Subgroups — #16927131

Order 4352.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528936	C₂	1	1	yes	maximal, characteristic
2	#14528901	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762555	C₂	1	2	yes	maximal
4	#14526667	C₂	1	2	yes	maximal
5	C₄×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×C₄⋊#3622	C₂²	1	2	yes	—
9	C₂×C₄⋊#3622	C₂²	1	2	yes	—
10	#11762564	—	2	1	no	—
11	#11762564	—	2	1	no	—
12	#11762555	—	2	4	no	—
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
15	C₁₇×#391	—	2	1	no	—
16	C₁₇×#389	—	2	1	no	—
17	C₁₇×#389	C₄	1	2	yes	—
18	C₁₇×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
21	C₄³×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₇	—	2	1	no	—
23	C₁₇×C₄⋊Q₈	—	2	1	no	—
24	C₁₇×C₄⋊Q₈	—	2	1	no	—
25	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
26	C₁₇×C₄⋊Q₈	Q₈	1	2	yes	—
27	C₄×Q₈×C₁₇	—	4	2	no	—
28	C₄×Q₈×C₁₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₇	—	2	2	no	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	D₄	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	C₂³	1	1	yes	characteristic
43	C₂²×#3622	—	2	2	no	—
44	C₂²×#3622	—	2	2	no	—
45	C₁₇×#95	—	2	2	no	—
46	C₁₇×#95	—	2	2	no	—
47	C₄⋊#3622	—	2	2	no	—
48	C₄⋊#3622	—	2	2	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄⋊#3622	—	4	2	no	—
52	C₄⋊#3622	—	4	2	no	—
53	C₂×Q₈×C₁₇	—	2	2	no	—
54	C₂×Q₈×C₁₇	—	2	2	no	—
55	C₂×Q₈×C₁₇	—	4	1	no	—
56	C₂×Q₈×C₁₇	—	4	1	no	—
57	C₂×Q₈×C₁₇	—	4	1	no	—
58	C₂×Q₈×C₁₇	—	4	1	no	—
59	C₂²×C₄×C₁₇	—	2	2	no	—
60	C₂²×C₄×C₁₇	—	2	2	no	—
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₇	—	2	2	no	—
69	C₄⋊C₄×C₁₇	—	2	2	no	—
70	C₄⋊C₄×C₁₇	—	2	2	no	—
71	C₄⋊C₄×C₁₇	—	4	2	no	—
72	C₄⋊C₄×C₁₇	—	4	4	no	—
73	C₄⋊C₄×C₁₇	—	4	1	no	—
74	C₄⋊C₄×C₁₇	—	4	1	no	—
75	C₄⋊C₄×C₁₇	—	4	2	no	—
76	C₄⋊C₄×C₁₇	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₇	—	2	2	no	—
78	C₄²×C₁₇	—	2	2	no	—
79	C₄²×C₁₇	—	2	1	no	—
80	C₄²×C₁₇	—	2	2	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	—	4	1	no	—
83	C₄²×C₁₇	—	4	4	no	—
84	C₄²×C₁₇	D₈	1	2	yes	—
85	C₄²×C₁₇	SD₁₆	1	2	yes	—
86	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₁₇	#41	1	1	yes	characteristic
89	C₄²×C₁₇	#41	1	1	yes	characteristic
90	C₂×#3622	—	4	2	no	—
91	C₂×#3622	—	4	2	no	—
92	C₂×#3622	—	4	2	no	—
93	C₂×#3622	—	4	2	no	—
94	C₂×#3622	—	4	2	no	—
95	C₂×#3622	—	4	2	no	—
96	#8589	—	17	1	no	maximal, Sylow
97	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Q₈×C₁₇	—	4	2	no	—
99	Q₈×C₁₇	—	4	2	no	—
100	Q₈×C₁₇	—	4	2	no	—
101	Q₈×C₁₇	—	4	2	no	—
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	4	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	—	4	2	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	—	4	2	no	—
117	C₂×C₄×C₁₇	—	4	2	no	—
118	C₂×C₄×C₁₇	—	4	2	no	—
119	C₂×C₄×C₁₇	#101	1	2	yes	—
120	C₂×C₄×C₁₇	#102	1	2	yes	—
121	C₂×C₄×C₁₇	#102	1	2	yes	—
122	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₇	#135	1	2	yes	—
131	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₇	#137	1	2	yes	—
133	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
134	#3622	—	8	2	no	—
135	#3622	—	8	2	no	—
136	C₄×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
137	#1842	—	17	1	no	—
138	#1831	—	17	1	no	—
139	C₂×#332	—	17	2	no	—
140	#1296	—	17	2	no	—
141	C₂²×C₁₇	#380	1	1	yes	characteristic
142	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
143	C₂²×C₁₇	#413	1	1	yes	characteristic
144	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
145	C₂²×C₁₇	#416	1	1	yes	characteristic
146	C₂²×C₁₇	#417	1	1	yes	characteristic
147	C₂²×C₁₇	#419	1	1	yes	characteristic
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	2	no	—
152	C₄×C₁₇	—	2	2	no	—
153	C₄×C₁₇	—	4	4	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	8	2	no	—
156	C₄×C₁₇	#459	1	1	yes	characteristic
157	C₄×C₁₇	#461	1	1	yes	characteristic
158	C₄×C₁₇	#464	1	2	yes	—
159	C₄×C₁₇	#474	1	1	yes	characteristic
