Subgroups — #14540912

Order 2176.

392 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	#11763499	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763480	C₂	1	1	yes	maximal, characteristic
3	#11763174	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763134	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763128	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763055	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763010	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762973	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762973	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762811	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762806	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	C₂²	1	1	yes	characteristic
17	C₂×#63252	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
21	#10587626	—	2	1	no	—
22	#10587621	—	2	1	no	—
23	#10587616	C₂²	1	1	yes	characteristic
24	#10587616	C₂²	1	1	yes	characteristic
25	#10587612	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
27	#10587610	C₂²	1	1	yes	characteristic
28	#10587610	C₂²	1	1	yes	characteristic
29	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
30	C₂×#63219	C₂²	1	1	yes	characteristic
31	C₂×#63219	C₂²	1	1	yes	characteristic
32	#10587579	—	2	1	no	—
33	#10587575	—	2	1	no	—
34	#10587570	—	2	1	no	—
35	#10587570	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	#10587563	—	2	1	no	—
38	#10587560	C₂²	1	1	yes	characteristic
39	#10587556	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×#97	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587489	C₂²	1	1	yes	characteristic
53	#10587488	C₂²	1	1	yes	characteristic
54	#10587488	C₂²	1	1	yes	characteristic
55	#10587487	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	#63252	—	2	2	no	—
68	#63252	—	2	2	no	—
69	#63252	—	4	1	no	—
70	#63252	—	4	1	no	—
71	C₂×D₆₈	—	2	1	no	—
72	C₂×D₆₈	—	2	1	no	—
73	C₂×D₆₈	C₂³	1	1	yes	characteristic
74	C₂×D₆₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₇	—	2	1	no	—
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₇	—	4	1	no	—
80	C₂×C₈×C₁₇	—	4	1	no	—
81	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
88	C₄²×C₁₇	C₂³	1	1	yes	characteristic
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	C₂²⋊Dic₁₇	—	2	1	no	—
91	C₂²⋊Dic₁₇	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	—	2	1	no	—
108	Dic₆₈	—	2	2	no	—
109	#63219	—	4	1	no	—
110	#63219	—	4	1	no	—
111	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	—	2	1	no	—
114	C₂²×D₁₇	—	2	1	no	—
115	D₄×C₁₇	—	4	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₇	—	4	2	no	—
126	C₂²⋊D₁₇	—	4	2	no	—
127	C₂²⋊D₁₇	—	8	1	no	—
128	C₂²⋊D₁₇	—	8	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	4	2	no	—
136	D₆₈	—	2	1	no	—
137	D₆₈	—	2	1	no	—
138	D₆₈	—	2	1	no	—
139	D₆₈	—	2	1	no	—
140	D₆₈	—	4	2	no	—
141	D₆₈	—	8	1	no	—
142	C₄×D₁₇	—	4	2	no	—
143	C₄×D₁₇	—	4	2	no	—
144	Dic₃₄	—	2	1	no	—
145	Dic₃₄	—	2	1	no	—
146	Dic₃₄	—	2	1	no	—
147	Dic₃₄	#41	1	2	yes	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	C₈×C₁₇	—	2	1	no	—
151	C₈×C₁₇	—	2	1	no	—
152	#3317	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	—	4	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₇	—	2	2	no	—
173	Dic₁₇	—	4	1	no	—
174	Dic₁₇	—	4	1	no	—
175	Dic₁₇	—	4	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	#552	—	17	1	no	—
179	#550	—	17	1	no	—
180	#490	—	17	1	no	—
181	#482	—	17	1	no	—
182	#481	—	17	1	no	—
183	#479	—	17	1	no	—
184	#468	—	17	1	no	—
185	#465	—	17	1	no	—
186	#460	—	17	1	no	—
187	#452	—	17	1	no	—
188	#452	—	17	1	no	—
189	#441	—	17	1	no	—
190	#435	D₁₇	1	1	yes	characteristic, complemented
191	#417	—	17	1	no	—
192	#416	—	17	1	no	—
193	C₂×C₁₇	—	4	1	no	—
194	C₂×C₁₇	—	4	1	no	—
195	C₂×C₁₇	#546	1	1	yes	characteristic
196	C₂×C₁₇	#575	1	1	yes	characteristic
197	C₂×C₁₇	#577	1	1	yes	characteristic
198	D₁₇	—	8	1	no	—
199	D₁₇	—	8	1	no	—
200	C₂×#41	—	17	1	no	—
201	C₂×#41	—	17	1	no	—
202	C₂×Q₁₆	—	17	1	no	—
203	C₂×SD₁₆	—	17	1	no	—
204	C₂×SD₁₆	—	17	1	no	—
