Subgroups — #14541539

Order 2176.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763544	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763519	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763174	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763169	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763164	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763090	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763005	C₂	1	1	yes	maximal, characteristic
12	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587633	—	2	1	no	—
18	#10587632	—	2	1	no	—
19	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
20	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
21	C₂×#63219	C₂²	1	1	yes	characteristic
22	#10587582	—	2	1	no	—
23	#10587582	—	2	1	no	—
24	#10587580	—	2	1	no	—
25	#10587580	—	2	1	no	—
26	#10587579	—	2	1	no	—
27	C₄⋊C₄×D₁₇	—	2	1	no	—
28	#10587561	C₂²	1	1	yes	characteristic
29	#10587561	C₂²	1	1	yes	characteristic
30	#10587560	C₂²	1	1	yes	characteristic
31	C₄×D₆₈	C₂²	1	1	yes	characteristic
32	C₄×D₆₈	C₂²	1	1	yes	characteristic
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	C₄×D₆₈	C₂²	1	1	yes	characteristic
35	C₄×D₆₈	C₂²	1	1	yes	characteristic
36	#10587556	C₂²	1	1	yes	characteristic
37	#10587556	C₂²	1	1	yes	characteristic
38	#10587555	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	C₁₇×#97	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587492	C₂²	1	1	yes	characteristic
53	#10587489	C₂²	1	1	yes	characteristic
54	#10587489	C₂²	1	1	yes	characteristic
55	#10587488	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	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
60	C₂×D₆₈	—	2	1	no	—
61	C₂×D₆₈	—	2	1	no	—
62	C₂×D₆₈	C₂³	1	1	yes	characteristic
63	C₂×D₆₈	C₂³	1	1	yes	characteristic
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
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	—	2	1	no	—
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	C₄⋊Dic₁₇	—	2	1	no	—
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	D₁₃₆	—	2	2	no	—
103	D₁₃₆	—	2	2	no	—
104	#63219	—	2	2	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	D₆₈	—	2	1	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	—	2	1	no	—
124	D₆₈	—	2	1	no	—
125	D₆₈	—	2	1	no	—
126	D₆₈	—	4	2	no	—
127	D₆₈	—	8	1	no	—
128	D₆₈	#41	1	2	yes	—
129	C₄×D₁₇	—	4	2	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	C₄×D₁₇	—	8	1	no	—
133	C₄×D₁₇	—	8	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	8	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	—	2	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	#3370	—	17	1	no	maximal, Sylow
142	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₇	—	2	2	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₂×D₁₇	—	4	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	#559	—	17	1	no	—
166	#554	—	17	1	no	—
167	#490	—	17	1	no	—
168	#489	—	17	1	no	—
169	#488	—	17	1	no	—
170	#479	—	17	1	no	—
171	#478	—	17	1	no	—
172	#474	—	17	1	no	—
173	#463	—	17	1	no	—
174	#463	—	17	1	no	—
175	#459	—	17	1	no	—
176	#446	D₁₇	1	1	yes	characteristic, complemented
177	#442	—	17	1	no	—
178	#440	—	17	1	no	—
179	C₄×D₈	—	17	1	no	—
180	C₂×C₁₇	#548	1	1	yes	characteristic
181	C₂×C₁₇	#575	1	1	yes	characteristic
182	C₂×C₁₇	#576	1	1	yes	characteristic
183	D₁₇	—	4	2	no	—
184	D₁₇	—	8	1	no	—
185	D₁₇	—	8	1	no	—
186	C₂×SD₁₆	—	17	1	no	—
187	C₂×D₈	—	17	1	no	—
188	C₂×D₈	—	17	1	no	—
189	C₄⋊Q₈	—	17	1	no	—
190	#127	—	17	1	no	—
191	#125	—	17	1	no	—
192	#125	—	17	1	no	—
193	#124	—	17	1	no	—
194	#124	—	17	1	no	—
195	#124	—	34	1	no	—
196	#123	—	34	1	no	—
