Subgroups — #17026364

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541794	C₂	1	1	yes	maximal, characteristic
2	#14541598	C₂	1	2	yes	maximal, complemented
3	#14528377	C₂	1	1	yes	maximal, characteristic
4	#14528326	C₂	1	2	yes	maximal, complemented
5	#14528185	C₂	1	1	yes	maximal, characteristic, complemented
6	#14528144	C₂	1	1	yes	maximal, characteristic
7	#14527904	C₂	1	2	yes	maximal, complemented
8	#14527862	C₂	1	2	yes	maximal, complemented
9	#14527681	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#63231	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763544	C₂²	1	2	yes	—
13	#11763194	—	2	1	no	—
14	#11763186	C₂²	1	1	yes	characteristic
15	#11763183	C₂²	1	2	yes	—
16	#11763171	—	2	2	no	—
17	#11763157	C₂²	1	2	yes	—
18	#11763101	C₂²	1	2	yes	—
19	#11763086	—	2	2	no	—
20	#11763031	—	2	2	no	—
21	#11763027	—	2	2	no	—
22	#11763009	C₂²	1	2	yes	—
23	#11763009	C₂²	1	2	yes	—
24	#11762940	C₂²	1	2	yes	—
25	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
26	#11762907	—	2	2	no	—
27	C₄×#63231	C₂²	1	1	yes	characteristic
28	#11762891	—	2	1	no	—
29	#11762562	C₂²	1	1	yes	characteristic
30	#11762561	C₂²	1	1	yes	characteristic
31	#11762558	C₂²	1	2	yes	—
32	#11762552	C₂²	1	2	yes	—
33	Q₈⋊#3622	C₂²	1	1	yes	characteristic
34	Q₈⋊#3622	C₂²	1	2	yes	—
35	#11762539	C₂²	1	1	yes	characteristic
36	C₈×#3622	C₂²	1	1	yes	characteristic
37	Q₈²×C₁₇	C₂²	1	2	yes	—
38	C₁₇×#500	—	2	1	no	—
39	C₁₇×#500	C₂²	1	1	yes	characteristic
40	C₁₇×#477	—	2	2	no	—
41	C₁₇×#477	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	—	2	2	no	—
48	C₁₇×C₄⋊Q₈	—	4	2	no	—
49	C₁₇×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₄×Q₈×C₁₇	—	2	2	no	—
51	C₄×Q₈×C₁₇	—	2	2	no	—
52	C₄×Q₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
54	#10587633	—	2	2	no	—
55	C₂×#63231	D₄	1	1	yes	characteristic
56	C₂×#63230	—	2	2	no	—
57	C₂×#63230	—	2	2	no	—
58	C₂×#63230	—	4	2	no	—
59	C₂×D₁₃₆	—	2	1	no	—
60	C₂×D₁₃₆	—	2	1	no	—
61	#10587580	—	4	2	no	—
62	#10587579	—	2	2	no	—
63	#10587561	—	2	2	no	—
64	#10587560	C₂³	1	2	yes	—
65	C₄×D₆₈	—	2	2	no	—
66	C₄×Dic₃₄	C₂³	1	1	yes	characteristic
67	C₁₇×#107	—	2	1	no	—
68	C₁₇×#107	—	2	1	no	—
69	C₁₇×#107	—	2	2	no	—
70	C₁₇×#107	D₄	1	1	yes	characteristic
71	C₁₇×C₄⋊C₈	—	2	1	no	—
72	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₁₇×#103	—	2	2	no	—
74	C₁₇×#103	D₄	1	2	yes	—
75	C₄×C₈×C₁₇	—	2	1	no	—
76	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	#10587506	D₄	1	1	yes	characteristic
78	#10587485	—	2	1	no	—
79	C₈×Dic₁₇	—	2	1	no	—
80	#10587481	—	2	2	no	—
81	#10587481	D₄	1	1	yes	characteristic
82	#10587480	—	2	2	no	—
83	#10587480	—	2	2	no	—
84	#10587480	—	2	2	no	—
85	#10587480	—	2	2	no	—
86	#10587478	—	2	2	no	—
