Subgroups — #14537877

Order 2176.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763363	C₂	1	1	yes	maximal, characteristic
2	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762933	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762871	C₂	1	1	yes	maximal, characteristic
10	#11762869	C₂	1	1	yes	maximal, characteristic
11	#11762869	C₂	1	1	yes	maximal, characteristic
12	#11762868	C₂	1	1	yes	maximal, characteristic
13	#11762864	C₂	1	1	yes	maximal, characteristic
14	#11762861	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587632	C₄	1	1	yes	characteristic
18	#10587630	C₄	1	1	yes	characteristic
19	#10587583	C₄	1	1	yes	characteristic
20	#10587582	C₄	1	1	yes	characteristic
21	#10587582	C₄	1	1	yes	characteristic
22	#10587575	C₄	1	1	yes	characteristic
23	#10587575	C₄	1	1	yes	characteristic
24	#10587574	C₄	1	1	yes	characteristic
25	#10587558	C₂²	1	1	yes	characteristic
26	#10587558	C₂²	1	1	yes	characteristic
27	C₄²×D₁₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
31	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₇×#97	C₂²	1	1	yes	characteristic
35	C₁₇×#97	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
37	#10587491	C₂²	1	1	yes	characteristic
38	#10587491	C₂²	1	1	yes	characteristic
39	#10587491	C₂²	1	1	yes	characteristic
40	#10587491	C₂²	1	1	yes	characteristic
41	#10587486	C₂²	1	1	yes	characteristic
42	#10587486	C₂²	1	1	yes	characteristic
43	#10587485	C₂²	1	1	yes	characteristic
44	#10587485	C₂²	1	1	yes	characteristic
45	#10587485	C₂²	1	1	yes	characteristic
46	#10587485	C₂²	1	1	yes	characteristic
47	#10587485	C₂²	1	1	yes	characteristic
48	#10587485	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	C₄⋊#3622	C₂²	1	1	yes	characteristic
54	C₄⋊#3622	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₃₄	C₂×C₄	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	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂³	1	1	yes	characteristic
74	C₄²×C₁₇	C₂³	1	1	yes	characteristic
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	#63227	C₂×C₄	1	1	yes	characteristic
77	#63227	C₂×C₄	1	1	yes	characteristic
78	#63227	C₂×C₄	1	1	yes	characteristic
79	#63227	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
83	#63225	C₂×C₄	1	1	yes	characteristic
84	#63225	C₂×C₄	1	1	yes	characteristic
85	#63225	C₂×C₄	1	1	yes	characteristic
86	#63225	C₂×C₄	1	1	yes	characteristic
87	#63225	C₂×C₄	1	1	yes	characteristic
88	#63225	C₂×C₄	1	1	yes	characteristic
89	#63225	C₂×C₄	1	1	yes	characteristic
90	#63225	C₂×C₄	1	1	yes	characteristic
91	#63225	C₂×C₄	1	1	yes	characteristic
92	#63225	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
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	C₂×#3622	C₂³	1	1	yes	characteristic
99	C₂×#3622	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
103	Q₈×C₁₇	—	4	1	no	—
104	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₄×D₁₇	—	2	1	no	—
119	C₄×D₁₇	—	2	1	no	—
120	C₄×D₁₇	—	2	2	no	—
121	C₄×D₁₇	—	4	1	no	—
122	Dic₃₄	—	2	2	no	—
123	Dic₃₄	—	2	2	no	—
124	Dic₃₄	—	4	1	no	—
125	C₈×C₁₇	—	2	1	no	—
126	C₈×C₁₇	—	2	1	no	—
127	C₈×C₁₇	—	2	1	no	—
128	C₈×C₁₇	—	2	1	no	—
129	#3622	—	2	1	no	—
130	#3622	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3622	—	2	1	no	—
133	#2985	—	17	1	no	maximal, Sylow
134	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×D₁₇	—	2	1	no	—
136	C₂×D₁₇	—	2	1	no	—
137	C₂×D₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₁₇	—	2	1	no	—
147	Dic₁₇	—	2	1	no	—
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	#131	1	2	yes	—
153	#533	—	17	1	no	—
154	#446	—	17	1	no	—
155	#446	—	17	1	no	—
156	#446	—	17	1	no	—
157	#446	—	17	1	no	—
158	#446	—	17	1	no	—
159	#446	D₁₇	1	1	yes	characteristic, complemented
160	C₈×Q₈	—	17	1	no	—
161	C₈×Q₈	—	17	1	no	—
162	#433	—	17	1	no	—
163	#433	—	17	1	no	—
164	#433	—	17	1	no	—
