Subgroups — #14540682

Order 2176.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763507	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763470	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763165	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763149	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763127	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763118	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763045	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763041	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763027	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762957	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762948	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762804	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₁₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63256	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
20	#10587633	—	2	1	no	—
21	#10587633	C₂²	1	1	yes	characteristic
22	#10587632	—	2	1	no	—
23	#10587632	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
25	C₂×#63230	C₂²	1	1	yes	characteristic
26	C₂×#63230	C₂²	1	1	yes	characteristic
27	#10587616	—	2	1	no	—
28	#10587616	—	2	1	no	—
29	#10587616	C₂²	1	1	yes	characteristic
30	#10587616	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₇	—	2	1	no	—
32	C₄×C₂²⋊D₁₇	—	2	1	no	—
33	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
34	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
35	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
36	#10587584	—	2	1	no	—
37	#10587581	C₂²	1	1	yes	characteristic
38	#10587580	—	2	1	no	—
39	#10587579	C₂²	1	1	yes	characteristic
40	#10587578	C₂²	1	1	yes	characteristic
41	#10587576	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	C₂²⋊#3622	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587491	C₂²	1	1	yes	characteristic
52	#10587489	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587479	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	—	4	1	no	—
62	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₇	—	2	1	no	—
68	C₂×C₂²⋊D₁₇	—	2	1	no	—
69	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
70	#63256	—	2	2	no	—
71	#63256	—	4	1	no	—
72	C₂×D₆₈	—	2	1	no	—
73	C₂×D₆₈	—	2	1	no	—
74	C₂×D₆₈	C₂³	1	1	yes	characteristic
75	C₂×D₆₈	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	—	2	1	no	—
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×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⋊Dic₁₇	D₄	1	1	yes	characteristic
86	#63230	—	2	2	no	—
87	#63230	—	4	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	—	2	1	no	—
96	#63227	D₄	1	1	yes	characteristic
97	C₄⋊Dic₁₇	—	2	1	no	—
98	C₄⋊Dic₁₇	—	2	1	no	—
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	D₁₃₆	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	2	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	4	2	no	—
119	Q₈×C₁₇	#41	1	2	yes	—
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	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	4	2	no	—
127	C₂×C₄×C₁₇	—	8	1	no	—
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	2	2	no	—
131	C₂²⋊D₁₇	—	4	2	no	—
132	C₂²⋊D₁₇	—	4	2	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
138	D₆₈	—	2	1	no	—
139	D₆₈	—	2	1	no	—
140	D₆₈	—	2	1	no	—
141	D₆₈	—	2	1	no	—
142	D₆₈	—	4	2	no	—
143	D₆₈	—	8	1	no	—
144	C₄×D₁₇	—	2	1	no	—
145	C₄×D₁₇	—	2	1	no	—
146	C₄×D₁₇	—	4	2	no	—
147	C₄×D₁₇	—	8	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3622	—	2	1	no	—
152	#3301	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	#135	1	2	yes	—
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	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	—	4	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₄×C₁₇	—	2	2	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
172	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	2	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	—	4	1	no	—
178	#553	—	17	1	no	—
179	#548	—	17	1	no	—
180	#488	—	17	1	no	—
181	#485	—	17	1	no	—
182	#481	—	17	1	no	—
183	#479	—	17	1	no	—
184	#466	—	17	1	no	—
185	#465	—	17	1	no	—
186	#463	—	17	1	no	—
187	#449	—	17	1	no	—
188	#448	—	17	1	no	—
189	C₄×SD₁₆	—	17	1	no	—
190	C₈×D₄	—	17	1	no	—
191	#416	—	17	1	no	—