87	#10587478	D₄	1	1	yes	characteristic
88	C₄⋊#3622	—	2	2	no	—
89	C₄⋊#3622	C₂³	1	1	yes	characteristic
90	C₄⋊#3622	C₂³	1	2	yes	—
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	—	4	2	no	—
96	C₂×Q₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
98	C₂×D₆₈	—	2	2	no	—
99	C₂×D₆₈	—	2	2	no	—
100	C₂×D₆₈	—	2	2	no	—
101	C₂×C₄×D₁₇	—	4	2	no	—
102	C₂×Dic₃₄	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₁₇	—	2	2	no	—
104	C₂×C₈×C₁₇	—	2	1	no	—
105	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₁₇	—	2	1	no	—
108	C₄⋊C₄×C₁₇	—	2	2	no	—
109	C₄⋊C₄×C₁₇	—	2	2	no	—
110	C₄⋊C₄×C₁₇	—	2	2	no	—
111	C₄⋊C₄×C₁₇	—	4	2	no	—
112	C₄⋊C₄×C₁₇	—	4	2	no	—
113	C₄⋊C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
115	C₄²×C₁₇	—	2	1	no	—
116	C₄²×C₁₇	—	2	2	no	—
117	C₄²×C₁₇	—	4	2	no	—
118	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
119	#63231	—	2	2	no	—
120	#63230	—	4	4	no	—
121	#63230	—	8	4	no	—
122	#63230	—	8	2	no	—
123	#63227	—	4	2	no	—
124	#63227	—	4	2	no	—
125	#63227	—	4	2	no	—
126	C₄⋊Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	#63225	—	2	1	no	—
128	C₄×Dic₁₇	—	2	1	no	—
129	C₂×#3622	—	2	1	no	—
130	C₂×#3622	—	2	2	no	—
131	C₂×#3622	—	2	1	no	—
132	C₂×#3622	C₂×D₄	1	1	yes	characteristic
133	C₂×#3622	C₂×D₄	1	1	yes	characteristic
134	D₁₃₆	—	4	2	no	—
135	D₁₃₆	—	8	1	no	—
136	#19942	—	17	1	no	maximal, Sylow
137	C₂²×D₁₇	—	4	2	no	—
138	Q₈×C₁₇	—	2	1	no	—
139	Q₈×C₁₇	—	2	2	no	—
140	Q₈×C₁₇	—	2	2	no	—
141	Q₈×C₁₇	—	8	2	no	—
142	Q₈×C₁₇	—	16	2	no	—
143	Q₈×C₁₇	#135	1	2	yes	—
144	C₂×C₄×C₁₇	—	2	1	no	—
145	C₂×C₄×C₁₇	—	2	2	no	—
146	C₂×C₄×C₁₇	—	2	1	no	—
147	C₂×C₄×C₁₇	—	2	2	no	—
148	C₂×C₄×C₁₇	—	4	2	no	—
149	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₁₇	—	2	1	no	—
153	C₂×Dic₁₇	—	2	1	no	—
154	D₆₈	—	4	2	no	—
155	D₆₈	—	4	2	no	—
156	D₆₈	—	4	2	no	—
157	D₆₈	—	4	2	no	—
158	D₆₈	—	8	2	no	—
159	C₄×D₁₇	—	16	2	no	—
160	Dic₃₄	—	2	1	no	—
161	Dic₃₄	—	2	1	no	—
162	C₈×C₁₇	—	2	1	no	—
163	C₈×C₁₇	—	4	1	no	—
164	C₈×C₁₇	#137	1	2	yes	—
165	#3622	—	2	2	no	—
166	#3622	—	2	1	no	—
167	#3622	—	2	1	no	—
168	#3622	—	4	2	no	—
169	#3622	—	4	1	no	—
170	#3396	—	17	1	no	—
171	Q₈×Q₁₆	D₁₇	1	1	yes	characteristic, complemented
172	#3375	—	17	2	no	—
173	#1706	—	17	1	no	—
174	#1698	—	17	2	no	—
175	#1676	—	17	1	no	—
176	#1668	—	17	1	no	—
177	#1630	—	17	2	no	—
178	#1623	—	17	2	no	—
179	#1591	—	17	1	no	—
180	C₈×Q₁₆	—	17	1	no	—
181	C₂²×C₁₇	D₄²	1	1	yes	characteristic
182	C₂×D₁₇	—	8	2	no	—
183	C₂×D₁₇	—	8	2	no	—
184	C₂×D₁₇	—	8	2	no	—
185	C₄×C₁₇	—	2	2	no	—
186	C₄×C₁₇	—	2	1	no	—
187	C₄×C₁₇	—	4	2	no	—
188	C₄×C₁₇	—	4	2	no	—
189	C₄×C₁₇	—	4	1	no	—
190	C₄×C₁₇	—	8	2	no	—
191	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
