Subgroups — #14540821

Order 2176.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763527	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763453	C₂	1	1	yes	maximal, characteristic
3	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763067	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763037	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763030	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762972	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762960	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762906	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762878	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762802	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#418	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63254	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	1	yes	characteristic
20	#10587639	—	2	1	no	—
21	C₂×C₂²⋊Dic₁₇	—	2	1	no	—
22	#10587625	—	2	1	no	—
23	#10587624	C₂²	1	1	yes	characteristic
24	#10587622	C₂²	1	1	yes	characteristic
25	#10587621	—	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	#10587610	C₂²	1	1	yes	characteristic
31	#10587610	C₂²	1	1	yes	characteristic
32	C₂×Dic₆₈	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic, complemented
34	#10587583	C₂²	1	1	yes	characteristic, complemented
35	#10587578	C₂²	1	1	yes	characteristic, complemented
36	#10587574	C₂²	1	1	yes	characteristic, complemented
37	#10587573	C₂²	1	1	yes	characteristic, complemented
38	#10587571	—	2	1	no	—
39	#10587565	—	2	1	no	—
40	#10587562	—	2	1	no	—
41	C₄×Dic₃₄	—	2	1	no	—
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	C₂²⋊#3622	C₂²	1	1	yes	characteristic
49	#10587491	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	#10587486	C₂²	1	1	yes	characteristic, complemented
54	#10587485	C₂²	1	1	yes	characteristic, complemented
55	#10587481	C₂²	1	1	yes	characteristic, complemented
56	#10587481	C₂²	1	1	yes	characteristic, complemented
57	#10587479	C₂²	1	1	yes	characteristic, complemented
58	#10587478	C₂²	1	1	yes	characteristic, complemented
59	C₂⁴×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	2	no	—
61	C₂×D₄×C₁₇	—	2	1	no	—
62	C₂×D₄×C₁₇	—	4	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	C₂×C₂²⋊D₁₇	—	2	1	no	—
68	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₇	—	2	1	no	—
70	C₂²×Dic₁₇	—	2	1	no	—
71	#63254	—	2	2	no	—
72	#63254	—	4	1	no	—
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	M₁₆×C₁₇	—	4	1	no	—
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	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	—	2	1	no	—
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	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	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	Dic₆₈	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂³×C₁₇	—	2	1	no	—
113	C₂³×C₁₇	—	4	2	no	—
114	C₂³×C₁₇	—	4	2	no	—
115	C₂³×C₁₇	—	8	1	no	—
116	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
118	D₄×C₁₇	—	2	1	no	—
119	D₄×C₁₇	—	2	1	no	—
120	D₄×C₁₇	—	2	1	no	—
121	D₄×C₁₇	—	4	2	no	—
122	D₄×C₁₇	#41	1	2	yes	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	1	no	—
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₇	—	4	1	no	—
130	C₂²⋊D₁₇	—	4	2	no	—
131	C₂²⋊D₁₇	—	8	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	—	4	2	no	—
137	C₂×Dic₁₇	—	8	1	no	—
138	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	2	1	no	—
145	Dic₃₄	—	2	1	no	—
146	Dic₃₄	—	2	1	no	—
147	Dic₃₄	—	4	2	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3622	—	2	1	no	—
152	#3311	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	2	2	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	—	4	1	no	—
159	C₂²×C₁₇	—	4	1	no	—
160	C₂²×C₁₇	—	4	1	no	—
161	C₂²×C₁₇	—	4	1	no	—
162	C₂²×C₁₇	—	8	4	no	—
163	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₁₇	—	2	1	no	—
165	C₂×D₁₇	—	2	1	no	—
166	C₂×D₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₇	—	2	1	no	—
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	4	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	Dic₁₇	—	4	1	no	—
179	#556	—	17	1	no	—
180	#546	—	17	1	no	—
181	#489	—	17	1	no	—
182	#485	—	17	1	no	—
183	#482	—	17	1	no	—
184	#476	—	17	1	no	—
185	#470	—	17	1	no	—
186	#465	—	17	1	no	—
187	#464	—	17	1	no	—
188	#452	—	17	1	no	—
189	#450	—	17	1	no	—
190	#440	—	17	1	no	—
191	#435	—	17	1	no	—
192	#418	D₁₇	1	1	yes	characteristic, complemented
193	#416	—	17	1	no	—
194	C₂×C₁₇	—	4	1	no	—
195	C₂×C₁₇	—	4	2	no	—
196	C₂×C₁₇	—	8	1	no	—
197	C₂×C₁₇	#546	1	1	yes	characteristic
198	C₂×C₁₇	#575	1	1	yes	characteristic
199	C₂×C₁₇	#577	1	1	yes	characteristic
200	D₁₇	—	4	1	no	—
201	C₂×#41	—	17	1	no	—
