Subgroups — #14538696

Order 2176.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763319	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763121	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762846	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762844	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762802	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762802	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#121	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	1	yes	characteristic
20	#10587624	C₂²	1	1	yes	characteristic
21	#10587623	—	2	1	no	—
22	D₄×Dic₁₇	—	2	1	no	—
23	#10587612	C₂²	1	1	yes	characteristic
24	#10587610	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
27	#10587583	C₂²	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	#10587578	C₂²	1	1	yes	characteristic
30	#10587578	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587569	—	2	1	no	—
34	#10587567	—	2	1	no	—
35	C₂²⋊C₄×D₁₇	—	2	1	no	—
36	#10587563	—	2	1	no	—
37	#10587558	—	2	1	no	—
38	C₄²×D₁₇	—	2	1	no	—
39	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
40	C₁₇×#106	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×#102	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#102	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic, complemented
45	#10587503	C₂²	1	1	yes	characteristic, complemented
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic, complemented
50	#10587488	C₂²	1	1	yes	characteristic, complemented
51	#10587487	C₂²	1	1	yes	characteristic, complemented
52	#10587487	C₂²	1	1	yes	characteristic, complemented
53	#10587481	C₂²	1	1	yes	characteristic, complemented
54	#10587481	C₂²	1	1	yes	characteristic, complemented
55	#10587479	C₂²	1	1	yes	characteristic, complemented
56	#10587479	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	#10587478	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	D₄	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	—	2	1	no	—
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
67	#63254	—	2	2	no	—
68	#63254	—	2	2	no	—
69	C₂×C₄×D₁₇	—	2	2	no	—
70	C₂×C₄×D₁₇	—	2	2	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
77	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₈×D₁₇	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	—	2	2	no	—
111	C₂²×D₁₇	—	2	2	no	—
112	C₂²×D₁₇	—	4	1	no	—
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	D₄×C₁₇	—	2	1	no	—
116	D₄×C₁₇	—	2	1	no	—
117	D₄×C₁₇	—	4	2	no	—
118	C₂×C₄×C₁₇	—	2	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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₇	—	4	2	no	—
126	C₂²⋊D₁₇	—	4	2	no	—
127	C₂×Dic₁₇	—	2	2	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	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₄×D₁₇	—	2	2	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	2	2	no	—
139	C₄×D₁₇	—	4	2	no	—
140	C₄×D₁₇	—	4	2	no	—
141	C₄×D₁₇	#41	1	2	yes	—
142	C₄×D₁₇	#41	1	2	yes	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3091	—	17	1	no	maximal, Sylow
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	4	4	no	—
162	C₂×D₁₇	#123	1	2	yes	—
163	C₂×D₁₇	#135	1	2	yes	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	—	4	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	—	4	1	no	—
173	Dic₁₇	—	4	1	no	—
174	Dic₁₇	—	4	1	no	—
175	Dic₁₇	#123	1	2	yes	—
176	#525	—	17	1	no	—
177	C₂×#117	—	17	1	no	—
178	#485	—	17	1	no	—
179	#485	—	17	1	no	—
180	#484	—	17	1	no	—
181	#483	—	17	1	no	—
182	#482	—	17	1	no	—
183	#482	D₁₇	1	1	yes	characteristic, complemented
184	#481	—	17	1	no	—
185	#480	—	17	1	no	—
186	#427	—	17	1	no	—
187	#427	—	17	1	no	—
188	#416	—	17	1	no	—
189	#416	—	17	1	no	—
190	C₂×C₂²⋊C₈	—	17	1	no	—
191	C₂×C₁₇	—	4	1	no	—
192	C₂×C₁₇	—	8	1	no	—
193	C₂×C₁₇	C₂×#123	1	1	yes	characteristic
194	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
195	C₂×C₁₇	#575	1	1	yes	characteristic
196	D₁₇	—	2	2	no	—
197	D₁₇	—	2	2	no	—
198	C₂×#41	—	17	1	no	—
199	C₂³×C₄	—	17	1	no	—
200	C₂²×C₈	—	17	1	no	—
201	C₂²×C₈	—	17	1	no	—
202	#123	—	34	1	no	—
203	C₂²⋊Q₈	—	17	1	no	—
204	C₂²⋊Q₈	—	17	1	no	—
205	#121	—	17	1	no	—
206	#121	C₂×D₁₇	1	1	yes	characteristic
207	#120	—	34	1	no	—
208	C₄×D₄	—	34	1	no	—
209	C₄×D₄	—	34	1	no	—
210	#117	—	17	1	no	—
211	#117	—	17	1	no	—
