Subgroups — #14540054

Order 2176.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763437	C₂	1	1	yes	maximal, characteristic
2	#11763421	C₂	1	1	yes	maximal, characteristic
3	#11763218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763178	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763160	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763151	C₂	1	1	yes	maximal, characteristic
8	#11763151	C₂	1	1	yes	maximal, characteristic
9	#11763145	C₂	1	1	yes	maximal, characteristic
10	#11763140	C₂	1	1	yes	maximal, characteristic
11	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763105	C₂	1	1	yes	maximal, characteristic, complemented
14	#11763095	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#433	C₂	1	1	yes	maximal, characteristic
16	C₁₇×#117	C₂²	1	1	yes	characteristic
17	#10587612	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
19	#10587610	C₂²	1	1	yes	characteristic
20	#10587610	C₂²	1	1	yes	characteristic
21	#10587576	—	2	1	no	—
22	#10587575	—	2	1	no	—
23	#10587575	—	2	1	no	—
24	#10587574	—	2	1	no	—
25	#10587565	—	2	1	no	—
26	#10587565	—	2	1	no	—
27	#10587564	—	2	1	no	—
28	#10587564	—	2	1	no	—
29	#10587556	C₂²	1	1	yes	characteristic
30	#10587556	C₂²	1	1	yes	characteristic
31	#10587556	C₂²	1	1	yes	characteristic
32	#10587555	C₂²	1	1	yes	characteristic
33	#10587555	C₂²	1	1	yes	characteristic
34	#10587555	C₂²	1	1	yes	characteristic
35	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×#97	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
43	#10587489	C₂²	1	1	yes	characteristic
44	#10587489	C₂²	1	1	yes	characteristic
45	#10587489	C₂²	1	1	yes	characteristic
46	#10587489	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587488	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587487	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	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₇	—	2	1	no	—
61	C₂×Dic₃₄	—	2	1	no	—
62	C₂×Dic₃₄	—	2	1	no	—
63	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
64	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₇	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	D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂³	1	1	yes	characteristic
74	C₄²×C₁₇	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₁₇	—	2	1	no	—
76	C₂²⋊Dic₁₇	—	2	1	no	—
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	C₄⋊Dic₁₇	—	2	1	no	—
79	C₄⋊Dic₁₇	—	2	1	no	—
80	C₄⋊Dic₁₇	—	2	1	no	—
81	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₇	—	2	1	no	—
112	C₂×Dic₁₇	—	2	1	no	—
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	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	8	1	no	—
120	Dic₃₄	—	2	1	no	—
121	Dic₃₄	—	2	1	no	—
122	Dic₃₄	—	2	1	no	—
123	Dic₃₄	—	2	1	no	—
124	Dic₃₄	—	8	1	no	—
125	Dic₃₄	—	8	1	no	—
126	C₈×C₁₇	—	2	1	no	—
127	C₈×C₁₇	—	2	1	no	—
128	C₈×C₁₇	—	2	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	#3242	—	17	1	no	maximal, Sylow
131	C₂²×C₁₇	—	2	1	no	—
132	C₂²×C₁₇	—	2	1	no	—
133	C₂²×C₁₇	—	2	1	no	—
134	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
141	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₁₇	—	4	1	no	—
143	Dic₁₇	—	4	1	no	—
144	Dic₁₇	—	4	1	no	—
145	Dic₁₇	—	4	1	no	—
146	Dic₁₇	—	4	1	no	—
147	Dic₁₇	—	4	1	no	—
148	Dic₁₇	—	4	1	no	—
149	Dic₁₇	—	4	1	no	—
150	#544	—	17	1	no	—
151	#541	—	17	1	no	—
152	#501	—	17	1	no	—
153	#499	—	17	1	no	—
154	#491	—	17	1	no	—
155	#487	—	17	1	no	—
156	#485	—	17	1	no	—
157	#485	—	17	1	no	—
158	#484	—	17	1	no	—
159	#483	—	17	1	no	—
160	#479	—	17	1	no	—
161	#479	—	17	1	no	—
162	#477	—	17	1	no	—
163	#475	—	17	1	no	—
164	#433	D₁₇	1	1	yes	characteristic
165	C₂×C₁₇	—	4	1	no	—
166	C₂×C₁₇	#536	1	1	yes	characteristic
167	C₂×C₁₇	#577	1	1	yes	characteristic
168	C₂×C₁₇	#578	1	1	yes	characteristic
169	C₄⋊Q₈	—	17	1	no	—
170	C₄⋊Q₈	—	17	1	no	—