192	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
193	C₄×C₁₇	#577	1	1	yes	characteristic
194	C₄×C₁₇	#578	1	1	yes	characteristic
195	Dic₁₇	—	4	1	no	—
196	Dic₁₇	—	4	1	no	—
197	#559	—	17	2	no	—
198	Q₈²	C₂×D₁₇	1	2	yes	—
199	#500	—	2	1	no	—
200	#500	C₂×D₁₇	1	1	yes	characteristic
201	#495	—	34	1	no	—
202	#493	—	17	1	no	—
203	#492	—	17	2	no	—
204	#489	—	34	2	no	—
205	#486	—	17	2	no	—
206	#477	—	2	2	no	—
207	#477	—	17	2	no	—
208	#477	C₂×D₁₇	1	2	yes	—
209	#474	—	34	2	no	—
210	#464	—	34	2	no	—
211	#463	—	34	2	no	—
212	#460	—	17	2	no	—
213	#460	—	17	2	no	—
214	#446	—	17	2	no	—
215	C₈×Q₈	—	17	1	no	—
216	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
217	#440	—	34	2	no	—
218	C₄×Q₁₆	—	2	1	no	—
219	C₄×Q₁₆	—	17	1	no	—
220	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
221	C₄×D₈	—	34	1	no	—
222	#335	—	17	1	no	—
223	#335	—	17	1	no	—
224	#333	—	17	2	no	—
225	#331	—	17	2	no	—
226	Q₈⋊C₈	—	17	1	no	—
227	Q₈⋊C₈	—	17	2	no	—
228	Q₈⋊C₈	—	17	1	no	—
229	C₈²	—	17	1	no	—
230	C₂×C₁₇	#3278	1	1	yes	characteristic
231	C₂×C₁₇	#3285	1	1	yes	characteristic
232	C₂×C₁₇	#3292	1	1	yes	characteristic
233	D₁₇	—	16	2	no	—
234	C₂×Q₁₆	—	17	1	no	—
235	C₂×Q₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
236	C₂×SD₁₆	—	34	2	no	—
237	C₂×SD₁₆	—	34	2	no	—
238	C₂×SD₁₆	—	68	2	no	—
239	C₂×D₈	—	34	1	no	—
240	C₂×D₈	—	34	1	no	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	2	2	no	—
243	C₄⋊Q₈	—	4	2	no	—
244	C₄⋊Q₈	C₂²×D₁₇	1	2	yes	—
245	#127	—	17	2	no	—
246	#124	—	34	2	no	—
247	#124	—	34	2	no	—
248	#123	—	68	2	no	—
249	C₄×Q₈	—	2	2	no	—
250	C₄×Q₈	—	2	2	no	—
251	C₄×Q₈	—	17	1	no	—
252	C₄×Q₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
254	C₄×D₄	—	34	2	no	—
255	C₄×D₄	—	34	2	no	—
256	#107	—	2	1	no	—
257	#107	—	2	1	no	—
258	#107	—	2	2	no	—
259	#107	—	17	1	no	—
260	#107	—	34	2	no	—
261	#107	C₂²⋊D₁₇	1	1	yes	characteristic
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	17	2	no	—
264	C₄⋊C₈	—	17	1	no	—
265	C₄⋊C₈	—	34	2	no	—
266	C₄⋊C₈	—	34	1	no	—
267	C₄⋊C₈	C₂²×D₁₇	1	1	yes	characteristic
268	#103	—	2	2	no	—
269	#103	—	17	1	no	—
270	#103	—	17	1	no	—
271	#103	—	34	2	no	—
272	#103	C₂²⋊D₁₇	1	2	yes	—
273	#102	—	34	2	no	—
274	#102	—	34	2	no	—
275	#102	—	34	2	no	—
276	#102	—	34	2	no	—
277	#97	—	34	2	no	—
278	C₄×C₈	—	2	1	no	—
279	C₄×C₈	—	17	1	no	—
280	C₄×C₈	—	17	1	no	—
281	C₄×C₈	—	34	1	no	—
282	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
283	C₁₇	#19942	1	1	yes	characteristic, complemented, Sylow
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	4	2	no	—
286	C₂×Q₈	—	17	1	no	—
287	C₂×Q₈	D₄×D₁₇	1	1	yes	characteristic
288	C₂×Q₈	C₂×C₂²⋊D₁₇	1	2	yes	—
289	C₂×D₄	—	34	2	no	—
290	C₂×D₄	—	34	2	no	—
291	C₂×D₄	—	34	2	no	—
