Subgroups — #14537729

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763363	C₂	1	1	yes	maximal, characteristic
2	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762933	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762834	C₂	1	1	yes	maximal, characteristic
10	#11762834	C₂	1	1	yes	maximal, characteristic
11	C₄⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
12	#11762766	C₂	1	1	yes	maximal, characteristic
13	#11762763	C₂	1	1	yes	maximal, characteristic
14	#11762761	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587632	C₄	1	1	yes	characteristic
18	#10587630	C₄	1	1	yes	characteristic
19	C₂×#63218	C₂²	1	1	yes	characteristic
20	C₂×#63218	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	#10587583	C₄	1	1	yes	characteristic
24	#10587582	C₄	1	1	yes	characteristic
25	#10587582	C₄	1	1	yes	characteristic
26	#10587575	C₄	1	1	yes	characteristic
27	#10587575	C₄	1	1	yes	characteristic
28	#10587574	C₄	1	1	yes	characteristic
29	#10587558	C₂²	1	1	yes	characteristic
30	#10587558	C₂²	1	1	yes	characteristic
31	C₄²×D₁₇	C₂²	1	1	yes	characteristic
32	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
35	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×#97	C₂²	1	1	yes	characteristic
39	C₁₇×#97	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	#10587486	C₂²	1	1	yes	characteristic
42	#10587486	C₂²	1	1	yes	characteristic
43	#10587485	C₂²	1	1	yes	characteristic
44	#10587485	C₂²	1	1	yes	characteristic
45	#10587485	C₂²	1	1	yes	characteristic
46	#10587485	C₂²	1	1	yes	characteristic
47	#10587485	C₂²	1	1	yes	characteristic
48	#10587485	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	C₄⋊#3622	C₂²	1	1	yes	characteristic
54	C₄⋊#3622	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	#10587474	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₁₇	C₂³	1	1	yes	characteristic
74	C₄²×C₁₇	C₂³	1	1	yes	characteristic
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	#63227	C₂×C₄	1	1	yes	characteristic
77	#63227	C₂×C₄	1	1	yes	characteristic
78	#63227	C₂×C₄	1	1	yes	characteristic
79	#63227	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
83	#63225	C₂×C₄	1	1	yes	characteristic
84	#63225	C₂×C₄	1	1	yes	characteristic
85	#63225	C₂×C₄	1	1	yes	characteristic
86	#63225	C₂×C₄	1	1	yes	characteristic
87	#63225	C₂×C₄	1	1	yes	characteristic
88	#63225	C₂×C₄	1	1	yes	characteristic
89	#63225	C₂×C₄	1	1	yes	characteristic
90	#63225	C₂×C₄	1	1	yes	characteristic
91	#63225	C₂×C₄	1	1	yes	characteristic
92	#63225	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
98	C₂×#3622	C₂³	1	1	yes	characteristic
99	C₂×#3622	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	#63218	—	2	2	no	—
103	#63218	Q₈	1	4	yes	complemented
104	C₈×D₁₇	—	2	2	no	—
105	C₈×D₁₇	—	2	2	no	—
106	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
122	C₄×D₁₇	—	2	2	no	—
123	C₄×D₁₇	—	4	1	no	—
124	C₄×D₁₇	C₂×Q₈	1	2	yes	—
125	C₄×D₁₇	#41	1	2	yes	—
126	Dic₃₄	—	2	2	no	—
127	Dic₃₄	—	2	2	no	—
128	Dic₃₄	—	4	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	1	no	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	C₂×Q₈	1	2	yes	—
133	#3622	—	2	1	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	C₂×Q₈	1	2	yes	—
137	#2962	—	17	1	no	maximal, Sylow
138	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₁₇	—	2	1	no	—
140	C₂×D₁₇	—	2	1	no	—
141	C₂×D₁₇	C₄×Q₈	1	2	yes	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	—	2	1	no	—
153	Dic₁₇	—	2	1	no	—
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	C₄×Q₈	1	2	yes	—
156	Dic₁₇	#131	1	2	yes	—
157	#533	—	17	1	no	—
158	#446	—	17	1	no	—
159	#446	—	17	1	no	—
160	#446	—	17	1	no	—
161	#446	—	17	1	no	—
162	#446	—	17	1	no	—
163	#446	D₁₇	1	1	yes	characteristic, complemented
164	C₈×Q₈	—	17	1	no	—
