Subgroups — #14541323

Order 2176.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763529	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763219	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763139	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763099	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763053	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763036	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762881	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762849	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762845	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#553	C₂	1	1	yes	maximal, characteristic
16	C₁₇×#126	—	2	1	no	—
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₁₇×#123	—	2	1	no	—
19	C₁₇×#121	—	2	1	no	—
20	C₁₇×#121	C₂²	1	1	yes	characteristic
21	C₁₇×#121	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₇	—	2	1	no	—
23	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#117	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
27	#10587622	—	2	1	no	—
28	C₂×#63229	C₂²	1	1	yes	characteristic
29	#10587612	C₂²	1	1	yes	characteristic
30	#10587612	C₂²	1	1	yes	characteristic
31	#10587610	C₂²	1	1	yes	characteristic
32	#10587610	C₂²	1	1	yes	characteristic
33	C₂×#63223	C₂²	1	1	yes	characteristic
34	C₂²×#3622	C₂²	1	1	yes	characteristic
35	#10587573	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	#10587565	—	2	1	no	—
38	#10587555	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	#10587503	C₂²	1	1	yes	characteristic
41	#10587503	C₂²	1	1	yes	characteristic
42	#10587503	C₂²	1	1	yes	characteristic
43	C₂²⋊#3622	C₂²	1	1	yes	characteristic
44	C₂²⋊#3622	C₂²	1	1	yes	characteristic
45	#10587481	C₂²	1	1	yes	characteristic
46	#10587481	C₂²	1	1	yes	characteristic
47	#10587481	C₂²	1	1	yes	characteristic
48	#10587479	C₂²	1	1	yes	characteristic
49	#10587479	C₂²	1	1	yes	characteristic
50	#10587479	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	C₄⋊#3622	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
82	C₄²×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63229	—	2	2	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	C₂³	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	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	#63223	—	2	2	no	—
100	C₂×#3622	—	2	2	no	—
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂³×C₁₇	—	2	1	no	—
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
108	D₄×C₁₇	—	2	1	no	—
109	D₄×C₁₇	—	2	1	no	—
110	D₄×C₁₇	—	4	1	no	—
111	D₄×C₁₇	—	4	2	no	—
112	D₄×C₁₇	—	8	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×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	—	8	1	no	—
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	Dic₃₄	—	2	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	4	2	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	#41	1	2	yes	—
138	#3350	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	#556	—	17	1	no	—
163	#553	D₁₇	1	1	yes	characteristic
164	#501	—	17	1	no	—
165	#485	—	17	1	no	—
166	#483	—	17	1	no	—
167	#482	—	17	1	no	—
168	#482	—	17	1	no	—
169	#479	—	17	1	no	—
170	#476	—	17	1	no	—
171	#468	—	17	1	no	—
172	#465	—	17	1	no	—
173	#440	—	17	1	no	—
174	#435	—	17	1	no	—
175	#428	—	17	1	no	—
176	#427	—	17	1	no	—
177	C₂×C₁₇	—	4	1	no	—
178	C₂×C₁₇	—	4	1	no	—
179	C₂×C₁₇	—	8	1	no	—
180	C₂×C₁₇	#547	1	1	yes	characteristic
181	C₂×C₁₇	#575	1	1	yes	characteristic
182	C₂×C₁₇	#577	1	1	yes	characteristic
183	C₂×SD₁₆	—	17	1	no	—
184	C₂×M₁₆	—	17	1	no	—
185	C₂²×C₈	—	17	1	no	—
186	C₄⋊Q₈	—	17	1	no	—
187	#126	—	2	1	no	—
188	#126	—	34	1	no	—
189	#126	—	34	1	no	—
190	#125	C₂×D₁₇	1	1	yes	characteristic
191	#124	—	34	1	no	—
192	#123	—	2	1	no	—
193	C₂²⋊Q₈	—	17	1	no	—
194	C₂²⋊Q₈	—	17	1	no	—
195	#121	—	2	1	no	—
196	#121	C₂×D₁₇	1	1	yes	characteristic
197	#121	C₂×D₁₇	1	1	yes	characteristic
198	C₄×Q₈	—	17	1	no	—
199	C₄×Q₈	—	34	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
202	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
203	#117	—	17	1	no	—
204	#117	—	17	1	no	—
205	#117	C₂×D₁₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	#106	—	17	1	no	—
214	#106	—	17	1	no	—
215	#106	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	#97	—	17	1	no	—
226	C₁₇	#3350	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	17	1	no	—
228	C₂×Q₈	—	34	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
232	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
237	SD₁₆	—	34	2	no	—
238	M₁₆	—	34	2	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	17	1	no	—
242	C₂×C₈	—	17	1	no	—
243	C₂×C₈	—	34	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	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	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
267	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
268	C₄²	—	2	1	no	—
269	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	—	34	1	no	—
272	C₄²	C₂²×D₁₇	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
276	Q₈	—	34	1	no	—
277	Q₈	—	34	1	no	—
278	Q₈	—	68	2	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	4	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
302	C₈	—	17	2	no	—
303	C₈	—	34	1	no	—
304	C₈	—	34	1	no	—
305	C₈	—	34	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	68	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	—	68	1	no	—
327	C₄	#41×D₁₇	1	1	yes	characteristic
328	C₄	#10587697	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	8	1	no	—
332	C₂	#11763458	1	1	yes	characteristic
333	C₂	#11763607	1	1	yes	characteristic
334	C₂	#11763620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting, prime radical	C₁₇×#553	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#553	D₁₇
hypocenter	C₁₇	#3350
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #553

Complements

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

N	H ≅ G/N	type
1. #11763529	C₂	semidirect
2. #11763219	C₂	semidirect
3. #11763150	C₂	semidirect
4. #11763139	C₂	semidirect
5. #11763130	C₂	semidirect
6. #11763130	C₂	semidirect
7. #11763113	C₂	semidirect
8. #11763099	C₂	semidirect
9. #11763053	C₂	semidirect
10. #11763036	C₂	semidirect
11. #11762909	C₂	semidirect
12. #11762881	C₂	semidirect
13. #11762849	C₂	semidirect
14. #11762845	C₂	semidirect
226. C₁₇	#3350	semidirect

Maximal subgroups

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

subgroup	index
#11763529	2
#11763219	2
#11763150	2
#11763139	2
#11763130	2
#11763113	2
#11763099	2
#11763053	2
#11763036	2
#11762909	2
#11762881	2
#11762849	2
#11762845	2
C₁₇×#553	2
#3350	17