102	D₆₈	1	1	yes	characteristic
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	34	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
288	C₁₇	#19905	1	1	yes	characteristic, complemented, Sylow
289	C₂×Q₈	—	34	2	no	—
290	C₂×Q₈	—	34	2	no	—
291	C₂×Q₈	—	68	2	no	—
292	C₂×Q₈	—	68	4	no	—
293	C₂×Q₈	—	136	2	no	—
294	C₂×Q₈	C₂×D₆₈	1	1	yes	characteristic
295	C₂×D₄	—	17	2	no	—
296	C₂×D₄	—	68	2	no	—
297	C₂×D₄	C₂×D₆₈	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	34	2	no	—
300	C₂²×C₄	—	68	2	no	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	136	2	no	—
303	D₈	—	34	2	no	—
304	M₁₆	—	136	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	34	2	no	—
310	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
311	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
312	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
313	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	17	2	no	—
316	C₄⋊C₄	—	34	2	no	—
317	C₄⋊C₄	—	34	2	no	—
318	C₄⋊C₄	—	34	2	no	—
319	C₄⋊C₄	—	34	2	no	—
320	C₄⋊C₄	—	68	2	no	—
321	C₄⋊C₄	C₂×D₆₈	1	1	yes	characteristic
322	C₄⋊C₄	C₂×D₆₈	1	1	yes	characteristic
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	34	2	no	—
325	C₂²⋊C₄	—	68	2	no	—
326	C₂²⋊C₄	—	68	2	no	—
327	C₂²⋊C₄	—	68	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	68	2	no	—
330	C₄²	C₂³×D₁₇	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	34	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	68	2	no	—
335	Q₈	—	68	2	no	—
336	Q₈	—	136	4	no	—
337	Q₈	—	136	2	no	—
338	Q₈	#10587590	1	2	yes	—
339	D₄	—	2	1	no	—
340	D₄	—	2	1	no	—
341	D₄	—	272	2	no	—
342	D₄	—	34	2	no	—
343	D₄	—	34	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	34	2	no	—
350	C₂×C₄	—	68	2	no	—
351	C₂×C₄	—	68	2	no	—
352	C₂×C₄	—	68	2	no	—
353	C₂×C₄	—	68	2	no	—
354	C₂×C₄	—	136	4	no	—
355	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
357	C₂×C₄	#10587696	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	68	2	no	—
365	C₈	#10587601	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	68	2	no	—
370	C₂²	—	68	2	no	—
371	C₂²	—	68	2	no	—
372	C₂²	D₄×D₆₈	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	68	2	no	—
378	C₄	—	136	2	no	—
379	C₄	—	136	2	no	—
380	C₄	C₂×#10587590	1	1	yes	characteristic
381	C₄	C₂×#10587601	1	1	yes	characteristic
382	C₄	#11763614	1	1	yes	characteristic
383	C₄	#11763623	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	136	2	no	—
386	C₂	#14540431	1	1	yes	characteristic
387	C₂	#14540471	1	1	yes	characteristic
388	C₂	#14540571	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₆₈
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₈×SD₁₆×C₁₇	C₂
socle	C₂²×C₁₇	D₄²
prime radical	#14541207	C₂
hypercenter	C₈×SD₁₆	D₁₇
hypocenter	C₁₇	#19905
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₃₆	C₂	semidirect
3. #14541080	C₂	semidirect
4. #14528394	C₂	semidirect
5. #14528360	C₂	semidirect
6. #14528188	C₂	semidirect
7. #14528173	C₂	semidirect
8. #14528014	C₂	semidirect
9. #14527951	C₂	semidirect
10. #14527670	C₂	semidirect
11. C₈×SD₁₆×C₁₇	C₂	semidirect
182. C₈×SD₁₆	D₁₇	semidirect
288. C₁₇	#19905	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₁₃₆	2
#14541207	2
#14541080	2
#14528394	2
#14528360	2
#14528188	2
#14528173	2
#14528014	2
#14527951	2
#14527670	2
C₈×SD₁₆×C₁₇	2
#19905	17