Subgroups — #14539082

Order 2176.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763420	C₂	1	1	yes	maximal, characteristic
2	#11763285	C₂	1	1	yes	maximal, characteristic
3	#11763149	C₂	1	1	yes	maximal, characteristic
4	#11763148	C₂	1	1	yes	maximal, characteristic
5	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763016	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762963	C₂	1	1	yes	maximal, characteristic
9	#11762963	C₂	1	1	yes	maximal, characteristic
10	#11762916	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762913	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762901	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762869	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	#10587632	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₂×#63231	C₂²	1	1	yes	characteristic
27	C₂×#63231	C₂²	1	1	yes	characteristic
28	#10587584	—	2	1	no	—
29	#10587584	—	2	1	no	—
30	#10587578	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	#10587575	—	2	1	no	—
33	#10587575	—	2	1	no	—
34	#10587573	—	2	1	no	—
35	#10587573	C₂²	1	1	yes	characteristic
36	#10587558	—	2	1	no	—
37	#10587558	C₂²	1	1	yes	characteristic
38	#10587556	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	—	2	1	no	—
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	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	#10587491	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587486	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₇	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	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	Q₈×D₁₇	—	2	2	no	—
63	Q₈×D₁₇	—	2	2	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	Q₁₆×C₁₇	—	2	2	no	—
69	Q₁₆×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	#63231	—	2	2	no	—
79	#63231	—	2	2	no	—
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	D₄	1	1	yes	characteristic
84	#63227	D₄	1	1	yes	characteristic
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
98	#63225	D₄	1	1	yes	characteristic
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	#41	1	2	yes	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₇	—	2	4	no	—
130	C₄×D₁₇	—	2	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	Dic₃₄	—	2	4	no	—
133	Dic₃₄	—	2	1	no	—
134	Dic₃₄	#41	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	#3127	—	17	1	no	maximal, Sylow
140	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₁₇	—	2	1	no	—
142	C₂×D₁₇	—	2	1	no	—
143	C₂×D₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	2	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
152	Dic₁₇	—	2	1	no	—
153	Dic₁₇	—	2	2	no	—
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	#541	—	17	1	no	—
160	#519	—	17	1	no	—
161	#485	—	17	1	no	—
162	#485	—	17	1	no	—
163	#479	—	17	1	no	—
164	#479	—	17	1	no	—
165	#462	—	17	1	no	—
166	#462	D₁₇	1	1	yes	characteristic, complemented
167	#451	—	17	1	no	—
168	#451	—	17	1	no	—
169	#441	—	17	1	no	—
170	#441	—	17	1	no	—
171	C₄×Q₁₆	—	17	1	no	—
172	C₄×Q₁₆	—	17	1	no	—
173	#433	—	17	1	no	—
174	C₂×C₁₇	#525	1	1	yes	characteristic
175	C₂×C₁₇	#578	1	1	yes	characteristic
176	C₂×C₁₇	#578	1	1	yes	characteristic
177	D₁₇	—	4	1	no	—
178	C₂²×Q₈	—	17	1	no	—
179	C₂×Q₁₆	—	17	1	no	—
180	C₂×Q₁₆	—	17	1	no	—
181	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
182	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
183	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
184	#126	—	34	1	no	—
185	#126	—	34	1	no	—
186	#125	—	17	1	no	—
187	#125	—	34	1	no	—
188	#125	—	34	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₄×Q₈	—	17	1	no	—
192	C₄×Q₈	—	17	1	no	—
193	C₄×Q₈	—	17	1	no	—
194	C₄×Q₈	—	17	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	34	1	no	—
197	C₄×Q₈	—	34	1	no	—
198	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
199	#117	—	17	1	no	—
200	#117	—	17	1	no	—
201	#117	—	34	1	no	—
202	#117	—	34	1	no	—
203	#107	—	17	1	no	—
204	#107	—	17	1	no	—
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	C₄⋊C₈	—	17	1	no	—
208	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
209	#103	—	17	1	no	—
210	#103	—	17	1	no	—
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	C₂×D₁₇	1	1	yes	characteristic
216	#103	C₂×D₁₇	1	1	yes	characteristic
217	C₂²⋊C₈	—	17	1	no	—
218	C₂²⋊C₈	—	17	1	no	—
219	#97	—	17	1	no	—
220	C₄×C₈	—	17	1	no	—
221	C₁₇	#3127	1	1	yes	characteristic, complemented, Sylow
222	C₂×Q₈	—	17	1	no	—
223	C₂×Q₈	—	34	1	no	—
224	C₂×Q₈	—	34	2	no	—
225	C₂×Q₈	—	34	2	no	—
226	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
229	C₂²×C₄	—	17	1	no	—
230	C₂²×C₄	—	34	1	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	34	2	no	—
234	Q₁₆	—	34	2	no	—
235	C₂×C₈	—	17	1	no	—
236	C₂×C₈	—	17	1	no	—
237	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	34	1	no	—
248	C₄⋊C₄	—	34	1	no	—
249	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	17	1	no	—
258	C₂²⋊C₄	—	17	1	no	—
259	C₂²⋊C₄	—	34	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	17	1	no	—
264	C₄²	—	34	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	—	34	1	no	—
268	C₄²	—	34	1	no	—
269	C₄²	—	34	1	no	—
270	C₄²	C₂²×D₁₇	1	1	yes	characteristic
271	C₂³	—	17	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	17	2	no	—
278	Q₈	—	34	4	no	—
279	Q₈	—	34	1	no	—
280	Q₈	#63254	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	17	1	no	—
287	C₂×C₄	—	17	1	no	—
288	C₂×C₄	—	34	4	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	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	D₄×D₁₇	1	1	yes	characteristic
296	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₂²	—	34	1	no	—
303	C₂²	—	34	1	no	—
304	C₂²	—	34	1	no	—
305	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	34	1	no	—
313	C₄	—	34	2	no	—
314	C₄	—	34	1	no	—
315	C₄	—	34	1	no	—
316	C₄	—	68	1	no	—
317	C₄	—	68	1	no	—
318	C₄	—	68	1	no	—
319	C₄	C₂×#63254	1	1	yes	characteristic
320	C₄	#41×D₁₇	1	1	yes	characteristic
321	C₂	—	68	1	no	—
322	C₂	#11763319	1	1	yes	characteristic
323	C₂	#11763622	1	1	yes	characteristic
324	C₂	#11763625	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×#462	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	#462	D₁₇
hypocenter	C₁₇	#3127
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
5. #11763117	C₂	semidirect
6. #11763116	C₂	semidirect
7. #11763016	C₂	semidirect
10. #11762916	C₂	semidirect
11. #11762913	C₂	semidirect
12. Q₁₆×Dic₁₇	C₂	semidirect
13. #11762901	C₂	semidirect
15. C₁₇×#462	C₂	semidirect
166. #462	D₁₇	semidirect
221. C₁₇	#3127	semidirect

Maximal subgroups

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

subgroup	index
#11763420	2
#11763285	2
#11763149	2
#11763148	2
#11763117	2
#11763116	2
#11763016	2
#11762963	2
#11762916	2
#11762913	2
Q₁₆×Dic₁₇	2
#11762901	2
#11762869	2
C₁₇×#462	2
#3127	17