171	C₄⋊Q₈	—	17	1	no	—
172	C₄⋊Q₈	—	34	1	no	—
173	#126	—	34	1	no	—
174	#126	—	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	—	34	1	no	—
181	C₂²⋊Q₈	—	17	1	no	—
182	C₂²⋊Q₈	—	17	1	no	—
183	C₂²⋊Q₈	—	34	1	no	—
184	C₂²⋊Q₈	—	34	1	no	—
185	C₄×Q₈	—	17	1	no	—
186	C₄×Q₈	—	17	1	no	—
187	#117	—	17	1	no	—
188	#117	C₂×D₁₇	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	17	1	no	—
190	#107	—	17	1	no	—
191	#107	—	17	1	no	—
192	#107	—	17	1	no	—
193	#107	—	17	1	no	—
194	#106	—	17	1	no	—
195	#106	—	17	1	no	—
196	#106	—	17	1	no	—
197	#106	—	17	1	no	—
198	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
200	#103	—	17	1	no	—
201	#103	—	17	1	no	—
202	#103	—	17	1	no	—
203	#103	—	17	1	no	—
204	#103	—	17	1	no	—
205	#103	—	17	1	no	—
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
209	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
210	#97	C₂×D₁₇	1	1	yes	characteristic
211	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
212	C₁₇	#3242	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	17	1	no	—
214	C₂×Q₈	—	17	1	no	—
215	C₂×Q₈	—	34	1	no	—
216	C₂×Q₈	—	34	1	no	—
217	C₂²×C₄	—	34	1	no	—
218	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
219	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
223	C₄⋊C₄	—	17	1	no	—
224	C₄⋊C₄	—	17	1	no	—
225	C₄⋊C₄	—	17	1	no	—
226	C₄⋊C₄	—	17	1	no	—
227	C₄⋊C₄	—	17	1	no	—
228	C₄⋊C₄	—	17	1	no	—
229	C₄⋊C₄	—	34	1	no	—
230	C₄⋊C₄	—	34	1	no	—
231	C₄⋊C₄	—	34	1	no	—
232	C₄⋊C₄	—	34	1	no	—
233	C₄⋊C₄	—	34	1	no	—
234	C₄⋊C₄	—	34	1	no	—
235	C₄⋊C₄	—	34	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	D₆₈	1	1	yes	characteristic
245	C₄⋊C₄	D₆₈	1	1	yes	characteristic
246	C₂²⋊C₄	—	34	1	no	—
247	C₂²⋊C₄	—	34	1	no	—
248	C₂²⋊C₄	—	34	1	no	—
249	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
250	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
251	C₄²	—	34	1	no	—
252	C₄²	—	34	1	no	—
253	C₄²	—	34	1	no	—
254	C₄²	C₂²×D₁₇	1	1	yes	characteristic
255	C₄²	C₂²×D₁₇	1	1	yes	characteristic
256	C₂³	C₂×D₆₈	1	1	yes	characteristic
257	Q₈	—	34	1	no	—
258	Q₈	—	34	1	no	—
259	Q₈	—	34	1	no	—
260	Q₈	—	34	1	no	—
261	Q₈	—	136	1	no	—
262	Q₈	—	136	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	34	1	no	—
266	C₂×C₄	—	34	1	no	—
267	C₂×C₄	—	34	1	no	—
268	C₂×C₄	—	34	1	no	—
269	C₂×C₄	—	34	1	no	—
270	C₂×C₄	—	34	1	no	—
271	C₂×C₄	—	34	1	no	—
272	C₂×C₄	—	34	1	no	—
273	C₂×C₄	—	136	1	no	—
274	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
275	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
276	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
277	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
278	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂²×D₆₈	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	68	1	no	—
294	C₄	—	68	1	no	—
295	C₄	—	68	1	no	—
296	C₄	—	68	1	no	—
297	C₄	—	68	1	no	—
298	C₄	—	68	1	no	—
299	C₄	—	68	1	no	—
300	C₄	—	68	1	no	—
301	C₄	#10587697	1	1	yes	characteristic
302	C₄	#10587697	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#11763385	1	1	yes	characteristic
305	C₂	#11763621	1	1	yes	characteristic
306	C₂	#11763626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#433	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#433	D₁₇
hypocenter	C₁₇	#3242
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
3. #11763218	C₂	semidirect
4. #11763211	C₂	semidirect
5. #11763178	C₂	semidirect
6. #11763160	C₂	semidirect
11. #11763117	C₂	semidirect
12. #11763117	C₂	semidirect
13. #11763105	C₂	semidirect
14. #11763095	C₂	semidirect
212. C₁₇	#3242	semidirect

Maximal subgroups

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

subgroup	index
#11763437	2
#11763421	2
#11763218	2
#11763211	2
#11763178	2
#11763160	2
#11763151	2
#11763145	2
#11763140	2
#11763117	2
#11763105	2
#11763095	2
C₁₇×#433	2
#3242	17