Subgroups — #16934331

Order 4352.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587555	C₂	1	1	yes	maximal, characteristic
2	#14528936	C₂	1	1	yes	maximal, characteristic
3	#14526624	C₂	1	4	yes	maximal
4	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₃₄	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×C₄⋊#3622	—	2	1	no	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762715	—	2	1	no	—
11	#11762707	—	2	1	no	—
12	#11762707	C₄	1	2	yes	—
13	#11762679	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₁₇	—	2	1	no	—
15	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
19	C₄³×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₃₄	—	2	1	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×#63225	—	2	2	no	—
35	C₂×C₄×Dic₁₇	—	2	2	no	—
36	C₂²×#3622	—	2	2	no	—
37	#10587555	—	2	2	no	—
38	#10587555	C₈	1	4	yes	complemented
39	C₄×Dic₃₄	—	4	2	no	—
40	C₄×Dic₃₄	—	4	2	no	—
41	C₄×C₈×C₁₇	—	2	4	no	—
42	C₄×C₈×C₁₇	—	2	4	no	—
43	C₄×C₈×C₁₇	—	2	2	no	—
44	C₄×C₈×C₁₇	Q₈	1	4	yes	complemented
45	#10587502	—	2	2	no	—
46	#10587502	—	2	2	no	—
47	C₄⋊#3622	—	2	4	no	—
48	C₄⋊#3622	—	4	2	no	—
49	C₄⋊#3622	—	4	2	no	—
50	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₇	—	2	2	no	—
58	C₂²×Dic₁₇	—	2	2	no	—
59	C₂×Dic₃₄	—	2	4	no	—
60	C₂×Dic₃₄	—	4	2	no	—
61	C₂×Dic₃₄	—	4	2	no	—
62	C₂×C₈×C₁₇	—	2	4	no	—
63	C₂×C₈×C₁₇	—	2	4	no	—
64	C₂×C₈×C₁₇	—	2	4	no	—
65	C₄²×C₁₇	—	2	2	no	—
66	C₄²×C₁₇	—	2	2	no	—
67	C₄²×C₁₇	—	2	2	no	—
68	C₄²×C₁₇	C₂×C₈	1	2	yes	—
69	C₄²×C₁₇	M₁₆	1	2	yes	—
70	C₄²×C₁₇	D₈	1	2	yes	—
71	C₄²×C₁₇	SD₁₆	1	2	yes	—
72	C₄²×C₁₇	SD₁₆	1	2	yes	—
73	C₄²×C₁₇	Q₁₆	1	2	yes	—
74	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
75	C₄²×C₁₇	#41	1	2	yes	—
76	C₄⋊Dic₁₇	—	2	4	no	—
77	C₄⋊Dic₁₇	—	2	2	no	—
78	C₄⋊Dic₁₇	—	2	4	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	C₄⋊Dic₁₇	—	4	2	no	—
81	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
82	#63225	—	4	4	no	—
83	C₄×Dic₁₇	—	4	2	no	—
84	C₄×Dic₁₇	—	4	2	no	—
85	C₂×#3622	—	4	2	no	—
86	C₂×#3622	—	4	2	no	—
87	C₂×#3622	—	4	2	no	—
88	#9870	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₇	#102	1	2	yes	—
99	C₂×C₄×C₁₇	#102	1	2	yes	—
100	C₂×C₄×C₁₇	#103	1	2	yes	—
101	C₂×C₄×C₁₇	#103	1	2	yes	—
102	C₂×C₄×C₁₇	#104	1	2	yes	—
103	C₂×C₄×C₁₇	#104	1	2	yes	—
104	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
105	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#131	1	2	yes	—
111	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×Dic₁₇	—	2	4	no	—
116	C₂×Dic₁₇	—	4	2	no	—
117	C₂×Dic₁₇	—	4	2	no	—
118	C₂×Dic₁₇	—	4	2	no	—
119	C₂×Dic₁₇	—	4	2	no	—
120	C₂×Dic₁₇	—	4	2	no	—
121	Dic₃₄	—	4	8	no	—
122	C₈×C₁₇	—	2	8	no	—
123	#3622	—	8	2	no	—
124	C₄×C₄⋊Q₈	—	17	1	no	—
