Subgroups — #14541546

Order 2176.

344 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	#11763537	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763173	C₂	1	1	yes	maximal, characteristic
3	#11763168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763163	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763099	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763028	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763028	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763006	C₂	1	1	yes	maximal, characteristic
11	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63229	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#559	C₂	1	1	yes	maximal, characteristic
16	C₁₇×#127	C₂²	1	1	yes	characteristic
17	C₁₇×#124	—	2	1	no	—
18	C₁₇×#124	C₂²	1	1	yes	characteristic
19	C₁₇×#124	C₂²	1	1	yes	characteristic
20	C₁₇×#123	—	2	1	no	—
21	C₁₇×#123	—	2	1	no	—
22	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	—	2	1	no	—
25	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
27	#10587630	—	2	1	no	—
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63229	C₂²	1	1	yes	characteristic
30	C₂×#63229	C₂²	1	1	yes	characteristic
31	#10587575	—	2	1	no	—
32	#10587575	—	2	1	no	—
33	#10587573	—	2	1	no	—
34	#10587556	C₂²	1	1	yes	characteristic
35	#10587556	C₂²	1	1	yes	characteristic
36	#10587555	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	#10587506	C₂²	1	1	yes	characteristic
41	#10587506	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587503	C₂²	1	1	yes	characteristic
49	#10587481	C₂²	1	1	yes	characteristic
50	#10587479	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587478	C₂²	1	1	yes	characteristic
53	C₄⋊#3622	C₂²	1	1	yes	characteristic
54	C₄⋊#3622	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	—	2	1	no	—
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₇	—	2	1	no	—
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	C₄²×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	#63231	—	2	2	no	—
85	#63229	—	2	2	no	—
86	#63229	—	2	2	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂³	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	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	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	—	2	1	no	—
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	4	1	no	—
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	4	2	no	—
115	D₄×C₁₇	—	8	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	8	1	no	—
119	C₂×C₄×C₁₇	—	8	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
133	Dic₃₄	#41	1	2	yes	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3371	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	—	4	1	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₇	—	2	2	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	#559	D₁₇	1	1	yes	characteristic
161	#557	—	17	1	no	—
162	#490	—	17	1	no	—
163	#489	—	17	1	no	—
164	#488	—	17	1	no	—
165	#479	—	17	1	no	—
166	#478	—	17	1	no	—
167	#476	—	17	1	no	—
168	#464	—	17	1	no	—
169	#464	—	17	1	no	—
170	#460	—	17	1	no	—
171	#446	—	17	1	no	—
172	#441	—	17	1	no	—
173	#440	—	17	1	no	—
174	C₄×SD₁₆	—	17	1	no	—
175	C₂×C₁₇	—	8	1	no	—
176	C₂×C₁₇	—	8	1	no	—
177	C₂×C₁₇	#548	1	1	yes	characteristic
178	C₂×C₁₇	#575	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	C₂×Q₁₆	—	17	1	no	—
181	C₂×SD₁₆	—	17	1	no	—
182	C₂×SD₁₆	—	17	1	no	—
183	C₄⋊Q₈	—	17	1	no	—
184	C₄⋊Q₈	—	34	1	no	—
185	#127	C₂×D₁₇	1	1	yes	characteristic
186	#125	—	17	1	no	—
187	#125	—	17	1	no	—
188	#125	—	34	1	no	—
189	#125	—	34	1	no	—
190	#124	—	2	1	no	—
191	#124	C₂×D₁₇	1	1	yes	characteristic
192	#124	C₂×D₁₇	1	1	yes	characteristic
193	#123	—	2	1	no	—
194	#123	—	2	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	17	1	no	—
197	C₄×Q₈	—	17	1	no	—
198	C₄×Q₈	—	34	1	no	—
199	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
203	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
204	#107	—	17	1	no	—
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	C₄⋊C₈	—	17	1	no	—
208	C₄⋊C₈	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	#103	—	17	1	no	—
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	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	—	17	1	no	—
222	C₄×C₈	—	17	1	no	—
223	C₁₇	#3371	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	17	1	no	—
225	C₂×Q₈	—	34	1	no	—
226	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	Q₁₆	—	34	2	no	—
235	SD₁₆	—	34	2	no	—
236	SD₁₆	—	34	2	no	—
237	C₂×C₈	—	17	1	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	—	34	1	no	—
248	C₄⋊C₄	—	34	1	no	—
249	C₄⋊C₄	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
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	C₂²⋊C₄	—	2	1	no	—
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	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	34	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	C₄²	C₂²×D₁₇	1	1	yes	characteristic
269	C₄²	C₂²×D₁₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	4	1	no	—
275	Q₈	—	17	2	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	68	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	2	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	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
299	C₈	—	34	1	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₈	—	34	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	34	2	no	—
319	C₄	—	68	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	#41×D₁₇	1	1	yes	characteristic
323	C₄	#10587697	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	8	1	no	—
326	C₂	#11763466	1	1	yes	characteristic
327	C₂	#11763607	1	1	yes	characteristic
328	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	C₁₇×#559	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₁₇×#127	C₂²
hypercenter	#559	D₁₇
hypocenter	C₁₇	#3371
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #559

Complements

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

N	H ≅ G/N	type
1. #11763537	C₂	semidirect
3. #11763168	C₂	semidirect
4. #11763163	C₂	semidirect
5. #11763116	C₂	semidirect
6. #11763109	C₂	semidirect
7. #11763099	C₂	semidirect
8. #11763028	C₂	semidirect
9. #11763028	C₂	semidirect
11. #11762940	C₂	semidirect
12. #11762914	C₂	semidirect
13. #11762909	C₂	semidirect
14. C₄×#63229	C₂	semidirect
223. C₁₇	#3371	semidirect

Maximal subgroups

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

subgroup	index
#11763537	2
#11763173	2
#11763168	2
#11763163	2
#11763116	2
#11763109	2
#11763099	2
#11763028	2
#11763006	2
#11762940	2
#11762914	2
#11762909	2
C₄×#63229	2
C₁₇×#559	2
#3371	17