Subgroups — #14540840

Order 2176.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763500	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₈	C₂	1	1	yes	maximal, characteristic
3	#11763178	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763066	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763049	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763019	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762963	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762963	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762882	C₂	1	1	yes	maximal, characteristic
13	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	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	C₂×#63231	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	C₂×#63218	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
27	#10587583	—	2	1	no	—
28	#10587583	—	2	1	no	—
29	#10587583	C₂²	1	1	yes	characteristic
30	#10587583	C₂²	1	1	yes	characteristic
31	#10587580	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	#10587575	—	2	1	no	—
35	#10587562	—	2	1	no	—
36	C₄×D₆₈	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	#10587558	—	2	1	no	—
39	#10587556	C₂²	1	1	yes	characteristic
40	#10587555	—	2	1	no	—
41	#10587555	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#97	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	#10587487	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	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	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	Q₈×D₁₇	—	2	2	no	—
62	Q₈×D₁₇	—	2	2	no	—
63	Q₈×D₁₇	—	4	1	no	—
64	Q₈×D₁₇	—	4	1	no	—
65	C₂×D₆₈	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	—	2	1	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
75	Q₁₆×C₁₇	—	2	2	no	—
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄²×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	#63231	—	4	1	no	—
86	#63231	—	4	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	D₄	1	1	yes	characteristic
93	#63227	D₄	1	1	yes	characteristic
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	—	2	1	no	—
98	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	#63225	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	#63218	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	#41	1	2	yes	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	D₆₈	—	2	2	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	4	2	no	—
137	C₄×D₁₇	—	4	2	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	4	2	no	—
143	Dic₃₄	—	4	2	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3312	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	#137	1	2	yes	—
156	C₄×C₁₇	—	2	2	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	#552	—	17	1	no	—
171	D₄×Q₈	—	17	1	no	—
172	#491	—	17	1	no	—
173	#485	—	17	1	no	—
174	#483	—	17	1	no	—
175	#479	—	17	1	no	—
176	#470	—	17	1	no	—
177	#467	—	17	1	no	—
178	#462	—	17	1	no	—
179	#451	—	17	1	no	—
180	#451	—	17	1	no	—
181	#441	D₁₇	1	1	yes	characteristic, complemented
182	#435	—	17	1	no	—
183	#419	—	17	1	no	—
184	C₂×#103	—	17	1	no	—
185	C₂×C₁₇	#546	1	1	yes	characteristic
186	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
187	C₂×C₁₇	#578	1	1	yes	characteristic
188	D₁₇	—	2	2	no	—
189	D₁₇	—	4	1	no	—
190	C₂²×Q₈	—	17	1	no	—
191	C₂²×Q₈	—	17	1	no	—
192	C₂×Q₁₆	—	17	1	no	—
193	C₂×Q₁₆	—	17	1	no	—
194	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
195	C₂×M₁₆	—	17	1	no	—
196	C₂²×C₈	—	17	1	no	—
197	C₄⋊Q₈	—	17	1	no	—
198	C₄⋊Q₈	—	34	1	no	—
199	#126	—	34	1	no	—
200	#125	—	17	1	no	—
201	#125	—	34	1	no	—
202	#123	—	34	1	no	—
203	C₂²⋊Q₈	—	17	1	no	—
204	C₂²⋊Q₈	—	17	1	no	—
205	C₂²⋊Q₈	—	17	1	no	—
206	C₂²⋊Q₈	—	17	1	no	—
207	C₂²⋊Q₈	—	34	1	no	—
208	C₂²⋊Q₈	—	34	1	no	—
209	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
211	C₄×D₄	—	17	1	no	—
212	C₄×D₄	—	34	1	no	—
213	#117	—	17	1	no	—
214	#117	—	34	1	no	—
215	C₂×C₄⋊C₄	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	C₂×D₁₇	1	1	yes	characteristic
219	C₄⋊C₈	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	C₂×D₁₇	1	1	yes	characteristic
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	C₂²⋊C₈	—	17	1	no	—
231	C₂²⋊C₈	—	17	1	no	—
232	#97	C₂×D₁₇	1	1	yes	characteristic
233	C₁₇	#3312	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₈	—	17	1	no	—
235	C₂×Q₈	—	17	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×Q₈	—	34	1	no	—
238	C₂×Q₈	—	34	2	no	—
239	C₂×Q₈	—	34	2	no	—
240	C₂×Q₈	—	68	1	no	—
241	C₂×Q₈	—	68	1	no	—
242	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₂×D₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	68	1	no	—
252	Q₁₆	—	68	1	no	—
253	M₁₆	—	68	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	68	1	no	—
257	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	17	1	no	—
276	C₂²⋊C₄	—	17	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	C₂²×D₁₇	1	1	yes	characteristic
286	C₂³	—	17	1	no	—
287	C₂³	—	17	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	34	1	no	—
292	Q₈	—	34	1	no	—
293	Q₈	—	34	1	no	—
294	Q₈	—	34	1	no	—
295	Q₈	—	68	2	no	—
296	Q₈	—	68	2	no	—
297	Q₈	#63252	1	2	yes	—
298	D₄	—	34	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	—	68	2	no	—
315	C₂×C₄	—	68	2	no	—
316	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
317	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	34	1	no	—
322	C₈	—	34	1	no	—
323	C₂²	—	17	2	no	—
324	C₂²	—	34	1	no	—
325	C₂²	—	34	1	no	—
326	C₂²	—	34	1	no	—
327	C₂²	—	34	1	no	—
328	C₂²	—	34	1	no	—
329	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	34	1	no	—
338	C₄	—	68	1	no	—
339	C₄	—	68	1	no	—
340	C₄	—	68	1	no	—
341	C₄	—	68	1	no	—
342	C₄	C₂×#63252	1	1	yes	characteristic
343	C₄	#10587696	1	1	yes	characteristic
344	C₂	—	34	2	no	—
345	C₂	—	68	1	no	—
346	C₂	#11763455	1	1	yes	characteristic
347	C₂	#137×D₁₇	1	1	yes	characteristic
348	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×#441	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	D₄
hypercenter	#441	D₁₇
hypocenter	C₁₇	#3312
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. #11763178	C₂	semidirect
4. #11763153	C₂	semidirect
5. #11763141	C₂	semidirect
6. #11763116	C₂	semidirect
7. #11763066	C₂	semidirect
8. #11763049	C₂	semidirect
9. #11763019	C₂	semidirect
10. #11762963	C₂	semidirect
11. #11762963	C₂	semidirect
13. #11762816	C₂	semidirect
14. #103×D₁₇	C₂	semidirect
15. C₁₇×#441	C₂	semidirect
181. #441	D₁₇	semidirect
233. C₁₇	#3312	semidirect

Maximal subgroups

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

subgroup	index
#11763500	2
Q₈×D₆₈	2
#11763178	2
#11763153	2
#11763141	2
#11763116	2
#11763066	2
#11763049	2
#11763019	2
#11762963	2
#11762882	2
#11762816	2
#103×D₁₇	2
C₁₇×#441	2
#3312	17