125	#1842	—	17	1	no	—
126	C₄²×C₈	D₁₇	1	1	yes	characteristic
127	#1279	—	17	4	no	—
128	C₂²×C₁₇	#391	1	1	yes	characteristic
129	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₁₇	#408	1	1	yes	characteristic
131	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
132	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
133	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
134	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
135	C₄×C₁₇	—	2	2	no	—
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	D₄⋊C₈	1	4	yes	—
139	C₄×C₁₇	Q₈⋊C₈	1	4	yes	—
140	C₄×C₁₇	C₈×Q₈	1	2	yes	—
141	C₄×C₁₇	#446	1	2	yes	—
142	C₄×C₁₇	#486	1	1	yes	characteristic
143	C₄×C₁₇	#487	1	1	yes	characteristic
144	C₄×C₁₇	#498	1	1	yes	characteristic
145	C₄×C₁₇	#500	1	1	yes	characteristic
146	Dic₁₇	—	4	4	no	—
147	Dic₁₇	—	8	2	no	—
148	C₂×C₄⋊Q₈	—	17	1	no	—
149	C₂×C₄×Q₈	—	34	1	no	—
150	C₂×C₄⋊C₈	—	17	2	no	—
151	C₂×C₄⋊C₈	—	34	1	no	—
152	C₂×C₄⋊C₈	—	34	1	no	—
153	C₂×C₄×C₈	—	2	1	no	—
154	C₂×C₄×C₈	—	2	1	no	—
155	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
156	#391	—	34	1	no	—
157	#389	—	17	2	no	—
158	#389	—	34	1	no	—
159	#384	—	17	1	no	—
160	C₄×C₄⋊C₄	—	17	2	no	—
161	C₄×C₄⋊C₄	—	34	1	no	—
162	C₄³	C₂×D₁₇	1	1	yes	characteristic
163	C₂×C₁₇	C₂×D₄⋊C₈	1	1	yes	characteristic
164	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
165	C₂×C₁₇	#1492	1	1	yes	characteristic
166	C₂×C₁₇	#1493	1	1	yes	characteristic
167	C₂×C₁₇	#1984	1	1	yes	characteristic
168	C₂×C₁₇	#1990	1	1	yes	characteristic
169	C₂×C₁₇	#1994	1	1	yes	characteristic
170	C₂²×Q₈	—	34	1	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	34	2	no	—
173	C₄⋊Q₈	—	17	4	no	—
174	C₄⋊Q₈	—	34	2	no	—
175	C₄×Q₈	—	68	2	no	—
176	C₄×Q₈	—	68	2	no	—
177	C₂×C₄⋊C₄	—	17	2	no	—
178	C₂×C₄⋊C₄	—	17	2	no	—
179	C₂×C₄⋊C₄	—	34	1	no	—
180	C₂×C₄⋊C₄	—	34	1	no	—
181	C₂×C₄⋊C₄	—	34	2	no	—
182	C₂×C₄⋊C₄	—	34	1	no	—
183	C₂×C₄²	—	34	2	no	—
184	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
186	C₂×C₄²	D₆₈	1	1	yes	characteristic
187	C₂×C₄²	D₆₈	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
191	C₄⋊C₈	—	34	4	no	—
192	C₄⋊C₈	—	68	2	no	—
193	C₄⋊C₈	—	68	2	no	—
194	C₄×C₈	—	2	4	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	Dic₃₄	1	4	yes	complemented
198	#95	—	34	2	no	—
199	#95	—	34	2	no	—
200	C₁₇	#9870	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₈	—	34	4	no	—
202	C₂×Q₈	—	68	2	no	—
203	C₂×Q₈	—	68	2	no	—
204	C₂²×C₄	—	34	2	no	—
205	C₂²×C₄	—	34	2	no	—
206	C₂²×C₄	#63227	1	1	yes	characteristic
207	C₂²×C₄	#63227	1	1	yes	characteristic
208	C₂²×C₄	#63227	1	1	yes	characteristic
209	C₂²×C₄	#63227	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	68	2	no	—
217	C₂×C₈	—	68	2	no	—
218	C₂×C₈	—	68	2	no	—
