Subgroups — #14541519

Order 2176.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763528	C₂	1	1	yes	maximal, characteristic
3	#11763168	C₂	1	1	yes	maximal, characteristic
4	#11763167	C₂	1	1	yes	maximal, characteristic
5	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763097	C₂	1	1	yes	maximal, characteristic
7	#11763095	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763029	C₂	1	1	yes	maximal, characteristic
10	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762916	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#464	C₂	1	1	yes	maximal, characteristic
16	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#124	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	#10587630	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	#10587622	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
27	#10587574	—	2	1	no	—
28	#10587574	—	2	1	no	—
29	#10587574	C₂²	1	1	yes	characteristic
30	#10587574	C₂²	1	1	yes	characteristic
31	#10587565	—	2	1	no	—
32	#10587565	—	2	1	no	—
33	#10587564	—	2	1	no	—
34	#10587563	—	2	1	no	—
35	#10587555	—	2	1	no	—
36	#10587555	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	—	2	1	no	—
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587488	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587486	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587485	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587474	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×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×Dic₃₄	—	2	1	no	—
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×Dic₃₄	—	2	1	no	—
66	C₂×Dic₃₄	D₄	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	Q₁₆×C₁₇	—	2	2	no	—
69	SD₁₆×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	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₇	—	2	1	no	—
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	#63231	—	2	2	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
96	#63225	D₄	1	1	yes	characteristic
97	C₄×Dic₁₇	—	2	1	no	—
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	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂³×C₁₇	—	2	1	no	—
106	Q₈×C₁₇	—	2	1	no	—
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	Q₈×C₁₇	#41	1	2	yes	—
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	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	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	4	2	no	—
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	Dic₃₄	—	2	2	no	—
128	Dic₃₄	—	2	1	no	—
129	Dic₃₄	—	2	1	no	—
130	Dic₃₄	—	8	1	no	—
131	Dic₃₄	—	8	1	no	—
132	Dic₃₄	—	8	1	no	—
133	C₈×C₁₇	—	2	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3369	—	17	1	no	maximal, Sylow
138	C₂²×C₁₇	—	4	1	no	—
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	#137	1	2	yes	—
157	Q₈²	—	17	1	no	—
158	#556	—	17	1	no	—
159	#489	—	17	1	no	—
160	#489	—	17	1	no	—
161	#487	—	17	1	no	—
162	#476	—	17	1	no	—
163	#475	—	17	1	no	—
164	#475	—	17	1	no	—
165	#464	—	17	1	no	—
166	#464	D₁₇	1	1	yes	characteristic
167	#462	—	17	1	no	—
168	#446	—	17	1	no	—
169	#441	—	17	1	no	—
170	#441	—	17	1	no	—
171	C₄×SD₁₆	—	17	1	no	—
172	C₂×C₁₇	—	8	1	no	—
173	C₂×C₁₇	#548	1	1	yes	characteristic
174	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
175	C₂×C₁₇	#577	1	1	yes	characteristic
176	C₂×Q₁₆	—	17	1	no	—
177	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
178	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
179	C₄⋊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	—	34	1	no	—
184	C₄⋊Q₈	—	34	1	no	—
185	C₄⋊Q₈	—	34	1	no	—
186	#126	—	34	1	no	—
187	#126	—	34	1	no	—
188	#124	—	17	1	no	—
189	#124	C₂×D₁₇	1	1	yes	characteristic
190	C₂²⋊Q₈	—	34	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	C₂×D₁₇	1	1	yes	characteristic
198	C₄×D₄	—	17	1	no	—
199	#117	—	34	1	no	—
200	#106	—	17	1	no	—
201	#106	—	17	1	no	—
202	#106	—	17	1	no	—
203	C₄⋊C₈	—	17	1	no	—
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#103	—	17	1	no	—
209	#103	—	17	1	no	—
210	#103	—	17	1	no	—
211	#103	—	17	1	no	—
212	#103	C₂×D₁₇	1	1	yes	characteristic
213	#102	—	17	1	no	—
214	#102	—	17	1	no	—
215	#102	C₂×D₁₇	1	1	yes	characteristic
216	#97	—	17	1	no	—
217	#97	—	17	1	no	—
218	C₄×C₈	—	17	1	no	—
219	C₁₇	#3369	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	17	1	no	—
221	C₂×Q₈	—	17	1	no	—
222	C₂×Q₈	—	34	1	no	—
223	C₂×Q₈	—	34	1	no	—
224	C₂×Q₈	—	34	1	no	—
225	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
228	C₂²×C₄	—	34	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	34	2	no	—
231	SD₁₆	—	2	2	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	17	1	no	—
234	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
236	C₄⋊C₄	—	2	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	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	34	1	no	—
256	C₂²⋊C₄	—	34	1	no	—
257	C₂²⋊C₄	—	34	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	17	1	no	—
261	C₄²	—	34	1	no	—
262	C₄²	—	34	1	no	—
263	C₄²	—	34	1	no	—
264	C₄²	—	34	1	no	—
265	C₄²	C₂²×D₁₇	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	34	2	no	—
271	Q₈	—	34	1	no	—
272	Q₈	—	34	1	no	—
273	Q₈	—	136	1	no	—
274	Q₈	—	136	1	no	—
275	Q₈	—	136	1	no	—
276	Q₈	#63254	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	17	1	no	—
283	C₂×C₄	—	17	1	no	—
284	C₂×C₄	—	17	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	68	2	no	—
291	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	34	1	no	—
297	C₈	—	34	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	17	2	no	—
308	C₄	—	34	1	no	—
309	C₄	—	34	1	no	—
310	C₄	—	34	1	no	—
311	C₄	—	68	1	no	—
312	C₄	—	68	1	no	—
313	C₄	—	68	1	no	—
314	C₄	—	68	1	no	—
315	C₄	C₂×#63254	1	1	yes	characteristic
316	C₄	#10587697	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#11763465	1	1	yes	characteristic
319	C₂	#137×D₁₇	1	1	yes	characteristic
320	C₂	#11763616	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×#464	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×C₁₇	C₂³
hypercenter	#464	D₁₇
hypocenter	C₁₇	#3369
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₄	C₂	semidirect
5. #11763158	C₂	semidirect
7. #11763095	C₂	semidirect
8. #11763092	C₂	semidirect
10. #11763017	C₂	semidirect
11. #11762938	C₂	semidirect
12. #11762916	C₂	semidirect
13. #11762914	C₂	semidirect
219. C₁₇	#3369	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₄	2
#11763528	2
#11763168	2
#11763167	2
#11763158	2
#11763097	2
#11763095	2
#11763092	2
#11763029	2
#11763017	2
#11762938	2
#11762916	2
#11762914	2
SD₁₆×Dic₁₇	2
C₁₇×#464	2
#3369	17