205	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
206	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
207	#127	—	17	1	no	—
208	#127	—	34	1	no	—
209	#126	—	34	1	no	—
210	#125	—	17	1	no	—
211	#125	—	34	1	no	—
212	#123	—	34	1	no	—
213	C₂²⋊Q₈	—	17	1	no	—
214	C₂²⋊Q₈	—	17	1	no	—
215	#121	—	17	1	no	—
216	#121	—	17	1	no	—
217	#121	—	34	1	no	—
218	#121	—	34	1	no	—
219	C₄×Q₈	—	17	1	no	—
220	C₄×Q₈	—	17	1	no	—
221	C₄×D₄	—	34	1	no	—
222	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
223	#117	—	17	1	no	—
224	#117	—	34	1	no	—
225	C₂×C₄⋊C₄	—	17	1	no	—
226	#107	—	17	1	no	—
227	#106	—	17	1	no	—
228	#106	—	17	1	no	—
229	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
241	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
242	#97	C₂×D₁₇	1	1	yes	characteristic
243	C₁₇	#3317	1	1	yes	characteristic, complemented, Sylow
244	#41	—	34	2	no	—
245	#41	—	34	2	no	—
246	#41	—	68	1	no	—
247	#41	—	68	1	no	—
248	C₂×Q₈	—	17	1	no	—
249	C₂×Q₈	—	17	1	no	—
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	17	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂×D₄	—	34	1	no	—
254	C₂×D₄	—	34	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂×D₄	D₆₈	1	1	yes	characteristic
259	C₂²×C₄	—	34	1	no	—
260	C₂²×C₄	—	34	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
264	Q₁₆	—	34	2	no	—
265	SD₁₆	—	68	1	no	—
266	SD₁₆	—	68	1	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	17	1	no	—
274	C₄⋊C₄	—	17	1	no	—
275	C₄⋊C₄	—	17	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	—	34	1	no	—
283	C₄⋊C₄	—	34	1	no	—
284	C₄⋊C₄	D₆₈	1	1	yes	characteristic
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
291	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
292	C₄²	—	34	1	no	—
293	C₄²	—	34	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	C₂²×D₁₇	1	1	yes	characteristic
296	C₂³	—	34	1	no	—
297	C₂³	—	34	1	no	—
298	C₂³	C₂×D₆₈	1	1	yes	characteristic
299	C₂³	C₂×D₆₈	1	1	yes	characteristic
300	Q₈	—	17	2	no	—
301	Q₈	—	34	1	no	—
302	Q₈	—	34	1	no	—
303	Q₈	—	34	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	34	1	no	—
306	D₄	—	34	1	no	—
307	D₄	—	34	1	no	—
308	D₄	—	34	1	no	—
309	D₄	—	68	2	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	136	1	no	—
313	D₄	—	136	1	no	—
314	D₄	—	136	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	1	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	34	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
329	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
330	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
331	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	68	1	no	—
345	C₂²	—	68	1	no	—
346	C₂²	—	68	1	no	—
347	C₂²	—	68	1	no	—
348	C₂²	—	68	1	no	—
349	C₂²	C₂²×D₆₈	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	34	2	no	—
355	C₄	—	68	1	no	—
356	C₄	—	68	1	no	—
357	C₄	—	68	1	no	—
358	C₄	—	68	1	no	—
359	C₄	—	68	1	no	—
360	C₄	#10587691	1	1	yes	characteristic
361	C₄	#41×D₁₇	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	136	1	no	—
365	C₂	—	136	1	no	—
366	C₂	#11763454	1	1	yes	characteristic
367	C₂	#11763609	1	1	yes	characteristic
368	C₂	#11763621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#435	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763480	C₂
hypercenter	#435	D₁₇
hypocenter	C₁₇	#3317
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #11763499	C₂	semidirect
3. #11763174	C₂	semidirect
4. #11763134	C₂	semidirect
5. #11763128	C₂	semidirect
6. #11763117	C₂	semidirect
7. #11763055	C₂	semidirect
8. #11763038	C₂	semidirect
9. #11763010	C₂	semidirect
10. #11762973	C₂	semidirect
11. #11762973	C₂	semidirect
12. #11762917	C₂	semidirect
13. #11762811	C₂	semidirect
14. #11762806	C₂	semidirect
15. C₁₇×#435	C₂	semidirect
190. #435	D₁₇	semidirect
243. C₁₇	#3317	semidirect

Maximal subgroups

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

subgroup	index
#11763499	2
#11763480	2
#11763174	2
#11763134	2
#11763128	2
#11763117	2
#11763055	2
#11763038	2
#11763010	2
#11762973	2
#11762917	2
#11762811	2
#11762806	2
C₁₇×#435	2
#3317	17