Subgroups — #16927100

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528637	C₂	1	1	yes	maximal, characteristic
2	#14528637	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762551	C₂	1	2	yes	maximal
4	#14526665	C₂	1	2	yes	maximal
5	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×#10587474	C₂²	1	2	yes	—
9	C₂×#10587474	C₂²	1	2	yes	—
10	#11762564	—	2	1	no	—
11	#11762564	—	2	1	no	—
12	#11762551	—	2	4	no	—
13	C₂×C₁₇×#125	C₂²	1	1	yes	characteristic
14	C₁₇×#389	C₄	1	1	yes	characteristic
15	C₁₇×#389	C₄	1	1	yes	characteristic
16	C₁₇×#387	—	2	1	no	—
17	C₁₇×#387	—	2	1	no	—
18	C₁₇×#382	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
23	C₄³×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#125	—	2	1	no	—
25	C₁₇×#125	—	2	1	no	—
26	C₁₇×#125	D₄	1	2	yes	—
27	C₁₇×#125	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	—	2	1	no	—
37	C₂×C₄²×C₁₇	—	2	1	no	—
38	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
45	C₂²×#3622	—	2	2	no	—
46	C₂²×#3622	—	2	2	no	—
47	C₁₇×#95	—	2	1	no	—
48	C₁₇×#95	—	2	1	no	—
49	C₁₇×#95	—	2	1	no	—
50	C₁₇×#95	—	2	1	no	—
51	C₁₇×#95	—	2	1	no	—
52	C₁₇×#95	—	2	1	no	—
53	C₄⋊#3622	—	2	4	no	—
54	C₄⋊#3622	—	2	4	no	—
55	#10587474	—	2	2	no	—
56	#10587474	—	2	2	no	—
57	#10587474	—	2	2	no	—
58	#10587474	—	2	2	no	—
59	C₂²×C₄×C₁₇	—	2	1	no	—
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	2	no	—
73	C₄⋊C₄×C₁₇	—	2	2	no	—
74	C₄⋊C₄×C₁₇	—	2	2	no	—
75	C₄⋊C₄×C₁₇	—	2	2	no	—
76	C₄⋊C₄×C₁₇	—	4	1	no	—
77	C₄⋊C₄×C₁₇	—	4	1	no	—
78	C₄⋊C₄×C₁₇	—	4	2	no	—
79	C₄⋊C₄×C₁₇	—	4	2	no	—
80	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
81	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₁₇	—	2	2	no	—
83	C₄²×C₁₇	—	2	2	no	—
84	C₄²×C₁₇	—	2	2	no	—
85	C₄²×C₁₇	—	4	1	no	—
86	C₄²×C₁₇	—	4	1	no	—
87	C₄²×C₁₇	—	4	1	no	—
88	C₄²×C₁₇	—	4	2	no	—
89	C₄²×C₁₇	—	4	2	no	—
90	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₁₇	#41	1	1	yes	characteristic
93	C₄²×C₁₇	#41	1	1	yes	characteristic
94	C₂×#3622	—	2	4	no	—
95	C₂×#3622	—	2	4	no	—
96	C₂×#3622	—	4	2	no	—
97	C₂×#3622	—	4	2	no	—
98	C₂×#3622	—	4	2	no	—
99	C₂×#3622	—	4	2	no	—
100	#8582	—	17	1	no	maximal, Sylow
101	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	2	no	—
115	C₂×C₄×C₁₇	—	2	2	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	—	4	1	no	—
120	C₂×C₄×C₁₇	—	4	1	no	—
121	C₂×C₄×C₁₇	—	4	1	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	—	4	1	no	—
126	C₂×C₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	#101	1	2	yes	—
130	C₂×C₄×C₁₇	#104	1	2	yes	—
131	C₂×C₄×C₁₇	#104	1	2	yes	—
132	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
134	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
138	#3622	—	4	4	no	—
139	#3622	—	4	4	no	—
140	C₄×#125	D₁₇	1	1	yes	characteristic
141	#1767	—	17	1	no	—
142	#1767	—	17	1	no	—
143	C₂×#330	—	17	2	no	—
144	#1295	—	17	2	no	—
145	C₂²×C₁₇	#380	1	1	yes	characteristic
146	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
147	C₂²×C₁₇	#413	1	1	yes	characteristic
148	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
149	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
150	C₂²×C₁₇	#421	1	1	yes	characteristic
151	C₂²×C₁₇	#421	1	1	yes	characteristic