160	C₄×C₁₇	#476	1	1	yes	characteristic
161	C₄×C₁₇	#479	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
163	C₂×C₄×Q₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	17	2	no	—
165	C₂×C₄⋊C₈	—	17	2	no	—
166	C₂×C₄⋊C₈	—	34	1	no	—
167	C₂×C₄⋊C₈	—	34	1	no	—
168	#391	—	2	1	no	—
169	#389	—	2	1	no	—
170	#389	Dic₁₇	1	2	yes	—
171	#384	C₂×D₁₇	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	Dic₁₇	1	2	yes	—
174	C₄³	C₂×D₁₇	1	1	yes	characteristic
175	#336	—	34	1	no	—
176	#336	—	34	1	no	—
177	#332	—	34	4	no	—
178	C₂×C₁₇	#1540	1	1	yes	characteristic
179	C₂×C₁₇	#1546	1	1	yes	characteristic
180	C₂×C₁₇	#1552	1	1	yes	characteristic
181	C₂×C₁₇	#1559	1	1	yes	characteristic
182	C₂×C₁₇	#1801	1	1	yes	characteristic
183	C₂×C₁₇	#1802	1	1	yes	characteristic
184	C₂×C₁₇	#1805	1	1	yes	characteristic
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×C₈	—	34	2	no	—
187	C₂²×C₈	—	34	2	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	Dic₃₄	1	2	yes	—
191	C₄⋊Q₈	D₆₈	1	2	yes	—
192	C₄×Q₈	—	4	2	no	—
193	C₄×Q₈	—	4	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
208	C₄⋊C₈	—	34	2	no	—
209	C₄⋊C₈	—	34	2	no	—
210	C₄⋊C₈	—	34	2	no	—
211	C₄⋊C₈	—	34	2	no	—
212	C₄⋊C₈	—	68	2	no	—
213	C₄⋊C₈	—	68	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	C₁₇	#8589	1	1	yes	characteristic, complemented, Sylow
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
232	C₂×C₈	—	68	2	no	—
233	C₂×C₈	—	68	2	no	—
234	C₂×C₈	—	68	2	no	—
235	C₂×C₈	—	68	2	no	—
236	C₂×C₈	—	68	2	no	—
237	C₂×C₈	—	68	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₄⋊Dic₁₇	1	4	yes	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	4	no	—
254	C₄²	#63228	1	2	yes	—
255	C₄²	#63229	1	2	yes	—
256	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
257	C₄²	C₂×D₆₈	1	1	yes	characteristic
258	C₄²	#63252	1	1	yes	characteristic
259	C₄²	#63252	1	1	yes	characteristic
260	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
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	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	#10587503	1	2	yes	—
283	C₂×C₄	#10587503	1	2	yes	—
284	C₂×C₄	#10587507	1	2	yes	—
285	C₂×C₄	#10587610	1	1	yes	characteristic
286	C₂×C₄	#10587610	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
288	C₂×C₄	#10587612	1	1	yes	characteristic
289	C₂×C₄	#10587616	1	1	yes	characteristic
290	C₂×C₄	#10587616	1	1	yes	characteristic
291	C₂×C₄	C₂×#63228	1	1	yes	characteristic
292	C₂×C₄	C₂×#63229	1	1	yes	characteristic
293	C₂×C₄	#10587628	1	2	yes	—
294	C₂×C₄	#10587635	1	1	yes	characteristic
295	C₂×C₄	#10587636	1	2	yes	—
296	C₂×C₄	#10587637	1	1	yes	characteristic
297	C₈	—	136	2	no	—
298	C₈	—	136	2	no	—
299	C₂²	#11762669	1	1	yes	characteristic
300	C₂²	C₂×#10587507	1	1	yes	characteristic
301	C₂²	#11762793	1	1	yes	characteristic
302	C₂²	C₂×#10587503	1	1	yes	characteristic
303	C₂²	#11762805	1	1	yes	characteristic
304	C₂²	#11762810	1	1	yes	characteristic
305	C₂²	#11762818	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	1	no	—
313	C₄	—	8	2	no	—
314	C₄	#11763004	1	1	yes	characteristic
315	C₄	#11763014	1	1	yes	characteristic
316	C₄	#11763031	1	2	yes	—
317	C₄	#11763089	1	1	yes	characteristic
318	C₄	#11763099	1	1	yes	characteristic
319	C₄	#11763116	1	2	yes	—
320	C₂	#14527433	1	1	yes	characteristic
321	C₂	#14527456	1	1	yes	characteristic
322	C₂	#14527484	1	1	yes	characteristic
323	C₂	#14527514	1	1	yes	characteristic
324	C₂	#14528780	1	1	yes	characteristic
325	C₂	#14528784	1	1	yes	characteristic
326	C₂	#14528799	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×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#8589
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
216. C₁₇	#8589	semidirect

Maximal subgroups

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

subgroup	index
#14528936	2
#14528901	2
C₂×#11762555	2
#14526667	2
C₄×C₁₇×C₄⋊Q₈	2
#8589	17