212	#117	—	17	1	no	—
213	#117	—	17	1	no	—
214	#117	—	34	1	no	—
215	#117	—	34	1	no	—
216	#117	C₂×D₁₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	17	1	no	—
218	C₂×C₄⋊C₄	—	17	1	no	—
219	C₂×C₂²⋊C₄	—	34	1	no	—
220	C₂×C₄²	—	34	1	no	—
221	#107	—	17	1	no	—
222	#107	—	17	1	no	—
223	#107	—	17	1	no	—
224	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
225	#106	—	17	1	no	—
226	#106	—	17	1	no	—
227	#106	—	17	1	no	—
228	#106	C₂×D₁₇	1	1	yes	characteristic, complemented
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#102	—	17	1	no	—
234	#102	—	17	1	no	—
235	#102	C₂×D₁₇	1	1	yes	characteristic, complemented
236	#102	C₂×D₁₇	1	1	yes	characteristic, complemented
237	C₂²⋊C₈	—	17	1	no	—
238	C₂²⋊C₈	—	17	1	no	—
239	C₂²⋊C₈	—	17	1	no	—
240	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
241	C₁₇	#3091	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	17	1	no	—
243	#41	—	34	2	no	—
244	#41	—	34	2	no	—
245	C₂×Q₈	—	17	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²×C₄	—	17	1	no	—
251	C₂²×C₄	—	17	1	no	—
252	C₂²×C₄	—	17	1	no	—
253	C₂²×C₄	—	17	1	no	—
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	34	2	no	—
256	C₂²×C₄	—	34	2	no	—
257	C₂²×C₄	—	34	1	no	—
258	C₂²×C₄	—	34	1	no	—
259	C₂²×C₄	—	34	1	no	—
260	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	34	2	no	—
264	C₂×C₈	—	34	2	no	—
265	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
267	C₄⋊C₄	—	17	1	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	—	17	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	34	1	no	—
288	C₄²	—	34	1	no	—
289	C₄²	—	34	1	no	—
290	C₄²	—	34	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	17	1	no	—
293	C₂³	—	17	1	no	—
294	C₂³	—	34	2	no	—
295	C₂³	—	34	2	no	—
296	C₂³	—	68	1	no	—
297	C₂³	D₄×D₁₇	1	1	yes	characteristic
298	Q₈	—	34	1	no	—
299	Q₈	—	34	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	68	2	no	—
304	D₄	—	68	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	17	1	no	—
311	C₂×C₄	—	17	2	no	—
312	C₂×C₄	—	17	2	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	2	no	—
315	C₂×C₄	—	34	2	no	—
316	C₂×C₄	—	34	2	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	—	17	2	no	—
339	C₂²	—	17	2	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	34	1	no	—
342	C₂²	—	34	1	no	—
343	C₂²	—	34	1	no	—
344	C₂²	—	68	4	no	—
345	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	17	2	no	—
351	C₄	—	34	1	no	—
352	C₄	—	68	1	no	—
353	C₄	—	68	1	no	—
354	C₄	—	68	1	no	—
355	C₄	—	68	1	no	—
356	C₄	#41×D₁₇	1	1	yes	characteristic
357	C₄	#41×D₁₇	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	34	2	no	—
361	C₂	—	34	2	no	—
362	C₂	#123×D₁₇	1	1	yes	characteristic
363	C₂	#135×D₁₇	1	1	yes	characteristic
364	C₂	#11763605	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×#482	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763319	C₂
hypercenter	#482	D₁₇
hypocenter	C₁₇	#3091
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #117×D₁₇	C₂	semidirect
3. #11763151	C₂	semidirect
4. #11763150	C₂	semidirect
5. #11763146	C₂	semidirect
6. #11763141	C₂	semidirect
7. #11763130	C₂	semidirect
8. #11763126	C₂	semidirect
9. #11763121	C₂	semidirect
10. #11762846	C₂	semidirect
11. #11762844	C₂	semidirect
12. #11762802	C₂	semidirect
13. #11762802	C₂	semidirect
14. C₂²⋊C₈×D₁₇	C₂	semidirect
15. C₁₇×#482	C₂	semidirect
39. C₁₇×#107	C₂²	semidirect
40. C₁₇×#106	C₂²	semidirect
41. C₁₇×#102	C₂²	semidirect
42. C₁₇×#102	C₂²	semidirect
44. #10587503	C₂²	semidirect
45. #10587503	C₂²	semidirect
49. #10587489	C₂²	semidirect
50. #10587488	C₂²	semidirect
51. #10587487	C₂²	semidirect
52. #10587487	C₂²	semidirect
53. #10587481	C₂²	semidirect
54. #10587481	C₂²	semidirect
55. #10587479	C₂²	semidirect
56. #10587479	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. #10587478	C₂²	semidirect
183. #482	D₁₇	semidirect
224. #107	C₂×D₁₇	semidirect
228. #106	C₂×D₁₇	semidirect
235. #102	C₂×D₁₇	semidirect
236. #102	C₂×D₁₇	semidirect
241. C₁₇	#3091	semidirect

Maximal subgroups

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

subgroup	index
#11763319	2
#117×D₁₇	2
#11763151	2
#11763150	2
#11763146	2
#11763141	2
#11763130	2
#11763126	2
#11763121	2
#11762846	2
#11762844	2
#11762802	2
C₂²⋊C₈×D₁₇	2
C₁₇×#482	2
#3091	17