165	C₈×Q₈	—	17	1	no	—
166	#424	—	17	1	no	—
167	#424	—	17	1	no	—
168	C₂×C₄⋊C₈	—	17	1	no	—
169	#405	—	17	1	no	—
170	#405	—	17	1	no	—
171	C₄×M₁₆	—	17	1	no	—
172	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
174	C₂×C₁₇	#568	1	1	yes	characteristic
175	D₁₇	—	2	2	no	—
176	C₂×M₁₆	—	17	1	no	—
177	C₂×M₁₆	—	17	1	no	—
178	C₂²×C₈	—	17	1	no	—
179	C₂²×C₈	—	17	1	no	—
180	C₄⋊Q₈	—	17	1	no	—
181	C₄⋊Q₈	—	17	1	no	—
182	#125	—	17	1	no	—
183	#125	—	17	1	no	—
184	C₂²⋊Q₈	—	17	1	no	—
185	C₂²⋊Q₈	—	17	1	no	—
186	C₂²⋊Q₈	—	17	1	no	—
187	C₂²⋊Q₈	—	17	1	no	—
188	C₄×Q₈	—	17	1	no	—
189	C₄×Q₈	—	17	1	no	—
190	C₄×Q₈	—	17	1	no	—
191	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
192	#117	—	17	1	no	—
193	#117	—	17	1	no	—
194	C₂×C₄²	—	17	1	no	—
195	C₄⋊C₈	—	17	1	no	—
196	C₄⋊C₈	—	17	1	no	—
197	C₄⋊C₈	—	17	1	no	—
198	C₄⋊C₈	—	17	1	no	—
199	C₄⋊C₈	—	17	1	no	—
200	C₄⋊C₈	—	17	1	no	—
201	C₄⋊C₈	—	17	1	no	—
202	C₄⋊C₈	—	17	1	no	—
203	C₄⋊C₈	—	17	1	no	—
204	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
205	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
207	#97	—	17	1	no	—
208	#97	—	17	1	no	—
209	#97	—	17	1	no	—
210	#97	—	17	1	no	—
211	#97	C₂×D₁₇	1	1	yes	characteristic
212	#97	C₂×D₁₇	1	1	yes	characteristic
213	C₄×C₈	—	17	1	no	—
214	C₄×C₈	—	17	1	no	—
215	C₄×C₈	—	17	1	no	—
216	C₄×C₈	—	17	1	no	—
217	C₄×C₈	—	17	1	no	—
218	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
219	C₁₇	#2962	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	17	1	no	—
221	C₂×Q₈	—	17	1	no	—
222	C₂×Q₈	—	17	1	no	—
223	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
224	C₂²×C₄	—	17	1	no	—
225	C₂²×C₄	—	17	1	no	—
226	C₂²×C₄	—	17	1	no	—
227	M₁₆	—	17	4	no	—
228	M₁₆	—	34	2	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	34	2	no	—
234	C₂×C₈	—	34	2	no	—
235	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
255	C₂²⋊C₄	—	17	1	no	—
256	C₂²⋊C₄	—	17	1	no	—
257	C₂²⋊C₄	—	17	1	no	—
258	C₂²⋊C₄	—	17	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	17	1	no	—
261	C₄²	—	17	1	no	—
262	C₄²	—	17	1	no	—
263	C₄²	—	17	1	no	—
264	C₄²	C₂²×D₁₇	1	1	yes	characteristic
265	C₄²	C₂²×D₁₇	1	1	yes	characteristic
266	C₄²	C₂²×D₁₇	1	1	yes	characteristic
267	C₂³	—	17	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	34	2	no	—
270	Q₈	—	34	2	no	—
271	Q₈	—	68	1	no	—
272	C₂×C₄	—	17	1	no	—
273	C₂×C₄	—	17	1	no	—
274	C₂×C₄	—	17	2	no	—
275	C₂×C₄	—	17	2	no	—
276	C₂×C₄	—	17	1	no	—
277	C₂×C₄	—	17	1	no	—
278	C₂×C₄	—	17	1	no	—
279	C₂×C₄	—	17	1	no	—
280	C₂×C₄	—	17	1	no	—
281	C₂×C₄	—	34	2	no	—
282	C₂×C₄	—	68	1	no	—
283	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	17	2	no	—
294	C₈	—	34	1	no	—
295	C₈	—	34	1	no	—
296	C₈	—	34	1	no	—
297	C₈	Q₈×D₁₇	1	2	yes	—
298	C₂²	—	17	2	no	—
299	C₂²	—	34	1	no	—
300	C₂²	—	34	1	no	—
301	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	17	2	no	—
309	C₄	—	17	2	no	—
310	C₄	—	34	1	no	—
311	C₄	—	34	1	no	—
312	C₄	—	34	1	no	—
313	C₄	—	34	1	no	—
314	C₄	—	34	1	no	—
315	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
316	C₄	#41×D₁₇	1	1	yes	characteristic
317	C₂	—	34	2	no	—
318	C₂	C₄×Q₈×D₁₇	1	1	yes	characteristic
319	C₂	#131×D₁₇	1	1	yes	characteristic
320	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2962
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11762940	C₂	semidirect
3. #11762938	C₂	semidirect
4. #11762938	C₂	semidirect
5. #11762937	C₂	semidirect
6. #11762937	C₂	semidirect
7. C₈×Dic₃₄	C₂	semidirect
8. #11762933	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
103. #63218	Q₈	semidirect
163. #446	D₁₇	semidirect
219. C₁₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#11763363	2
#11762940	2
#11762938	2
#11762937	2
C₈×Dic₃₄	2
#11762933	2
#11762834	2
C₄⋊C₈×D₁₇	2
#11762766	2
#11762763	2
#11762761	2
C₁₇×#446	2
#2962	17