Subgroups — #14541680

Order 2176.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763515	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763474	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763112	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763098	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763098	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic
13	#11762920	C₂	1	1	yes	maximal, characteristic
14	#11762891	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₇×#125	C₂²	1	1	yes	characteristic
19	#10587632	—	2	1	no	—
20	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
21	#10587583	—	2	1	no	—
22	#10587582	—	2	1	no	—
23	#10587582	—	2	1	no	—
24	#10587582	—	2	1	no	—
25	#10587582	—	2	1	no	—
26	#10587579	C₂²	1	1	yes	characteristic
27	#10587579	C₂²	1	1	yes	characteristic
28	#10587579	C₂²	1	1	yes	characteristic
29	#10587579	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₇	—	2	1	no	—
31	#10587576	C₂²	1	1	yes	characteristic
32	#10587574	C₂²	1	1	yes	characteristic
33	#10587574	C₂²	1	1	yes	characteristic
34	#10587574	C₂²	1	1	yes	characteristic
35	C₄×D₆₈	C₂²	1	1	yes	characteristic
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×#107	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#97	C₂²	1	1	yes	characteristic
43	#10587492	C₂²	1	1	yes	characteristic
44	#10587492	C₂²	1	1	yes	characteristic
45	#10587488	C₂²	1	1	yes	characteristic
46	#10587486	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	#10587480	C₂²	1	1	yes	characteristic
51	#10587480	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	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	—	2	1	no	—
60	Q₈×D₁₇	—	4	4	no	—
61	C₂×D₆₈	C₂³	1	1	yes	characteristic
62	C₂×D₆₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
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×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×Dic₃₄	—	2	1	no	—
72	C₂×Dic₃₄	D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	D₁₃₆	Q₈	1	4	yes	complemented
104	C₂²×D₁₇	—	2	1	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	Q₈×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	D₆₈	—	2	1	no	—
121	D₆₈	—	2	1	no	—
122	D₆₈	C₂×Q₈	1	2	yes	—
123	D₆₈	C₂×Q₈	1	2	yes	complemented
124	C₄×D₁₇	—	4	2	no	—
125	C₄×D₁₇	—	4	2	no	—
126	C₄×D₁₇	—	4	2	no	—
127	C₄×D₁₇	—	4	2	no	—
128	C₄×D₁₇	—	8	1	no	—
129	C₄×D₁₇	—	8	1	no	—
130	Dic₃₄	—	2	2	no	—
131	Dic₃₄	—	4	2	no	—
132	Dic₃₄	—	4	2	no	—
133	Dic₃₄	—	8	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	C₂×Q₈	1	2	yes	complemented
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3384	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₇	—	2	2	no	—
141	C₂×D₁₇	—	2	2	no	—
142	C₂×D₁₇	—	4	1	no	—
143	C₂×D₁₇	—	4	1	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	2	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
160	#554	—	17	1	no	—
161	D₄×Q₈	—	17	1	no	—
162	#501	—	17	1	no	—
163	#501	D₁₇	1	1	yes	characteristic, complemented
164	#498	—	17	1	no	—
165	#478	—	17	1	no	—
166	#478	—	17	1	no	—
167	#476	—	17	1	no	—
168	#476	—	17	1	no	—
169	#476	—	17	1	no	—
170	#474	—	17	1	no	—
171	#446	—	17	1	no	—
172	#442	—	17	1	no	—
173	#442	—	17	1	no	—
174	C₄×D₈	—	17	1	no	—
175	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
176	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₁₇	#577	1	1	yes	characteristic
178	D₁₇	—	4	2	no	—
179	D₁₇	—	4	2	no	—
180	C₂²×Q₈	—	34	1	no	—
181	C₂×D₈	—	17	1	no	—
182	C₄⋊Q₈	—	17	1	no	—
183	C₄⋊Q₈	—	17	1	no	—
184	C₄⋊Q₈	—	17	1	no	—
