Subgroups — #14539000

Order 2176.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763421	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763289	C₂	1	1	yes	maximal, characteristic
3	#11763145	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763140	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763020	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763015	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762973	C₂	1	1	yes	maximal, characteristic
10	#11762973	C₂	1	1	yes	maximal, characteristic
11	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	C₂²	1	1	yes	characteristic
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
19	#10587610	C₂²	1	1	yes	characteristic
20	#10587610	C₂²	1	1	yes	characteristic
21	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
22	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
23	C₂×#63219	C₂²	1	1	yes	characteristic
24	C₂×#63219	C₂²	1	1	yes	characteristic
25	#10587578	—	2	1	no	—
26	C₄⋊C₄×D₁₇	—	2	1	no	—
27	#10587575	—	2	1	no	—
28	#10587575	—	2	1	no	—
29	#10587567	—	2	1	no	—
30	#10587567	—	2	1	no	—
31	#10587565	—	2	1	no	—
32	#10587565	—	2	1	no	—
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	C₄×D₆₈	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	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×#97	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587487	C₂²	1	1	yes	characteristic
55	#10587487	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₇	—	2	1	no	—
61	C₂²×Dic₁₇	—	2	1	no	—
62	#63252	—	2	2	no	—
63	#63252	—	2	2	no	—
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	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	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
99	#63225	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	Dic₆₈	—	2	2	no	—
105	Dic₆₈	—	2	2	no	—
106	#63219	—	2	2	no	—
107	#63219	—	2	2	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	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	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	2	4	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	8	1	no	—
127	D₆₈	—	2	1	no	—
128	D₆₈	#41	1	2	yes	—
129	C₄×D₁₇	—	2	4	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	2	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	#41	1	2	yes	—
138	C₈×C₁₇	—	2	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	#3120	—	17	1	no	maximal, Sylow
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	2	2	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
157	Dic₁₇	—	2	2	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	#541	—	17	1	no	—
165	#520	—	17	1	no	—
166	#484	—	17	1	no	—
167	#483	—	17	1	no	—
168	#479	—	17	1	no	—
169	#478	—	17	1	no	—
170	#462	—	17	1	no	—
171	#461	—	17	1	no	—
172	#452	—	17	1	no	—
173	#452	—	17	1	no	—
174	#441	—	17	1	no	—
175	#440	—	17	1	no	—
176	C₄×Q₁₆	—	17	1	no	—
177	C₄×SD₁₆	—	17	1	no	—
178	#433	D₁₇	1	1	yes	characteristic, complemented
179	C₂×C₁₇	—	4	1	no	—
180	C₂×C₁₇	#525	1	1	yes	characteristic
181	C₂×C₁₇	#577	1	1	yes	characteristic
182	C₂×C₁₇	#578	1	1	yes	characteristic
183	D₁₇	—	4	2	no	—
184	C₂×#41	—	17	1	no	—
185	C₂×Q₁₆	—	17	1	no	—
186	C₂×Q₁₆	—	17	1	no	—
187	C₂×SD₁₆	—	17	1	no	—
188	C₂×SD₁₆	—	17	1	no	—
189	C₄⋊Q₈	—	17	1	no	—
190	#126	—	34	1	no	—
191	#126	—	34	1	no	—
192	#125	—	17	1	no	—
193	#125	—	34	1	no	—
194	#125	—	34	1	no	—
195	C₂²⋊Q₈	—	17	1	no	—
196	C₂²⋊Q₈	—	17	1	no	—
197	C₄×Q₈	—	17	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×Q₈	—	17	1	no	—
200	C₄×Q₈	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	34	1	no	—
204	C₄×D₄	—	34	1	no	—
205	#117	—	34	1	no	—
206	#117	C₂×D₁₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	17	1	no	—
208	C₂×C₄⋊C₄	—	34	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
224	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
225	#97	C₂×D₁₇	1	1	yes	characteristic
226	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
227	C₁₇	#3120	1	1	yes	characteristic, complemented, Sylow
228	#41	—	34	2	no	—
229	#41	—	34	2	no	—
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	—	17	1	no	—
232	C₂×Q₈	—	17	1	no	—
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
240	Q₁₆	—	34	2	no	—
241	Q₁₆	—	34	2	no	—
242	SD₁₆	—	34	2	no	—
243	SD₁₆	—	34	2	no	—
244	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	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	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	D₆₈	1	1	yes	characteristic
264	C₄⋊C₄	D₆₈	1	1	yes	characteristic
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
270	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
271	C₄²	—	34	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	—	34	1	no	—
275	C₄²	C₂²×D₁₇	1	1	yes	characteristic
276	C₄²	C₂²×D₁₇	1	1	yes	characteristic
277	C₂³	—	34	1	no	—
278	C₂³	C₂×D₆₈	1	1	yes	characteristic
279	Q₈	—	17	2	no	—
280	Q₈	—	34	1	no	—
281	Q₈	—	34	1	no	—
282	Q₈	—	34	1	no	—
283	Q₈	—	34	1	no	—
284	Q₈	—	34	1	no	—
285	D₄	—	17	2	no	—
286	D₄	—	34	4	no	—
287	D₄	—	34	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	34	4	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	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	—	34	1	no	—
298	C₂×C₄	—	136	1	no	—
299	C₂×C₄	—	136	1	no	—
300	C₂×C₄	—	136	1	no	—
301	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
302	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
303	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
304	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
305	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	34	2	no	—
315	C₂²	—	68	1	no	—
316	C₂²	—	68	1	no	—
317	C₂²	C₂²×D₆₈	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	34	2	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	—	68	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	#41×D₁₇	1	1	yes	characteristic
331	C₄	#41×D₁₇	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	68	2	no	—
334	C₂	#11763321	1	1	yes	characteristic
335	C₂	#11763621	1	1	yes	characteristic
336	C₂	#11763626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#433	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63252	C₂²
hypercenter	#433	D₁₇
hypocenter	C₁₇	#3120
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #11763421	C₂	semidirect
3. #11763145	C₂	semidirect
4. #11763140	C₂	semidirect
5. #11763117	C₂	semidirect
6. #11763110	C₂	semidirect
7. #11763020	C₂	semidirect
8. #11763015	C₂	semidirect
11. #11762917	C₂	semidirect
12. #11762911	C₂	semidirect
13. C₄×Dic₆₈	C₂	semidirect
14. C₄×#63219	C₂	semidirect
15. C₁₇×#433	C₂	semidirect
178. #433	D₁₇	semidirect
227. C₁₇	#3120	semidirect

Maximal subgroups

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

subgroup	index
#11763421	2
#11763289	2
#11763145	2
#11763140	2
#11763117	2
#11763110	2
#11763020	2
#11763015	2
#11762973	2
#11762917	2
#11762911	2
C₄×Dic₆₈	2
C₄×#63219	2
C₁₇×#433	2
#3120	17