219	C₄⋊C₄	—	17	4	no	—
220	C₄⋊C₄	—	34	4	no	—
221	C₄⋊C₄	—	34	2	no	—
222	C₄⋊C₄	—	34	4	no	—
223	C₄⋊C₄	—	68	2	no	—
224	C₄⋊C₄	—	68	4	no	—
225	C₄⋊C₄	—	68	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	68	2	no	—
230	C₄²	—	68	2	no	—
231	C₄²	C₈×D₁₇	1	2	yes	—
232	C₄²	#63218	1	2	yes	—
233	C₄²	#63219	1	2	yes	—
234	C₄²	#63219	1	2	yes	—
235	C₄²	D₁₃₆	1	2	yes	—
236	C₄²	Dic₆₈	1	2	yes	—
237	C₄²	C₂×Dic₃₄	1	2	yes	—
238	C₄²	#63252	1	2	yes	—
239	C₂³	C₂×#63227	1	1	yes	characteristic
240	Q₈	—	68	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	34	4	no	—
248	C₂×C₄	—	68	2	no	—
249	C₂×C₄	—	68	2	no	—
250	C₂×C₄	—	68	2	no	—
251	C₂×C₄	—	68	2	no	—
252	C₂×C₄	—	68	2	no	—
253	C₂×C₄	#10587476	1	2	yes	—
254	C₂×C₄	#10587476	1	2	yes	—
255	C₂×C₄	#10587487	1	2	yes	—
256	C₂×C₄	#10587487	1	2	yes	—
257	C₂×C₄	#10587491	1	2	yes	—
258	C₂×C₄	#10587491	1	2	yes	—
259	C₂×C₄	#10587492	1	2	yes	—
260	C₂×C₄	#10587492	1	2	yes	—
261	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
262	C₂×C₄	#10587555	1	1	yes	characteristic
263	C₂×C₄	#10587561	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
265	C₂×C₄	C₂×#63218	1	1	yes	characteristic
266	C₂×C₄	#10587587	1	2	yes	—
267	C₂×C₄	C₂×#63219	1	1	yes	characteristic
268	C₂×C₄	C₂×#63219	1	1	yes	characteristic
269	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
270	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
271	C₂×C₄	#10587610	1	2	yes	—
272	C₈	—	2	8	no	—
273	C₈	—	136	2	no	—
274	C₂²	#11762715	1	1	yes	characteristic
275	C₂²	C₂×#10587491	1	1	yes	characteristic
276	C₂²	#11762775	1	1	yes	characteristic
277	C₂²	C₂×#10587492	1	1	yes	characteristic
278	C₂²	C₂×#10587487	1	1	yes	characteristic
279	C₂²	C₂×#10587476	1	1	yes	characteristic
280	C₂²	C₂×#10587476	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	68	4	no	—
285	C₄	—	136	2	no	—
286	C₄	#11762538	1	4	yes	—
287	C₄	#11762541	1	4	yes	—
288	C₄	C₈×Dic₃₄	1	2	yes	—
289	C₄	#11762939	1	2	yes	—
290	C₄	#11763156	1	1	yes	characteristic
291	C₄	#11763160	1	1	yes	characteristic
292	C₄	#11763207	1	1	yes	characteristic
293	C₄	#11763214	1	1	yes	characteristic
294	C₂	C₂×#11762538	1	1	yes	characteristic
295	C₂	C₂×#11762541	1	1	yes	characteristic
296	C₂	#14527236	1	1	yes	characteristic
297	C₂	#14527240	1	1	yes	characteristic
298	C₂	#14529720	1	1	yes	characteristic
299	C₂	#14529745	1	1	yes	characteristic
300	C₂	#14529761	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#9870
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
38. #10587555	C₈	semidirect
44. C₄×C₈×C₁₇	Q₈	semidirect
197. C₄×C₈	Dic₃₄	semidirect
200. C₁₇	#9870	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587555	2
#14528936	2
#14526624	2
C₄²×C₈×C₁₇	2
#9870	17