202	C₂²×D₄	C₂×D₁₇	1	1	yes	characteristic
203	C₂×Q₁₆	—	17	1	no	—
204	C₂×SD₁₆	—	17	1	no	—
205	C₂×SD₁₆	—	17	1	no	—
206	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
207	C₂²×C₈	—	17	1	no	—
208	C₄⋊Q₈	—	17	1	no	—
209	#126	—	34	1	no	—
210	#126	—	34	1	no	—
211	#124	—	17	1	no	—
212	#124	—	34	1	no	—
213	#123	—	34	1	no	—
214	C₂²⋊Q₈	—	17	1	no	—
215	C₂²⋊Q₈	—	17	1	no	—
216	C₂²⋊Q₈	—	17	1	no	—
217	#121	—	17	1	no	—
218	#121	—	34	1	no	—
219	#120	—	34	1	no	—
220	C₄×Q₈	—	17	1	no	—
221	C₄×Q₈	—	34	1	no	—
222	C₄×D₄	—	17	1	no	—
223	C₄×D₄	—	17	1	no	—
224	#117	—	17	1	no	—
225	#117	C₂×D₁₇	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	34	1	no	—
227	#107	—	17	1	no	—
228	#106	—	17	1	no	—
229	#106	—	17	1	no	—
230	C₄⋊C₈	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#103	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	C₂×D₁₇	1	1	yes	characteristic
238	#102	C₂×D₁₇	1	1	yes	characteristic
239	#102	C₂×D₁₇	1	1	yes	characteristic
240	#102	C₂×D₁₇	1	1	yes	characteristic
241	C₂²⋊C₈	—	17	1	no	—
242	C₂²⋊C₈	—	17	1	no	—
243	#97	—	17	1	no	—
244	C₁₇	#3311	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	2	1	no	—
246	#41	—	34	2	no	—
247	#41	—	68	1	no	—
248	C₂×Q₈	—	17	1	no	—
249	C₂×Q₈	—	17	1	no	—
250	C₂×Q₈	—	34	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	17	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
259	C₂²×C₄	—	17	1	no	—
260	C₂²×C₄	—	34	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
263	Q₁₆	—	68	1	no	—
264	SD₁₆	—	34	2	no	—
265	SD₁₆	—	68	1	no	—
266	M₁₆	—	4	1	no	—
267	C₂×C₈	—	17	1	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	68	1	no	—
270	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₄⋊C₄	—	17	1	no	—
273	C₄⋊C₄	—	17	1	no	—
274	C₄⋊C₄	—	17	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
281	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	17	1	no	—
284	C₂²⋊C₄	—	17	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	—	34	1	no	—
291	C₂²⋊C₄	—	34	1	no	—
292	C₂²⋊C₄	—	34	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	17	1	no	—
295	C₄²	—	34	1	no	—
296	C₄²	—	34	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	17	1	no	—
303	C₂³	D₄×D₁₇	1	1	yes	characteristic
304	Q₈	—	34	1	no	—
305	Q₈	—	34	1	no	—
306	Q₈	—	34	1	no	—
307	Q₈	—	34	1	no	—
308	Q₈	—	68	2	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	68	1	no	—
314	D₄	—	68	2	no	—
315	D₄	—	136	1	no	—
316	D₄	#63252	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	17	1	no	—
322	C₂×C₄	—	17	1	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	34	1	no	—
328	C₂×C₄	—	34	1	no	—
329	C₂×C₄	—	34	1	no	—
330	C₂×C₄	—	68	2	no	—
331	C₂×C₄	—	136	1	no	—
332	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	34	1	no	—
337	C₈	—	34	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	8	4	no	—
348	C₂²	—	34	1	no	—
349	C₂²	—	34	1	no	—
350	C₂²	—	34	1	no	—
351	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	34	1	no	—
356	C₄	—	34	1	no	—
357	C₄	—	34	1	no	—
358	C₄	—	68	1	no	—
359	C₄	—	68	1	no	—
360	C₄	—	68	1	no	—
361	C₄	—	68	1	no	—
362	C₄	C₂×#63252	1	1	yes	characteristic
363	C₄	#10587691	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	4	2	no	—
366	C₂	—	8	1	no	—
367	C₂	—	68	1	no	—
368	C₂	#11763449	1	1	yes	characteristic
369	C₂	#11763605	1	1	yes	characteristic
370	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×#418	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763453	C₂
hypercenter	#418	D₁₇
hypocenter	C₁₇	#3311
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
1. #11763527	C₂	semidirect
3. #11763167	C₂	semidirect
4. #11763150	C₂	semidirect
5. #11763132	C₂	semidirect
6. #11763097	C₂	semidirect
7. #11763067	C₂	semidirect
8. #11763037	C₂	semidirect
9. #11763030	C₂	semidirect
10. #11762972	C₂	semidirect
11. #11762960	C₂	semidirect
12. #11762906	C₂	semidirect
13. #11762878	C₂	semidirect
14. #11762802	C₂	semidirect
15. C₁₇×#418	C₂	semidirect
32. C₂×Dic₆₈	C₂²	semidirect
33. C₂×C₈×D₁₇	C₂²	semidirect
34. #10587583	C₂²	semidirect
35. #10587578	C₂²	semidirect
36. #10587574	C₂²	semidirect
37. #10587573	C₂²	semidirect
49. #10587491	C₂²	semidirect
50. #10587488	C₂²	semidirect
51. #10587487	C₂²	semidirect
52. #10587487	C₂²	semidirect
53. #10587486	C₂²	semidirect
54. #10587485	C₂²	semidirect
55. #10587481	C₂²	semidirect
56. #10587481	C₂²	semidirect
57. #10587479	C₂²	semidirect
58. #10587478	C₂²	semidirect
192. #418	D₁₇	semidirect
244. C₁₇	#3311	semidirect

Maximal subgroups

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

subgroup	index
#11763527	2
#11763453	2
#11763167	2
#11763150	2
#11763132	2
#11763097	2
#11763067	2
#11763037	2
#11763030	2
#11762972	2
#11762960	2
#11762906	2
#11762878	2
#11762802	2
C₁₇×#418	2
#3311	17