Subgroups — #14541073

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763523	C₂	1	1	yes	maximal, characteristic
2	#11763472	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763152	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763103	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763054	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763012	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762947	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762946	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762900	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762885	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762819	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×#121	C₂²	1	1	yes	characteristic
22	#10587630	C₂²	1	1	yes	characteristic
23	#10587625	—	2	1	no	—
24	#10587624	C₂²	1	1	yes	characteristic
25	#10587622	—	2	1	no	—
26	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
27	C₂×#63229	C₂²	1	1	yes	characteristic
28	C₂×#63229	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
30	#10587612	C₂²	1	1	yes	characteristic
31	C₂×#63218	C₂²	1	1	yes	characteristic
32	#10587583	C₂²	1	1	yes	characteristic
33	#10587582	—	2	1	no	—
34	#10587578	C₂²	1	1	yes	characteristic
35	#10587574	—	2	1	no	—
36	#10587574	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587571	—	2	1	no	—
39	#10587567	—	2	1	no	—
40	#10587564	—	2	1	no	—
41	#10587563	—	2	1	no	—
42	C₁₇×#107	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	C₂²⋊#3622	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	Q₈×D₁₇	—	2	2	no	—
68	Q₈×D₁₇	—	4	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	—	2	2	no	—
72	C₂×Dic₃₄	—	2	1	no	—
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	—	4	1	no	—
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
78	SD₁₆×C₁₇	—	4	1	no	—
79	M₁₆×C₁₇	—	4	1	no	—
80	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	C₂²⋊Dic₁₇	—	2	1	no	—
90	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
91	#63229	—	2	2	no	—
92	#63229	—	4	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	D₄	1	1	yes	characteristic
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	D₄	1	1	yes	characteristic
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	#63218	—	4	1	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	D₄×C₁₇	—	2	1	no	—
117	D₄×C₁₇	—	2	1	no	—
118	D₄×C₁₇	—	8	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₇	—	2	2	no	—
127	C₂²⋊D₁₇	—	8	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	4	2	no	—
133	C₂×Dic₁₇	—	4	2	no	—
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	2	1	no	—
139	C₄×D₁₇	—	4	2	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	4	2	no	—
144	Dic₃₄	—	4	2	no	—
145	Dic₃₄	—	8	1	no	—
146	Dic₃₄	#41	1	2	yes	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	#3622	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3329	—	17	1	no	maximal, Sylow
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	—	4	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	—	4	1	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₁₇	—	2	2	no	—
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	—	4	1	no	—
173	Dic₁₇	—	4	1	no	—
174	Dic₁₇	#137	1	2	yes	—
175	#555	—	17	1	no	—
176	D₄×Q₈	—	17	1	no	—
177	#487	—	17	1	no	—
178	#485	—	17	1	no	—
179	#485	—	17	1	no	—
180	#477	—	17	1	no	—
181	#468	—	17	1	no	—
182	#468	D₁₇	1	1	yes	characteristic, complemented
183	#461	—	17	1	no	—
184	#448	—	17	1	no	—
185	#448	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	#436	—	17	1	no	—
188	#419	—	17	1	no	—
189	#419	—	17	1	no	—
190	C₂×C₁₇	—	4	1	no	—
191	C₂×C₁₇	—	8	1	no	—
192	C₂×C₁₇	#546	1	1	yes	characteristic
193	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
194	C₂×C₁₇	#577	1	1	yes	characteristic
195	D₁₇	—	4	1	no	—
196	C₂²×Q₈	—	17	1	no	—
197	C₂²×Q₈	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
201	C₂×M₁₆	—	17	1	no	—
202	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	17	1	no	—