165	#433	—	17	1	no	—
166	#432	—	17	1	no	—
167	#431	—	17	1	no	—
168	C₂×C₁₇	#519	1	1	yes	characteristic
169	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
170	C₂×C₁₇	#568	1	1	yes	characteristic
171	D₁₇	—	4	1	no	—
172	C₄⋊Q₈	—	17	1	no	—
173	C₄⋊Q₈	—	17	1	no	—
174	#125	—	17	1	no	—
175	#125	—	17	1	no	—
176	C₂²⋊Q₈	—	17	1	no	—
177	C₂²⋊Q₈	—	17	1	no	—
178	C₂²⋊Q₈	—	17	1	no	—
179	C₂²⋊Q₈	—	17	1	no	—
180	C₄×Q₈	—	17	1	no	—
181	C₄×Q₈	—	17	1	no	—
182	C₄×Q₈	—	17	1	no	—
183	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
184	#117	—	17	1	no	—
185	#117	—	17	1	no	—
186	C₂×C₄²	—	17	1	no	—
187	C₄⋊C₈	—	17	1	no	—
188	C₄⋊C₈	—	17	1	no	—
189	C₄⋊C₈	—	17	1	no	—
190	C₄⋊C₈	—	17	1	no	—
191	C₄⋊C₈	—	17	1	no	—
192	C₄⋊C₈	—	17	1	no	—
193	C₄⋊C₈	—	17	1	no	—
194	C₄⋊C₈	—	17	1	no	—
195	C₄⋊C₈	—	17	1	no	—
196	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
197	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
198	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
199	C₂²⋊C₈	—	17	1	no	—
200	C₂²⋊C₈	—	17	1	no	—
201	C₂²⋊C₈	—	17	1	no	—
202	C₂²⋊C₈	—	17	1	no	—
203	#97	—	17	1	no	—
204	#97	—	17	1	no	—
205	#97	—	17	1	no	—
206	#97	—	17	1	no	—
207	#97	C₂×D₁₇	1	1	yes	characteristic
208	#97	C₂×D₁₇	1	1	yes	characteristic
209	C₄×C₈	—	17	1	no	—
210	C₄×C₈	—	17	1	no	—
211	C₄×C₈	—	17	1	no	—
212	C₄×C₈	—	17	1	no	—
213	C₄×C₈	—	17	1	no	—
214	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
215	C₁₇	#2985	1	1	yes	characteristic, complemented, Sylow
216	C₂×Q₈	—	17	1	no	—
217	C₂×Q₈	—	17	1	no	—
218	C₂×Q₈	—	17	1	no	—
219	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
220	C₂²×C₄	—	17	1	no	—
221	C₂²×C₄	—	17	1	no	—
222	C₂²×C₄	—	17	1	no	—
223	C₂×C₈	—	17	1	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	—	17	1	no	—
227	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
237	C₄⋊C₄	—	17	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
245	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
247	C₂²⋊C₄	—	17	1	no	—
248	C₂²⋊C₄	—	17	1	no	—
249	C₂²⋊C₄	—	17	1	no	—
250	C₂²⋊C₄	—	17	1	no	—
251	C₄²	—	17	1	no	—
252	C₄²	—	17	1	no	—
253	C₄²	—	17	1	no	—
254	C₄²	—	17	1	no	—
255	C₄²	—	17	1	no	—
256	C₄²	C₂²×D₁₇	1	1	yes	characteristic
257	C₄²	C₂²×D₁₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₇	1	1	yes	characteristic
259	C₂³	—	17	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	34	2	no	—
262	Q₈	—	34	2	no	—
263	Q₈	—	68	1	no	—
264	C₂×C₄	—	17	1	no	—
265	C₂×C₄	—	17	1	no	—
266	C₂×C₄	—	17	1	no	—
267	C₂×C₄	—	17	1	no	—
268	C₂×C₄	—	17	1	no	—
269	C₂×C₄	—	17	1	no	—
270	C₂×C₄	—	17	1	no	—
271	C₂×C₄	—	34	2	no	—
272	C₂×C₄	—	34	1	no	—
273	C₂×C₄	—	34	1	no	—
274	C₂×C₄	—	68	1	no	—
275	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	34	1	no	—
287	C₈	—	34	1	no	—
288	C₈	—	34	1	no	—
289	C₈	—	34	1	no	—
290	C₂²	—	34	1	no	—
291	C₂²	—	34	1	no	—
292	C₂²	—	34	1	no	—
293	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	17	2	no	—
301	C₄	—	34	1	no	—
302	C₄	—	34	1	no	—
303	C₄	—	34	1	no	—
304	C₄	—	34	1	no	—
305	C₄	—	34	1	no	—
306	C₄	—	34	1	no	—
307	C₄	#10587694	1	1	yes	characteristic
308	C₄	#10587697	1	1	yes	characteristic
309	C₂	—	68	1	no	—
310	C₂	#11763285	1	1	yes	characteristic
311	C₂	#131×D₁₇	1	1	yes	characteristic
312	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2985
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11762940	C₂	semidirect
3. #11762938	C₂	semidirect
4. #11762938	C₂	semidirect
5. #11762937	C₂	semidirect
6. #11762937	C₂	semidirect
7. C₈×Dic₃₄	C₂	semidirect
8. #11762933	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
159. #446	D₁₇	semidirect
215. C₁₇	#2985	semidirect

Maximal subgroups

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

subgroup	index
#11763363	2
#11762940	2
#11762938	2
#11762937	2
C₈×Dic₃₄	2
#11762933	2
#11762871	2
#11762869	2
#11762868	2
#11762864	2
#11762861	2
C₁₇×#446	2
#2985	17