Subgroups — #14541838

Order 2176.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763542	C₂	1	1	yes	maximal, characteristic
2	#11763537	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763203	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763200	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763180	C₂	1	1	yes	maximal, characteristic
6	#11763031	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763030	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763014	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763013	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763008	C₂	1	1	yes	maximal, characteristic
12	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762908	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762903	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587630	C₂²	1	1	yes	characteristic
20	#10587630	C₂²	1	1	yes	characteristic
21	C₂×#63231	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587626	C₂²	1	1	yes	characteristic
24	#10587622	C₂²	1	1	yes	characteristic
25	C₂×#63229	C₂²	1	1	yes	characteristic
26	C₂×#63229	C₂²	1	1	yes	characteristic
27	#10587615	—	2	1	no	—
28	C₄×C₂²⋊D₁₇	—	2	1	no	—
29	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
30	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
31	C₂×#63219	C₂²	1	1	yes	characteristic
32	C₂×#63219	C₂²	1	1	yes	characteristic
33	#10587579	C₂²	1	1	yes	characteristic
34	#10587575	—	2	1	no	—
35	#10587575	—	2	1	no	—
36	#10587573	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587573	C₂²	1	1	yes	characteristic
39	#10587571	—	2	1	no	—
40	#10587569	—	2	1	no	—
41	#10587561	C₂²	1	1	yes	characteristic
42	#10587556	—	2	1	no	—
43	#10587555	C₂²	1	1	yes	characteristic
44	C₄×Dic₃₄	—	2	1	no	—
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	C₁₇×#107	C₂²	1	1	yes	characteristic
47	C₁₇×#103	C₂²	1	1	yes	characteristic
48	C₁₇×#102	C₂²	1	1	yes	characteristic
49	C₁₇×#97	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587480	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	D₄	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	SD₁₆×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	#63231	—	4	1	no	—
82	#63230	—	4	1	no	—
83	#63229	—	4	1	no	—
84	#63229	—	4	1	no	—
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	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
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	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	Dic₆₈	—	2	2	no	—
108	Dic₆₈	—	4	1	no	—
109	#63219	—	2	2	no	—
110	#63219	—	4	1	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	2	1	no	—
116	D₄×C₁₇	—	2	1	no	—
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	—	8	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	C₂²⋊D₁₇	—	4	2	no	—
125	C₂²⋊D₁₇	—	8	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
133	D₆₈	—	2	1	no	—
134	D₆₈	—	2	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	Dic₃₄	—	2	2	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	#41	1	2	yes	—
145	#3622	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3400	—	17	1	no	maximal, Sylow
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	—	4	1	no	—
150	C₂²×C₁₇	—	4	1	no	—
151	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	#137	1	2	yes	—
169	#559	—	17	1	no	—
170	#557	—	17	1	no	—
171	#497	—	17	1	no	—
172	#497	—	17	1	no	—
173	#492	—	17	1	no	—
174	#464	—	17	1	no	—
175	#464	—	17	1	no	—
176	#462	—	17	1	no	—
177	#461	—	17	1	no	—
178	#461	—	17	1	no	—
179	#460	—	17	1	no	—
180	#446	—	17	1	no	—
181	#440	—	17	1	no	—
182	#440	D₁₇	1	1	yes	characteristic, complemented
183	C₄×Q₁₆	—	17	1	no	—
184	C₂×C₁₇	—	8	1	no	—
185	C₂×C₁₇	#550	1	1	yes	characteristic
186	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
187	C₂×C₁₇	#577	1	1	yes	characteristic
188	D₁₇	—	8	1	no	—
189	C₂×Q₁₆	—	17	1	no	—
190	C₂×Q₁₆	—	17	1	no	—