292	C₂²×C₄	—	68	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	34	2	no	—
295	SD₁₆	—	68	4	no	—
296	SD₁₆	—	136	4	no	—
297	SD₁₆	—	136	2	no	—
298	D₈	—	68	2	no	—
299	D₈	—	136	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	17	1	no	—
302	C₂×C₈	—	17	1	no	—
303	C₂×C₈	—	34	1	no	—
304	C₂×C₈	—	34	2	no	—
305	C₂×C₈	—	34	1	no	—
306	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	17	1	no	—
315	C₄⋊C₄	—	34	1	no	—
316	C₄⋊C₄	D₄×D₁₇	1	1	yes	characteristic
317	C₄⋊C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
318	C₂²⋊C₄	—	68	2	no	—
319	C₂²⋊C₄	—	68	2	no	—
320	C₂²⋊C₄	—	68	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	—	34	1	no	—
325	C₄²	C₂³×D₁₇	1	1	yes	characteristic
326	C₂³	—	68	2	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	2	2	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	8	2	no	—
331	Q₈	—	16	2	no	—
332	Q₈	—	34	1	no	—
333	Q₈	—	34	1	no	—
334	Q₈	#10587605	1	2	yes	—
335	D₄	—	68	2	no	—
336	D₄	—	68	2	no	—
337	D₄	—	68	2	no	—
338	D₄	—	68	2	no	—
339	D₄	—	136	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	272	2	no	—
346	C₂×C₄	—	34	1	no	—
347	C₂×C₄	—	34	1	no	—
348	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
349	C₂×C₄	#10587691	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
351	C₈	—	2	1	no	—
352	C₈	—	4	1	no	—
353	C₈	—	34	2	no	—
354	C₈	—	34	1	no	—
355	C₈	—	34	1	no	—
356	C₈	—	68	2	no	—
357	C₈	—	68	1	no	—
358	C₈	#10587628	1	2	yes	—
359	C₂²	—	136	2	no	—
360	C₂²	—	136	2	no	—
361	C₂²	—	136	2	no	—
362	C₂²	D₄×C₂²⋊D₁₇	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	8	2	no	—
369	C₄	—	68	1	no	—
370	C₄	—	68	1	no	—
371	C₄	C₂×#10587605	1	1	yes	characteristic
372	C₄	C₂×#10587628	1	1	yes	characteristic
373	C₄	#11763617	1	1	yes	characteristic
374	C₄	#11763622	1	1	yes	characteristic
375	C₂	—	272	2	no	—
376	C₂	#14540446	1	1	yes	characteristic
377	C₂	#14540525	1	1	yes	characteristic
378	C₂	#14540581	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₁₇
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	Q₈×Q₁₆×C₁₇	C₂
socle	C₂²×C₁₇	D₄²
prime radical	#11763186	C₂²
hypercenter	Q₈×Q₁₆	D₁₇
hypocenter	C₁₇	#19942
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #14541598	C₂	semidirect
4. #14528326	C₂	semidirect
5. #14528185	C₂	semidirect
7. #14527904	C₂	semidirect
8. #14527862	C₂	semidirect
9. #14527681	C₂	semidirect
10. C₈×#63231	C₂	semidirect
11. Q₈×Q₁₆×C₁₇	C₂	semidirect
171. Q₈×Q₁₆	D₁₇	semidirect
283. C₁₇	#19942	semidirect

Maximal subgroups

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

subgroup	index
#14541794	2
#14541598	2
#14528377	2
#14528326	2
#14528185	2
#14528144	2
#14527904	2
#14527862	2
#14527681	2
C₈×#63231	2
Q₈×Q₁₆×C₁₇	2
#19942	17