197	#123	—	34	1	no	—
198	C₂²⋊Q₈	—	34	1	no	—
199	C₂²⋊Q₈	—	34	1	no	—
200	C₂²⋊Q₈	—	34	1	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	—	17	1	no	—
208	C₄×D₄	—	34	1	no	—
209	C₂×C₄⋊C₄	—	34	1	no	—
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#106	—	17	1	no	—
213	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#102	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#97	C₂×D₁₇	1	1	yes	characteristic
227	#97	C₂×D₁₇	1	1	yes	characteristic
228	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
229	C₁₇	#3370	1	1	yes	characteristic, complemented, Sylow
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	—	34	1	no	—
232	C₂×Q₈	D₆₈	1	1	yes	characteristic
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	—	17	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	—	34	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	SD₁₆	—	34	2	no	—
244	D₈	—	34	2	no	—
245	D₈	—	34	2	no	—
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	17	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	D₆₈	1	1	yes	characteristic
261	C₄⋊C₄	D₆₈	1	1	yes	characteristic
262	C₄⋊C₄	D₆₈	1	1	yes	characteristic
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	C₂²×D₁₇	1	1	yes	characteristic
274	C₄²	C₂²×D₁₇	1	1	yes	characteristic
275	C₄²	C₂²×D₁₇	1	1	yes	characteristic
276	C₂³	—	34	1	no	—
277	C₂³	—	34	1	no	—
278	C₂³	—	34	1	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	34	1	no	—
281	Q₈	—	34	1	no	—
282	Q₈	—	136	1	no	—
283	D₄	—	17	2	no	—
284	D₄	—	34	1	no	—
285	D₄	—	34	1	no	—
286	D₄	—	34	1	no	—
287	D₄	—	34	1	no	—
288	D₄	—	34	1	no	—
289	D₄	—	68	2	no	—
290	D₄	—	136	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	—	34	1	no	—
296	C₂×C₄	—	68	2	no	—
297	C₂×C₄	—	136	1	no	—
298	C₂×C₄	—	136	1	no	—
299	C₂×C₄	—	136	1	no	—
300	C₂×C₄	—	136	1	no	—
301	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
303	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
304	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
305	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
306	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₂²	—	34	2	no	—
313	C₂²	—	68	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	—	68	1	no	—
316	C₂²	—	68	1	no	—
317	C₂²	—	68	1	no	—
318	C₂²	—	68	1	no	—
319	C₂²	—	68	1	no	—
320	C₂²	—	68	1	no	—
321	C₂²	C₂²×D₆₈	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	—	68	1	no	—
331	C₄	—	68	1	no	—
332	C₄	—	68	1	no	—
333	C₄	#10587694	1	1	yes	characteristic
334	C₄	#41×D₁₇	1	1	yes	characteristic
335	C₂	—	68	2	no	—
336	C₂	—	136	1	no	—
337	C₂	—	136	1	no	—
338	C₂	#11763471	1	1	yes	characteristic
339	C₂	#11763609	1	1	yes	characteristic
340	C₂	#11763614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#446	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763005	C₂
hypercenter	#446	D₁₇
hypocenter	C₁₇	#3370
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #11763544	C₂	semidirect
2. #11763519	C₂	semidirect
3. #11763174	C₂	semidirect
4. #11763169	C₂	semidirect
5. #11763164	C₂	semidirect
6. #11763117	C₂	semidirect
7. #11763110	C₂	semidirect
8. #11763090	C₂	semidirect
9. #11763025	C₂	semidirect
10. #11763025	C₂	semidirect
12. #11762922	C₂	semidirect
13. #11762911	C₂	semidirect
14. C₄×D₁₃₆	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
176. #446	D₁₇	semidirect
229. C₁₇	#3370	semidirect

Maximal subgroups

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

subgroup	index
#11763544	2
#11763519	2
#11763174	2
#11763169	2
#11763164	2
#11763117	2
#11763110	2
#11763090	2
#11763025	2
#11763005	2
#11762922	2
#11762911	2
C₄×D₁₃₆	2
C₁₇×#446	2
#3370	17