Subgroups — #14538998

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	#11763144	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763139	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763019	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763014	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762971	C₂	1	1	yes	maximal, characteristic
9	#11762971	C₂	1	1	yes	maximal, characteristic
10	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#63229	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762870	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#520	C₂	1	1	yes	maximal, characteristic
16	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₇	—	2	1	no	—
20	C₄×D₄×C₁₇	—	2	1	no	—
21	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#117	—	2	1	no	—
24	C₁₇×#117	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×#63231	C₂²	1	1	yes	characteristic
28	C₂×#63231	C₂²	1	1	yes	characteristic
29	C₂×#63229	C₂²	1	1	yes	characteristic
30	C₂×#63229	C₂²	1	1	yes	characteristic
31	#10587610	C₂²	1	1	yes	characteristic
32	#10587610	C₂²	1	1	yes	characteristic
33	#10587575	—	2	1	no	—
34	#10587575	—	2	1	no	—
35	#10587565	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	#10587556	C₂²	1	1	yes	characteristic
38	#10587555	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	#10587506	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	C₂²⋊#3622	C₂²	1	1	yes	characteristic
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic
47	#10587481	C₂²	1	1	yes	characteristic
48	#10587481	C₂²	1	1	yes	characteristic
49	#10587481	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	C₄⋊#3622	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	#41×C₁₇	—	2	2	no	—
60	#41×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	—	2	1	no	—
67	C₂²×C₄×C₁₇	—	2	1	no	—
68	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
82	C₄²×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63231	—	2	2	no	—
89	#63231	—	2	2	no	—
90	#63229	—	2	2	no	—
91	#63229	—	2	2	no	—
92	C₄⋊Dic₁₇	—	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	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	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	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	#41	1	2	yes	—
112	D₄×C₁₇	—	2	4	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	#41	1	2	yes	—
115	C₂×C₄×C₁₇	—	2	4	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	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	—	8	1	no	—
122	C₂×C₄×C₁₇	—	8	1	no	—
123	C₂×C₄×C₁₇	—	8	1	no	—
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	2	1	no	—
138	#3622	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3120	—	17	1	no	maximal, Sylow
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	2	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	2	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
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	D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
179	C₂×C₁₇	—	4	1	no	—
180	C₂×C₁₇	—	4	2	no	—
181	C₂×C₁₇	#525	1	1	yes	characteristic
182	C₂×C₁₇	#577	1	1	yes	characteristic
183	C₂×C₁₇	#578	1	1	yes	characteristic
184	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
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	C₂×D₁₇	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
204	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
205	#117	—	2	1	no	—
206	#117	C₂×D₁₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
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	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	#97	—	17	1	no	—
226	C₄×C₈	—	17	1	no	—
227	C₁₇	#3120	1	1	yes	characteristic, complemented, Sylow
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	—	17	1	no	—
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	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	—	17	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	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	C₂²⋊D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
271	C₄²	—	2	1	no	—
272	C₄²	—	2	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	—	2	1	no	—
278	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	34	1	no	—
281	Q₈	—	34	1	no	—
282	Q₈	—	34	1	no	—
283	Q₈	—	34	1	no	—
284	Q₈	#63252	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#63252	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
307	C₈	—	34	1	no	—
308	C₈	—	34	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	68	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	C₂×#63252	1	1	yes	characteristic
331	C₄	C₂×#63252	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#11763320	1	1	yes	characteristic
335	C₂	#11763620	1	1	yes	characteristic
336	C₂	#11763624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#520	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#41×C₁₇	C₂²
hypercenter	#520	D₁₇
hypocenter	C₁₇	#3120
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #11763421	C₂	semidirect
2. #11763144	C₂	semidirect
3. #11763139	C₂	semidirect
4. #11763116	C₂	semidirect
5. #11763109	C₂	semidirect
6. #11763019	C₂	semidirect
7. #11763014	C₂	semidirect
10. #11762914	C₂	semidirect
11. #11762909	C₂	semidirect
12. C₄×#63231	C₂	semidirect
13. C₄×#63229	C₂	semidirect
14. #11762870	C₂	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
#11763144	2
#11763139	2
#11763116	2
#11763109	2
#11763019	2
#11763014	2
#11762971	2
#11762914	2
#11762909	2
C₄×#63231	2
C₄×#63229	2
#11762870	2
C₁₇×#520	2
#3120	17