185	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
186	#125	—	17	1	no	—
187	#125	C₂×D₁₇	1	1	yes	characteristic
188	C₂²⋊Q₈	—	34	1	no	—
189	C₂²⋊Q₈	—	34	1	no	—
190	C₂²⋊Q₈	—	34	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	—	34	1	no	—
194	C₄×Q₈	—	17	1	no	—
195	C₄×D₄	—	17	1	no	—
196	C₄×D₄	—	17	1	no	—
197	C₄×D₄	—	17	1	no	—
198	C₄×D₄	—	17	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	17	1	no	—
201	C₂×C₄⋊C₄	—	34	1	no	—
202	#107	—	17	1	no	—
203	#107	C₂×D₁₇	1	1	yes	characteristic
204	#107	C₂×D₁₇	1	1	yes	characteristic
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	C₂×D₁₇	1	1	yes	characteristic
210	#106	C₂×D₁₇	1	1	yes	characteristic
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	#102	—	17	1	no	—
215	#102	—	17	1	no	—
216	#102	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	—	17	1	no	—
219	#102	—	17	1	no	—
220	#97	—	17	1	no	—
221	#97	C₂×D₁₇	1	1	yes	characteristic
222	C₄×C₈	—	17	1	no	—
223	C₁₇	#3384	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	17	1	no	—
226	C₂×Q₈	—	34	1	no	—
227	C₂×Q₈	—	34	1	no	—
228	C₂×Q₈	—	34	1	no	—
229	C₂×Q₈	—	68	4	no	—
230	C₂×D₄	—	17	1	no	—
231	C₂×D₄	—	17	1	no	—
232	C₂²×C₄	—	34	1	no	—
233	C₂²×C₄	—	34	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	D₈	—	17	4	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	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	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₄²	—	17	1	no	—
267	C₄²	—	17	1	no	—
268	C₄²	C₂²×D₁₇	1	1	yes	characteristic
269	C₂³	—	34	1	no	—
270	C₂³	—	34	1	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	34	2	no	—
273	Q₈	—	68	2	no	—
274	Q₈	—	68	2	no	—
275	Q₈	—	136	1	no	—
276	D₄	—	17	2	no	—
277	D₄	—	17	2	no	—
278	D₄	—	34	1	no	—
279	D₄	—	34	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	17	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	17	1	no	—
287	C₂×C₄	—	17	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	68	2	no	—
291	C₂×C₄	—	68	2	no	—
292	C₂×C₄	—	68	2	no	—
293	C₂×C₄	—	68	2	no	—
294	C₂×C₄	—	136	1	no	—
295	C₂×C₄	—	136	1	no	—
296	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₈	Q₈×D₁₇	1	2	yes	complemented
303	C₂²	—	34	2	no	—
304	C₂²	—	34	2	no	—
305	C₂²	—	68	1	no	—
306	C₂²	—	68	1	no	—
307	C₂²	—	68	1	no	—
308	C₂²	—	68	1	no	—
309	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	17	2	no	—
317	C₄	—	34	1	no	—
318	C₄	—	34	1	no	—
319	C₄	—	34	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
323	C₄	#10587694	1	1	yes	characteristic
324	C₂	—	68	2	no	—
325	C₂	—	68	2	no	—
326	C₂	#11763474	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
328	C₂	#11763618	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×#501	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₁₃₆	C₂²
hypercenter	#501	D₁₇
hypocenter	C₁₇	#3384
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #11763515	C₂	semidirect
2. #11763474	C₂	semidirect
3. #11763217	C₂	semidirect
4. #11763206	C₂	semidirect
5. #11763112	C₂	semidirect
6. #11763106	C₂	semidirect
7. #11763098	C₂	semidirect
8. #11763098	C₂	semidirect
9. #11763096	C₂	semidirect
10. #11763088	C₂	semidirect
11. #11762937	C₂	semidirect
15. C₁₇×#501	C₂	semidirect
103. D₁₃₆	Q₈	semidirect
123. D₆₈	C₂×Q₈	semidirect
135. C₈×C₁₇	C₂×Q₈	semidirect
163. #501	D₁₇	semidirect
223. C₁₇	#3384	semidirect
302. C₈	Q₈×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
#11763515	2
#11763474	2
#11763217	2
#11763206	2
#11763112	2
#11763106	2
#11763098	2
#11763096	2
#11763088	2
#11762937	2
#11762922	2
#11762920	2
#11762891	2
C₁₇×#501	2
#3384	17