191	C₂×Q₁₆	—	17	1	no	—
192	C₂×SD₁₆	—	17	1	no	—
193	C₂×SD₁₆	—	17	1	no	—
194	C₂×SD₁₆	—	17	1	no	—
195	C₂×SD₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	17	1	no	—
197	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
198	C₄⋊Q₈	—	17	1	no	—
199	C₄⋊Q₈	—	17	1	no	—
200	C₄⋊Q₈	—	17	1	no	—
201	#127	—	17	1	no	—
202	#125	—	34	1	no	—
203	#125	—	34	1	no	—
204	#125	—	34	1	no	—
205	#124	—	17	1	no	—
206	#124	—	17	1	no	—
207	#124	—	34	1	no	—
208	#123	—	34	1	no	—
209	#123	—	34	1	no	—
210	C₄×Q₈	—	17	1	no	—
211	C₄×Q₈	—	17	1	no	—
212	C₄×Q₈	—	17	1	no	—
213	C₄×Q₈	—	17	1	no	—
214	C₄×Q₈	—	34	1	no	—
215	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
216	C₄×D₄	—	17	1	no	—
217	C₄×D₄	—	34	1	no	—
218	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
219	#107	C₂×D₁₇	1	1	yes	characteristic
220	C₄⋊C₈	—	17	1	no	—
221	C₄⋊C₈	—	17	1	no	—
222	C₄⋊C₈	—	17	1	no	—
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	C₂×D₁₇	1	1	yes	characteristic
227	#102	—	17	1	no	—
228	#102	C₂×D₁₇	1	1	yes	characteristic
229	#97	—	17	1	no	—
230	#97	C₂×D₁₇	1	1	yes	characteristic
231	C₄×C₈	—	17	1	no	—
232	C₁₇	#3400	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₈	—	17	1	no	—
234	C₂×Q₈	—	17	1	no	—
235	C₂×Q₈	—	17	1	no	—
236	C₂×Q₈	—	17	1	no	—
237	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×D₄	—	17	1	no	—
239	C₂×D₄	—	34	1	no	—
240	C₂×D₄	—	34	1	no	—
241	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	34	1	no	—
244	Q₁₆	—	34	2	no	—
245	Q₁₆	—	68	1	no	—
246	Q₁₆	—	68	1	no	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	34	2	no	—
249	SD₁₆	—	68	1	no	—
250	SD₁₆	—	68	1	no	—
251	SD₁₆	—	68	1	no	—
252	SD₁₆	—	68	1	no	—
253	C₂×C₈	—	17	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	17	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	—	34	1	no	—
283	C₄²	—	34	1	no	—
284	C₄²	C₂²×D₁₇	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	34	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	34	2	no	—
290	Q₈	—	34	1	no	—
291	Q₈	—	34	1	no	—
292	Q₈	—	34	1	no	—
293	Q₈	—	34	1	no	—
294	Q₈	—	34	1	no	—
295	Q₈	—	34	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	34	1	no	—
299	D₄	—	34	1	no	—
300	D₄	—	68	2	no	—
301	D₄	—	136	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	17	1	no	—
307	C₂×C₄	—	17	1	no	—
308	C₂×C₄	—	17	1	no	—
309	C₂×C₄	—	17	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	136	1	no	—
314	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	34	1	no	—
319	C₈	—	34	1	no	—
320	C₈	#63252	1	2	yes	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	—	68	1	no	—
326	C₂²	—	68	1	no	—
327	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	17	2	no	—
334	C₄	—	34	1	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	68	1	no	—
338	C₄	—	68	1	no	—
339	C₄	—	68	1	no	—
340	C₄	C₂×#63252	1	1	yes	characteristic
341	C₄	#10587696	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	136	1	no	—
344	C₂	#11763480	1	1	yes	characteristic
345	C₂	#137×D₁₇	1	1	yes	characteristic
346	C₂	#11763619	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×#440	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587626	C₂²
hypercenter	#440	D₁₇
hypocenter	C₁₇	#3400
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #11763537	C₂	semidirect
3. #11763203	C₂	semidirect
4. #11763200	C₂	semidirect
6. #11763031	C₂	semidirect
7. #11763030	C₂	semidirect
8. #11763018	C₂	semidirect
9. #11763014	C₂	semidirect
10. #11763013	C₂	semidirect
12. #11762937	C₂	semidirect
13. #11762908	C₂	semidirect
14. #11762903	C₂	semidirect
15. C₁₇×#440	C₂	semidirect
182. #440	D₁₇	semidirect
232. C₁₇	#3400	semidirect

Maximal subgroups

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

subgroup	index
#11763542	2
#11763537	2
#11763203	2
#11763200	2
#11763180	2
#11763031	2
#11763030	2
#11763018	2
#11763014	2
#11763013	2
#11763008	2
#11762937	2
#11762908	2
#11762903	2
C₁₇×#440	2
#3400	17