Subgroups — #14540835

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 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
2	#11763177	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763137	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763065	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763048	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763016	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762965	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762965	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762881	C₂	1	1	yes	maximal, characteristic
13	#11762818	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10587506	C₂	1	1	yes	maximal, characteristic, complemented
15	D₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	—	2	1	no	—
19	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₇×C₂²⋊Q₈	—	2	1	no	—
21	C₁₇×C₂²⋊Q₈	—	2	1	no	—
22	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₇	—	2	1	no	—
26	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×#63231	C₂²	1	1	yes	characteristic
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63231	C₂²	1	1	yes	characteristic
30	#10587621	—	2	1	no	—
31	#10587612	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
33	#10587610	C₂²	1	1	yes	characteristic
34	#10587610	C₂²	1	1	yes	characteristic
35	C₂×#63223	C₂²	1	1	yes	characteristic
36	C₂²×#3622	C₂²	1	1	yes	characteristic
37	#10587575	—	2	1	no	—
38	#10587565	—	2	1	no	—
39	#10587563	—	2	1	no	—
40	#10587556	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	#10587506	C₂²	1	1	yes	characteristic
51	C₂²⋊#3622	C₂²	1	1	yes	characteristic
52	C₂²⋊#3622	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	C₄⋊#3622	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂×Q₈×C₁₇	—	2	2	no	—
62	C₂×Q₈×C₁₇	—	2	2	no	—
63	C₂×Q₈×C₁₇	—	4	1	no	—
64	C₂×Q₈×C₁₇	—	4	1	no	—
65	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	—	2	1	no	—
70	C₂²×C₄×C₁₇	—	2	1	no	—
71	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₂²×Dic₁₇	—	2	1	no	—
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	C₂²⋊Dic₁₇	—	2	1	no	—
91	#63231	—	2	2	no	—
92	#63231	—	4	1	no	—
93	#63231	—	4	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	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	#63223	—	4	1	no	—
106	C₂×#3622	—	4	1	no	—
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	C₂×#3622	C₂³	1	1	yes	characteristic
111	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	4	2	no	—
119	Q₈×C₁₇	—	4	2	no	—
120	Q₈×C₁₇	#41	1	2	yes	—
121	D₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	1	no	—
127	C₂×C₄×C₁₇	—	2	1	no	—
128	C₂×C₄×C₁₇	—	2	1	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	4	2	no	—
131	C₂×C₄×C₁₇	—	4	2	no	—
132	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
137	C₂×Dic₁₇	—	2	1	no	—
138	C₂×Dic₁₇	—	2	1	no	—
139	C₂×Dic₁₇	—	2	1	no	—
140	C₂×Dic₁₇	—	2	1	no	—
141	C₂×Dic₁₇	—	4	2	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3312	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	#137	1	2	yes	—
155	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	2	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	#552	—	17	1	no	—
171	D₄×Q₈	D₁₇	1	1	yes	characteristic
172	#491	—	17	1	no	—
173	#485	—	17	1	no	—
174	#483	—	17	1	no	—
175	#479	—	17	1	no	—
176	#470	—	17	1	no	—
177	#467	—	17	1	no	—
178	#462	—	17	1	no	—
179	#451	—	17	1	no	—
180	#451	—	17	1	no	—
181	#441	—	17	1	no	—
182	#435	—	17	1	no	—
183	#419	—	17	1	no	—
184	C₂×#103	—	17	1	no	—
185	C₂×C₁₇	—	2	2	no	—
186	C₂×C₁₇	—	4	1	no	—
187	C₂×C₁₇	#546	1	1	yes	characteristic
188	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
189	C₂×C₁₇	#578	1	1	yes	characteristic
190	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
191	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
192	C₂×Q₁₆	—	17	1	no	—
193	C₂×Q₁₆	—	17	1	no	—
194	C₂×Q₁₆	—	17	1	no	—
195	C₂×M₁₆	—	17	1	no	—
196	C₂²×C₈	—	17	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
199	#126	—	34	1	no	—
200	#125	—	17	1	no	—
201	#125	—	34	1	no	—
202	#123	—	34	1	no	—
203	C₂²⋊Q₈	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	17	1	no	—
206	C₂²⋊Q₈	—	17	1	no	—
207	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
208	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
209	C₄×Q₈	—	17	1	no	—
210	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
213	#117	—	17	1	no	—
214	#117	—	34	1	no	—
215	C₂×C₄⋊C₄	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	C₂²⋊C₈	—	17	1	no	—
231	C₂²⋊C₈	—	17	1	no	—
232	#97	—	17	1	no	—
233	C₁₇	#3312	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₂×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
250	Q₁₆	—	34	2	no	—
251	Q₁₆	—	68	1	no	—
252	Q₁₆	—	68	1	no	—
253	M₁₆	—	68	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	68	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
281	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
282	C₄²	—	2	1	no	—
283	C₄²	—	34	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	C₂²×D₁₇	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
287	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	#63254	1	2	yes	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
319	C₈	—	34	1	no	—
320	C₈	—	34	1	no	—
321	C₈	—	34	1	no	—
322	C₈	—	34	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	#10587628	1	2	yes	—
329	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	68	1	no	—
339	C₄	—	68	1	no	—
340	C₄	—	68	1	no	—
341	C₄	—	68	1	no	—
342	C₄	C₂×#63254	1	1	yes	characteristic
343	C₄	#10587691	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#11763453	1	1	yes	characteristic
347	C₂	C₂×#10587628	1	1	yes	characteristic
348	C₂	#11763624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	D₄×Q₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×C₁₇	D₄
hypercenter	D₄×Q₈	D₁₇
hypocenter	C₁₇	#3312
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #11763177	C₂	semidirect
3. #11763147	C₂	semidirect
4. #11763137	C₂	semidirect
5. #11763116	C₂	semidirect
6. #11763065	C₂	semidirect
7. #11763048	C₂	semidirect
8. #11763016	C₂	semidirect
9. #11762965	C₂	semidirect
10. #11762965	C₂	semidirect
11. #11762914	C₂	semidirect
13. #11762818	C₂	semidirect
14. C₂×#10587506	C₂	semidirect
233. C₁₇	#3312	semidirect

Maximal subgroups

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

subgroup	index
#11763499	2
#11763177	2
#11763147	2
#11763137	2
#11763116	2
#11763065	2
#11763048	2
#11763016	2
#11762965	2
#11762914	2
#11762881	2
#11762818	2
C₂×#10587506	2
D₄×Q₈×C₁₇	2
#3312	17