Subgroups — #14541733

Order 2176.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763536	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763518	C₂	1	1	yes	maximal, characteristic
3	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763115	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763107	C₂	1	1	yes	maximal, characteristic
9	#11763097	C₂	1	1	yes	maximal, characteristic
10	#11763093	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762936	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762906	C₂	1	1	yes	maximal, characteristic
13	#11762906	C₂	1	1	yes	maximal, characteristic
14	C₄×#63229	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
19	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×#63229	C₂²	1	1	yes	characteristic
21	#10587610	—	2	1	no	—
22	C₂×#63225	—	2	1	no	—
23	#10587576	C₂²	1	1	yes	characteristic
24	#10587576	C₂²	1	1	yes	characteristic
25	#10587575	—	2	1	no	—
26	#10587575	—	2	1	no	—
27	#10587574	—	2	1	no	—
28	#10587574	C₂²	1	1	yes	characteristic
29	#10587574	C₂²	1	1	yes	characteristic
30	#10587573	—	2	1	no	—
31	#10587573	C₂²	1	1	yes	characteristic
32	#10587573	C₂²	1	1	yes	characteristic
33	#10587564	—	2	1	no	—
34	#10587564	—	2	1	no	—
35	#10587556	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×#106	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587489	C₂²	1	1	yes	characteristic
45	#10587488	C₂²	1	1	yes	characteristic
46	#10587487	C₂²	1	1	yes	characteristic
47	#10587487	C₂²	1	1	yes	characteristic
48	#10587486	C₂²	1	1	yes	characteristic
49	#10587486	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	#10587479	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	#10587478	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×Dic₁₇	—	2	1	no	—
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×Dic₃₄	—	2	1	no	—
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×Dic₃₄	D₄	1	1	yes	characteristic
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₁₇	—	2	1	no	—
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	#63229	Q₈	1	4	yes	complemented
79	C₄⋊Dic₁₇	—	2	1	no	—
80	C₄⋊Dic₁₇	—	2	1	no	—
81	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
85	#63225	—	2	1	no	—
86	#63225	—	2	1	no	—
87	#63225	—	2	1	no	—
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	D₄	1	1	yes	characteristic
95	#63225	D₄	1	1	yes	characteristic
96	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂³×C₁₇	—	2	1	no	—
104	D₄×C₁₇	—	2	1	no	—
105	D₄×C₁₇	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	8	1	no	—
110	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₁₇	—	2	1	no	—
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	4	2	no	—
118	C₂×Dic₁₇	—	8	1	no	—
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
123	Dic₃₄	—	2	1	no	—
124	Dic₃₄	—	4	1	no	—
125	Dic₃₄	—	4	2	no	—
126	Dic₃₄	—	8	1	no	—
127	Dic₃₄	C₂×Q₈	1	2	yes	—
128	C₈×C₁₇	—	2	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	#3622	—	2	1	no	—
131	#3622	C₂×Q₈	1	2	yes	—
132	#3388	—	17	1	no	maximal, Sylow
133	C₂²×C₁₇	—	2	2	no	—
134	C₂²×C₁₇	—	4	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	—	4	1	no	—
142	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
143	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₁₇	—	2	2	no	—
145	Dic₁₇	—	2	1	no	—
146	Dic₁₇	—	2	1	no	—
147	Dic₁₇	—	2	1	no	—
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	—	4	1	no	—
150	Dic₁₇	—	4	1	no	—
151	Dic₁₇	—	4	1	no	—
152	#557	—	17	1	no	—
153	#554	—	17	1	no	—
154	#501	—	17	1	no	—
155	#501	—	17	1	no	—
156	#499	—	17	1	no	—
157	#479	—	17	1	no	—
158	#479	—	17	1	no	—
159	#478	—	17	1	no	—
160	#478	D₁₇	1	1	yes	characteristic
161	#476	—	17	1	no	—
162	#475	—	17	1	no	—
163	#446	—	17	1	no	—
164	#440	—	17	1	no	—
165	#440	—	17	1	no	—
166	C₄×SD₁₆	—	17	1	no	—
167	C₂×C₁₇	—	4	2	no	—