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	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	2	no	—
159	C₄×C₁₇	—	2	2	no	—
160	C₄×C₁₇	—	2	2	no	—
161	C₄×C₁₇	—	4	2	no	—
162	C₄×C₁₇	—	4	2	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	—	8	1	no	—
165	C₄×C₁₇	—	8	1	no	—
166	C₂×#125	C₂×D₁₇	1	1	yes	characteristic
167	C₂×C₄⋊C₈	—	34	1	no	—
168	C₂×C₄⋊C₈	—	34	1	no	—
169	C₂×#97	—	17	2	no	—
170	C₂×#97	—	17	2	no	—
171	#389	Dic₁₇	1	1	yes	characteristic
172	#389	Dic₁₇	1	1	yes	characteristic
173	#387	—	2	1	no	—
174	#387	—	2	1	no	—
175	#382	C₂×D₁₇	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	2	1	no	—
177	C₄×C₄⋊C₄	—	2	1	no	—
178	C₄×C₄⋊C₄	Dic₁₇	1	1	yes	characteristic
179	C₄×C₄⋊C₄	Dic₁₇	1	1	yes	characteristic
180	C₄³	C₂×D₁₇	1	1	yes	characteristic
181	#336	—	34	1	no	—
182	#336	—	34	1	no	—
183	#330	—	34	4	no	—
184	C₂×C₁₇	#1524	1	1	yes	characteristic
185	C₂×C₁₇	#1528	1	1	yes	characteristic
186	C₂×C₁₇	#1531	1	1	yes	characteristic
187	C₂×C₁₇	#1535	1	1	yes	characteristic
188	C₂×C₁₇	#1801	1	1	yes	characteristic
189	C₂×C₁₇	#1806	1	1	yes	characteristic
190	C₂×C₁₇	#1806	1	1	yes	characteristic
191	C₂²×C₈	—	34	2	no	—
192	C₂²×C₈	—	34	2	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	Dic₃₄	1	2	yes	—
196	#125	D₆₈	1	2	yes	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	—	34	4	no	—
215	C₄⋊C₈	—	34	4	no	—
216	#97	—	34	2	no	—
217	#97	—	34	2	no	—
218	#97	—	34	2	no	—
219	#97	—	34	2	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	C₁₇	#8582	1	1	yes	characteristic, complemented, Sylow
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
238	C₂×C₈	—	34	4	no	—
239	C₂×C₈	—	34	4	no	—
240	C₂×C₈	—	68	2	no	—
241	C₂×C₈	—	68	2	no	—
242	C₂×C₈	—	68	2	no	—
243	C₂×C₈	—	68	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
255	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
265	C₄²	C₂×D₆₈	1	1	yes	characteristic
266	C₄²	#63252	1	1	yes	characteristic
267	C₄²	#63252	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	#10587507	1	2	yes	—
297	C₂×C₄	#10587508	1	2	yes	—
298	C₂×C₄	#10587508	1	2	yes	—
299	C₂×C₄	#10587610	1	1	yes	characteristic
300	C₂×C₄	#10587610	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
302	C₂×C₄	#10587612	1	1	yes	characteristic
303	C₂×C₄	#10587616	1	1	yes	characteristic
304	C₂×C₄	#10587616	1	1	yes	characteristic
305	C₈	—	68	4	no	—
306	C₈	—	68	4	no	—
307	C₂²	#11762669	1	1	yes	characteristic
308	C₂²	C₂×#10587507	1	1	yes	characteristic
309	C₂²	#11762793	1	1	yes	characteristic
310	C₂²	C₂×#10587508	1	1	yes	characteristic
311	C₂²	C₂×#10587508	1	1	yes	characteristic
312	C₂²	#11762826	1	1	yes	characteristic
313	C₂²	#11762826	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₂	#14527372	1	1	yes	characteristic
329	C₂	#14527386	1	1	yes	characteristic
330	C₂	#14527399	1	1	yes	characteristic
331	C₂	#14527415	1	1	yes	characteristic
332	C₂	#14528780	1	1	yes	characteristic
333	C₂	#14528804	1	1	yes	characteristic
334	C₂	#14528804	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#8582
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
226. C₁₇	#8582	semidirect

Maximal subgroups

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

subgroup	index
#14528637	2
C₂×#11762551	2
#14526665	2
C₄×C₁₇×#125	2
#8582	17