204	C₄⋊Q₈	—	17	1	no	—
205	C₄⋊Q₈	—	34	1	no	—
206	#124	—	34	1	no	—
207	#124	—	34	1	no	—
208	C₂²⋊Q₈	—	17	1	no	—
209	C₂²⋊Q₈	—	34	1	no	—
210	C₂²⋊Q₈	—	34	1	no	—
211	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
212	#121	—	17	1	no	—
213	#121	—	34	1	no	—
214	#121	C₂×D₁₇	1	1	yes	characteristic
215	C₄×Q₈	—	17	1	no	—
216	C₄×D₄	—	17	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	34	1	no	—
219	#117	—	17	1	no	—
220	#117	—	17	1	no	—
221	#117	—	34	1	no	—
222	#107	—	17	1	no	—
223	#107	C₂×D₁₇	1	1	yes	characteristic
224	#106	—	17	1	no	—
225	C₄⋊C₈	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#102	C₂×D₁₇	1	1	yes	characteristic
236	C₂²⋊C₈	—	17	1	no	—
237	C₂²⋊C₈	—	17	1	no	—
238	C₄×C₈	—	17	1	no	—
239	C₁₇	#3329	1	1	yes	characteristic, complemented, Sylow
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	17	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	34	1	no	—
244	C₂×Q₈	—	34	2	no	—
245	C₂×Q₈	—	34	1	no	—
246	C₂×Q₈	—	34	2	no	—
247	C₂×Q₈	—	68	1	no	—
248	C₂×Q₈	—	68	1	no	—
249	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	17	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	34	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	C₂²×C₄	—	34	1	no	—
258	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
259	SD₁₆	—	4	1	no	—
260	SD₁₆	—	34	2	no	—
261	SD₁₆	—	68	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	68	1	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	—	17	1	no	—
266	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	17	1	no	—
270	C₄⋊C₄	—	17	1	no	—
271	C₄⋊C₄	—	17	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	1	no	—
277	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	17	1	no	—
282	C₂²⋊C₄	—	17	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₄²	—	17	1	no	—
289	C₄²	—	34	1	no	—
290	C₄²	—	34	1	no	—
291	C₄²	—	34	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	17	1	no	—
294	C₂³	D₄×D₁₇	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	17	2	no	—
298	Q₈	—	34	1	no	—
299	Q₈	—	34	1	no	—
300	Q₈	—	34	1	no	—
301	Q₈	—	68	2	no	—
302	Q₈	—	68	2	no	—
303	Q₈	—	136	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	34	2	no	—
308	D₄	—	136	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	17	1	no	—
315	C₂×C₄	—	17	1	no	—
316	C₂×C₄	—	17	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	34	1	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	68	2	no	—
324	C₂×C₄	—	68	2	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	34	1	no	—
331	C₈	—	34	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	34	1	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	17	2	no	—
347	C₄	—	34	1	no	—
348	C₄	—	34	1	no	—
349	C₄	—	34	2	no	—
350	C₄	—	68	1	no	—
351	C₄	—	68	1	no	—
352	C₄	—	68	1	no	—
353	C₄	#10587691	1	1	yes	characteristic
354	C₄	#41×D₁₇	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	8	1	no	—
357	C₂	—	68	1	no	—
358	C₂	#11763448	1	1	yes	characteristic
359	C₂	#137×D₁₇	1	1	yes	characteristic
360	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×#468	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763523	C₂
hypercenter	#468	D₁₇
hypocenter	C₁₇	#3329
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
2. #11763472	C₂	semidirect
3. #11763158	C₂	semidirect
4. #11763152	C₂	semidirect
5. #11763147	C₂	semidirect
6. #11763103	C₂	semidirect
7. #11763054	C₂	semidirect
8. #11763012	C₂	semidirect
9. #11762947	C₂	semidirect
10. #11762946	C₂	semidirect
11. #11762900	C₂	semidirect
12. #11762885	C₂	semidirect
13. #11762819	C₂	semidirect
14. #11762816	C₂	semidirect
15. C₁₇×#468	C₂	semidirect
182. #468	D₁₇	semidirect
239. C₁₇	#3329	semidirect

Maximal subgroups

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

subgroup	index
#11763523	2
#11763472	2
#11763158	2
#11763152	2
#11763147	2
#11763103	2
#11763054	2
#11763012	2
#11762947	2
#11762946	2
#11762900	2
#11762885	2
#11762819	2
#11762816	2
C₁₇×#468	2
#3329	17