168	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
169	C₂×C₁₇	#575	1	1	yes	characteristic
170	C₂×C₁₇	#577	1	1	yes	characteristic
171	C₂×SD₁₆	—	17	1	no	—
172	C₄⋊Q₈	—	17	1	no	—
173	C₄⋊Q₈	—	17	1	no	—
174	C₄⋊Q₈	—	34	1	no	—
175	#125	—	17	1	no	—
176	#125	—	17	1	no	—
177	#125	—	17	1	no	—
178	#125	—	34	1	no	—
179	#125	—	34	1	no	—
180	#125	C₂×D₁₇	1	1	yes	characteristic
181	C₂²⋊Q₈	—	34	1	no	—
182	C₂²⋊Q₈	—	34	1	no	—
183	C₂²⋊Q₈	—	34	1	no	—
184	C₄×Q₈	—	17	1	no	—
185	C₄×Q₈	—	17	1	no	—
186	C₄×Q₈	—	17	1	no	—
187	C₄×Q₈	—	17	1	no	—
188	C₄×Q₈	—	34	1	no	—
189	C₄×D₄	—	17	1	no	—
190	C₄×D₄	—	17	1	no	—
191	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	34	1	no	—
193	#107	—	17	1	no	—
194	#107	—	17	1	no	—
195	#107	—	17	1	no	—
196	#107	C₂×D₁₇	1	1	yes	characteristic
197	#106	—	17	1	no	—
198	#106	—	17	1	no	—
199	#106	—	17	1	no	—
200	#106	—	17	1	no	—
201	#106	C₂×D₁₇	1	1	yes	characteristic
202	C₄⋊C₈	—	17	1	no	—
203	C₄⋊C₈	—	17	1	no	—
204	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
205	#103	—	17	1	no	—
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#102	—	17	1	no	—
209	#102	C₂×D₁₇	1	1	yes	characteristic
210	#102	C₂×D₁₇	1	1	yes	characteristic
211	#97	—	17	1	no	—
212	#97	—	17	1	no	—
213	C₄×C₈	—	17	1	no	—
214	C₁₇	#3388	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₈	—	17	1	no	—
216	C₂×Q₈	—	17	1	no	—
217	C₂×Q₈	—	34	1	no	—
218	C₂×Q₈	—	34	1	no	—
219	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	34	1	no	—
222	C₂²×C₄	—	34	1	no	—
223	SD₁₆	—	17	4	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	17	1	no	—
231	C₄⋊C₄	—	17	1	no	—
232	C₄⋊C₄	—	17	1	no	—
233	C₄⋊C₄	—	17	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	17	1	no	—
236	C₄⋊C₄	—	34	1	no	—
237	C₄⋊C₄	—	34	1	no	—
238	C₄⋊C₄	—	34	1	no	—
239	C₄⋊C₄	—	34	1	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	34	1	no	—
252	C₂²⋊C₄	—	34	1	no	—
253	C₄²	—	17	1	no	—
254	C₄²	—	17	1	no	—
255	C₄²	—	34	1	no	—
256	C₄²	—	34	1	no	—
257	C₄²	—	34	1	no	—
258	C₄²	C₂²×D₁₇	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	17	2	no	—
261	Q₈	—	34	1	no	—
262	Q₈	—	68	1	no	—
263	Q₈	—	68	2	no	—
264	Q₈	—	136	1	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₁₇	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	17	1	no	—
272	C₂×C₄	—	17	1	no	—
273	C₂×C₄	—	17	1	no	—
274	C₂×C₄	—	17	1	no	—
275	C₂×C₄	—	34	1	no	—
276	C₂×C₄	—	34	1	no	—
277	C₂×C₄	—	34	1	no	—
278	C₂×C₄	—	34	1	no	—
279	C₂×C₄	—	68	2	no	—
280	C₂×C₄	—	136	1	no	—
281	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
282	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	17	2	no	—
287	C₈	—	34	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	34	2	no	—
298	C₄	—	34	1	no	—
299	C₄	—	34	1	no	—
300	C₄	—	34	1	no	—
301	C₄	—	34	1	no	—
302	C₄	—	68	1	no	—
303	C₄	—	68	1	no	—
304	C₄	—	68	1	no	—
305	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
306	C₄	#10587697	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	#11763472	1	1	yes	characteristic
309	C₂	#11763606	1	1	yes	characteristic
310	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×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×C₁₇	C₂³
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3388
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763536	C₂	semidirect
3. #11763217	C₂	semidirect
4. #11763216	C₂	semidirect
5. #11763208	C₂	semidirect
6. #11763117	C₂	semidirect
7. #11763115	C₂	semidirect
10. #11763093	C₂	semidirect
11. #11762936	C₂	semidirect
78. #63229	Q₈	semidirect
214. C₁₇	#3388	semidirect

Maximal subgroups

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

subgroup	index
#11763536	2
#11763518	2
#11763217	2
#11763216	2
#11763208	2
#11763117	2
#11763115	2
#11763107	2
#11763097	2
#11763093	2
#11762936	2
#11762906	2
C₄×#63229	2
C₁₇×#478	2
#3388	17