Subgroups — #14541686

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	D₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763219	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763111	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763100	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763098	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762919	C₂	1	1	yes	maximal, characteristic
13	#11762918	C₂	1	1	yes	maximal, characteristic
14	#11762889	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₃₄	—	2	1	no	—
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
21	C₂×#63228	C₂²	1	1	yes	characteristic
22	#10587610	—	2	1	no	—
23	#10587583	—	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	C₄⋊C₄×D₁₇	—	2	1	no	—
29	#10587576	C₂²	1	1	yes	characteristic
30	#10587574	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587574	C₂²	1	1	yes	characteristic
33	#10587564	—	2	1	no	—
34	#10587564	—	2	1	no	—
35	C₄×D₆₈	C₂²	1	1	yes	characteristic
36	#10587555	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×#106	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587492	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587488	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587486	C₂²	1	1	yes	characteristic
49	#10587485	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587479	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×Dic₁₇	—	2	1	no	—
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×D₆₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×Dic₃₄	—	4	4	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	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	#63228	Q₈	1	4	yes	complemented
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	C₄⋊Dic₁₇	—	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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	D₄	1	1	yes	characteristic
99	#63225	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂³×C₁₇	—	2	1	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	D₄×C₁₇	—	2	1	no	—
107	D₄×C₁₇	C₂×Q₈	1	2	yes	—
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	—	4	2	no	—
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	4	2	no	—
119	C₂×Dic₁₇	—	4	2	no	—
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	D₆₈	—	2	1	no	—
125	D₆₈	C₂×Q₈	1	2	yes	complemented
126	C₄×D₁₇	—	4	2	no	—
127	C₄×D₁₇	—	8	1	no	—
128	C₄×D₁₇	—	8	1	no	—
129	Dic₃₄	—	2	2	no	—
130	Dic₃₄	—	4	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	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	C₂×Q₈	1	2	yes	complemented
138	#3384	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	2	2	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
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	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	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₇	—	2	1	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	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	—	17	1	no	—
164	#498	—	17	1	no	—
165	#478	—	17	1	no	—
166	#478	D₁₇	1	1	yes	characteristic, complemented
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	—	4	2	no	—
176	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
177	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×C₁₇	#577	1	1	yes	characteristic
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	—	17	1	no	—
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	C₂×D₁₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	34	1	no	—
202	#107	—	17	1	no	—
203	#107	—	17	1	no	—
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	—	17	1	no	—
210	#106	C₂×D₁₇	1	1	yes	characteristic
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
214	#102	—	17	1	no	—
215	#102	—	17	1	no	—
216	#102	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	C₂×D₁₇	1	1	yes	characteristic
219	#102	C₂×D₁₇	1	1	yes	characteristic
220	#97	—	17	1	no	—
221	#97	—	17	1	no	—
222	C₄×C₈	—	17	1	no	—
223	C₁₇	#3384	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	17	1	no	—
225	C₂×Q₈	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	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	—	17	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	—	34	1	no	—
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	—	2	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	—	2	1	no	—
270	C₂³	—	34	1	no	—
271	Q₈	—	34	2	no	—
272	Q₈	—	68	2	no	—
273	Q₈	—	68	2	no	—
274	Q₈	—	68	2	no	—
275	Q₈	—	136	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	17	2	no	—
278	D₄	—	34	1	no	—
279	D₄	Q₈×D₁₇	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	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	—	34	1	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	—	2	1	no	—
301	C₈	—	17	2	no	—
302	C₈	—	34	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	34	2	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	—	17	2	no	—
316	C₄	—	34	1	no	—
317	C₄	—	34	1	no	—
318	C₄	—	34	1	no	—
319	C₄	—	68	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
323	C₄	#10587697	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	68	2	no	—
326	C₂	#11763472	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
328	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×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63228	C₂²
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3384
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763515	C₂	semidirect
2. D₄×Dic₃₄	C₂	semidirect
3. #11763219	C₂	semidirect
4. #11763217	C₂	semidirect
5. #11763206	C₂	semidirect
6. #11763111	C₂	semidirect
7. #11763100	C₂	semidirect
8. #11763098	C₂	semidirect
9. #11763097	C₂	semidirect
10. #11763088	C₂	semidirect
11. #11762938	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
84. #63228	Q₈	semidirect
125. D₆₈	C₂×Q₈	semidirect
137. #3622	C₂×Q₈	semidirect
166. #478	D₁₇	semidirect
223. C₁₇	#3384	semidirect

Maximal subgroups

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

subgroup	index
#11763515	2
D₄×Dic₃₄	2
#11763219	2
#11763217	2
#11763206	2
#11763111	2
#11763100	2
#11763098	2
#11763097	2
#11763088	2
#11762938	2
#11762919	2
#11762918	2
#11762889	2
C₁₇×#478	2
#3384	17