Subgroups — #14538641

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763436	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763288	C₂	1	1	yes	maximal, characteristic
3	#11763119	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763107	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763093	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763091	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762834	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762807	C₂	1	1	yes	maximal, characteristic
14	#11762802	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	C₂²	1	1	yes	characteristic
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587632	—	2	1	no	—
20	#10587630	—	2	1	no	—
21	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	C₂×#63225	—	2	1	no	—
24	C₂×#63218	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
26	#10587582	—	2	1	no	—
27	#10587578	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
30	#10587576	C₂²	1	1	yes	characteristic
31	#10587575	—	2	1	no	—
32	#10587574	C₂²	1	1	yes	characteristic
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587567	—	2	1	no	—
35	#10587563	—	2	1	no	—
36	#10587558	C₂²	1	1	yes	characteristic
37	#10587556	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₁₇×#106	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587489	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587481	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	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	#63254	—	2	2	no	—
66	#63254	Q₈	1	4	yes	complemented
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	D₄	1	1	yes	characteristic
83	#63227	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	#63225	—	2	1	no	—
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	D₄	1	1	yes	characteristic
97	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	#63218	—	2	2	no	—
105	C₈×D₁₇	—	2	2	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₁₇	—	8	1	no	—
109	D₄×C₁₇	—	2	1	no	—
110	D₄×C₁₇	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	8	1	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	2	4	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	4	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	8	1	no	—
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₇	—	8	1	no	—
129	C₄×D₁₇	—	8	1	no	—
130	C₄×D₁₇	C₂×Q₈	1	2	yes	—
131	C₄×D₁₇	#41	1	2	yes	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	8	1	no	—
134	Dic₃₄	C₂×Q₈	1	2	yes	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3622	—	2	1	no	—
139	#3084	—	17	1	no	maximal, Sylow
140	C₂²×C₁₇	—	2	2	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
154	Dic₁₇	—	2	2	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	2	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	C₂²⋊Q₈	1	2	yes	—
161	#544	—	17	1	no	—
162	#520	—	17	1	no	—
163	#479	—	17	1	no	—
164	#479	—	17	1	no	—
165	#478	—	17	1	no	—
166	#478	—	17	1	no	—
167	#476	—	17	1	no	—
168	#476	D₁₇	1	1	yes	characteristic, complemented
169	#475	—	17	1	no	—
170	#475	—	17	1	no	—
171	#428	—	17	1	no	—
172	C₂×#106	—	17	1	no	—
173	#424	—	17	1	no	—
174	#417	—	17	1	no	—
175	#416	—	17	1	no	—
176	C₂×C₁₇	—	4	2	no	—
177	C₂×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₁₇	#577	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	D₁₇	—	2	2	no	—
181	C₂×#41	—	17	1	no	—
182	C₂×M₁₆	—	17	1	no	—
183	C₂²×C₈	—	17	1	no	—
184	C₄⋊Q₈	—	17	1	no	—
185	C₄⋊Q₈	—	34	1	no	—
186	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
187	#125	—	17	1	no	—
188	#125	—	17	1	no	—
189	#125	—	34	1	no	—
190	C₂²⋊Q₈	—	34	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₄×Q₈	—	17	1	no	—
193	C₄×Q₈	—	17	1	no	—
194	C₄×D₄	—	17	1	no	—
195	C₄×D₄	—	34	1	no	—
196	C₄×D₄	—	34	1	no	—
197	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
198	#117	—	17	1	no	—
199	#117	—	17	1	no	—
200	#117	—	34	1	no	—
201	C₂×C₄⋊C₄	—	17	1	no	—
202	C₂×C₄⋊C₄	—	17	1	no	—
203	C₂×C₄⋊C₄	—	34	1	no	—
204	#107	—	17	1	no	—
205	#107	—	17	1	no	—
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	—	17	1	no	—
210	#106	C₂×D₁₇	1	1	yes	characteristic
211	#106	C₂×D₁₇	1	1	yes	characteristic
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	C₂×D₁₇	1	1	yes	characteristic
223	#102	C₂×D₁₇	1	1	yes	characteristic
224	C₁₇	#3084	1	1	yes	characteristic, complemented, Sylow
225	#41	—	17	4	no	—
226	#41	—	34	2	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	17	1	no	—
229	C₂×Q₈	—	34	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	M₁₆	—	34	2	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	34	2	no	—
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	17	1	no	—
264	C₂²⋊C₄	—	17	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₄²	—	17	1	no	—
268	C₄²	—	34	1	no	—
269	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	C₂²×D₁₇	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	17	1	no	—
274	Q₈	—	8	1	no	—
275	Q₈	—	17	2	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	136	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	34	4	no	—
280	D₄	Q₈×D₁₇	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	17	2	no	—
287	C₂×C₄	—	17	2	no	—
288	C₂×C₄	—	17	1	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	34	4	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	136	1	no	—
296	C₂×C₄	—	136	1	no	—
297	C₂×C₄	—	136	1	no	—
298	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	34	1	no	—
304	C₈	—	34	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	17	2	no	—
309	C₂²	—	34	1	no	—
310	C₂²	—	34	1	no	—
311	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	17	2	no	—
318	C₄	—	34	2	no	—
319	C₄	—	34	1	no	—
320	C₄	—	34	1	no	—
321	C₄	—	68	1	no	—
322	C₄	—	68	1	no	—
323	C₄	—	68	1	no	—
324	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
325	C₄	#41×D₁₇	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	34	2	no	—
328	C₂	C₂²⋊Q₈×D₁₇	1	1	yes	characteristic
329	C₂	#11763615	1	1	yes	characteristic
330	C₂	#11763619	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#476	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63254	C₂²
hypercenter	#476	D₁₇
hypocenter	C₁₇	#3084
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #11763436	C₂	semidirect
3. #11763119	C₂	semidirect
4. #11763117	C₂	semidirect
5. #11763109	C₂	semidirect
6. #11763107	C₂	semidirect
7. #11763097	C₂	semidirect
8. #11763093	C₂	semidirect
9. #11763091	C₂	semidirect
10. #11762848	C₂	semidirect
11. #106×D₁₇	C₂	semidirect
12. #11762834	C₂	semidirect
15. C₁₇×#476	C₂	semidirect
66. #63254	Q₈	semidirect
168. #476	D₁₇	semidirect
224. C₁₇	#3084	semidirect

Maximal subgroups

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

subgroup	index
#11763436	2
#11763288	2
#11763119	2
#11763117	2
#11763109	2
#11763107	2
#11763097	2
#11763093	2
#11763091	2
#11762848	2
#106×D₁₇	2
#11762834	2
#11762807	2
#11762802	2
C₁₇×#476	2
#3084	17