192	C₂×#103	D₁₇	1	1	yes	characteristic, complemented
193	C₂×C₁₇	—	2	2	no	—
194	C₂×C₁₇	#546	1	1	yes	characteristic
195	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
196	C₂×C₁₇	#577	1	1	yes	characteristic
197	D₁₇	—	4	1	no	—
198	D₁₇	—	8	1	no	—
199	D₁₇	—	8	1	no	—
200	C₂×#41	—	17	1	no	—
201	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
202	C₂×SD₁₆	—	17	1	no	—
203	C₂×SD₁₆	—	17	1	no	—
204	C₂×D₈	—	17	1	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
207	#126	—	34	1	no	—
208	#125	—	17	1	no	—
209	#124	—	17	1	no	—
210	#124	—	34	1	no	—
211	#123	—	34	1	no	—
212	C₂²⋊Q₈	—	17	1	no	—
213	C₂²⋊Q₈	—	34	1	no	—
214	#121	—	17	1	no	—
215	#121	—	17	1	no	—
216	#121	—	17	1	no	—
217	#121	—	34	1	no	—
218	#121	—	34	1	no	—
219	C₄×Q₈	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	17	1	no	—
222	C₄×D₄	—	34	1	no	—
223	C₄×D₄	—	34	1	no	—
224	#117	—	17	1	no	—
225	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
226	#107	—	17	1	no	—
227	#106	—	17	1	no	—
228	#106	—	17	1	no	—
229	C₄⋊C₈	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	C₂×D₁₇	1	1	yes	characteristic
233	#103	C₂×D₁₇	1	1	yes	characteristic
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#103	C₂×D₁₇	1	1	yes	characteristic
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₂²⋊C₈	—	17	1	no	—
242	C₄×C₈	—	17	1	no	—
243	C₁₇	#3301	1	1	yes	characteristic, complemented, Sylow
244	#41	—	34	2	no	—
245	#41	—	68	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
251	C₂×D₄	—	17	1	no	—
252	C₂×D₄	—	17	1	no	—
253	C₂×D₄	—	17	1	no	—
254	C₂×D₄	—	34	1	no	—
255	C₂×D₄	—	34	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	17	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	—	34	1	no	—
263	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
264	SD₁₆	—	34	2	no	—
265	SD₁₆	—	68	1	no	—
266	D₈	—	68	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	68	1	no	—
271	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	17	1	no	—
275	C₄⋊C₄	—	17	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	—	34	1	no	—
296	C₂³	—	17	1	no	—
297	C₂³	—	34	1	no	—
298	C₂³	—	34	1	no	—
299	C₂³	D₄×D₁₇	1	1	yes	characteristic
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	#63252	1	2	yes	—
305	D₄	—	34	2	no	—
306	D₄	—	34	1	no	—
307	D₄	—	34	1	no	—
308	D₄	—	34	1	no	—
309	D₄	—	34	1	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	68	2	no	—
313	D₄	—	136	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	17	1	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	17	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	—	68	2	no	—
330	C₂×C₄	—	136	1	no	—
331	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	34	1	no	—
336	C₈	—	34	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	34	1	no	—
342	C₂²	—	68	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	68	1	no	—
345	C₂²	—	68	1	no	—
346	C₂²	—	68	1	no	—
347	C₂²	—	68	1	no	—
348	C₂²	#10587605	1	2	yes	—
349	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	4	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	C₂×#63252	1	1	yes	characteristic
361	C₄	#10587691	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	68	1	no	—
364	C₂	—	136	1	no	—
365	C₂	—	136	1	no	—
366	C₂	#11763449	1	1	yes	characteristic
367	C₂	C₂×#10587605	1	1	yes	characteristic
368	C₂	#11763615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#103	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	C₂×#103	D₁₇
hypocenter	C₁₇	#3301
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
1. #11763507	C₂	semidirect
2. #11763470	C₂	semidirect
3. #11763165	C₂	semidirect
4. #11763149	C₂	semidirect
5. #11763127	C₂	semidirect
6. #11763118	C₂	semidirect
7. #11763045	C₂	semidirect
8. #11763041	C₂	semidirect
9. #11763027	C₂	semidirect
10. #11762957	C₂	semidirect
11. #11762948	C₂	semidirect
12. #11762894	C₂	semidirect
13. C₈×C₂²⋊D₁₇	C₂	semidirect
14. #11762804	C₂	semidirect
15. C₂×C₁₇×#103	C₂	semidirect
192. C₂×#103	D₁₇	semidirect
243. C₁₇	#3301	semidirect

Maximal subgroups

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

subgroup	index
#11763507	2
#11763470	2
#11763165	2
#11763149	2
#11763127	2
#11763118	2
#11763045	2
#11763041	2
#11763027	2
#11762957	2
#11762948	2
#11762894	2
C₈×C₂²⋊D₁₇	2
#11762804	2
C₂×C₁₇×#103	2
#3301	17