Subgroups — #16923839

Order 4352.

700 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#63223	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#63223	C₂	1	1	yes	maximal, characteristic
3	#14526620	C₂	1	4	yes	maximal, complemented
4	C₁₇×#2920	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	—	2	1	no	—
6	C₂²×#63223	C₄	1	2	yes	—
7	C₄×#63223	—	2	1	no	—
8	C₄×#63223	—	2	1	no	—
9	C₄×#63223	—	2	1	no	—
10	C₄×#63223	—	2	1	no	—
11	C₄×#63223	C₄	1	2	yes	—
12	C₄×#63223	C₄	1	2	yes	—
13	C₄×#63223	C₄	1	2	yes	—
14	C₄×#63223	C₂²	1	2	yes	—
15	C₂×#10587474	—	2	1	no	—
16	C₂×#10587474	—	2	1	no	—
17	C₂×C₄×#3622	—	2	1	no	—
18	C₂×C₄×#3622	C₂²	1	2	yes	—
19	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#432	—	2	1	no	—
21	C₁₇×#432	C₄	1	2	yes	—
22	C₁₇×#431	—	2	1	no	—
23	C₁₇×#431	C₂²	1	2	yes	—
24	C₁₇×#423	C₄	1	2	yes	—
25	C₁₇×#407	—	2	1	no	—
26	C₄×M₁₆×C₁₇	—	2	1	no	—
27	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₇	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₁₇	—	2	2	no	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	D₄	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	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
43	C₂×#63223	—	2	8	no	—
44	C₂×#63223	—	2	8	no	—
45	C₂×#63223	—	2	2	no	—
46	C₂×#63223	—	2	2	no	—
47	C₂×#63223	—	2	2	no	—
48	C₂×#63223	C₂×C₄	1	4	yes	—
49	C₂²×#3622	—	2	2	no	—
50	C₂²×#3622	—	2	2	no	—
51	C₁₇×C₄⋊C₈	—	2	2	no	—
52	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₇×C₂²⋊C₈	—	2	2	no	—
54	C₁₇×C₂²⋊C₈	—	2	2	no	—
55	C₁₇×#97	—	2	2	no	—
56	C₄×C₈×C₁₇	—	2	2	no	—
57	#10587474	—	2	2	no	—
58	#10587474	—	2	2	no	—
59	#10587474	—	2	2	no	—
60	#10587474	—	2	2	no	—
61	C₄×#3622	—	2	2	no	—
62	C₄×#3622	—	2	2	no	—
63	C₄×#3622	—	2	2	no	—
64	C₄×#3622	C₂×C₄	1	4	yes	—
65	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	—	2	4	no	—
67	C₂²×C₄×C₁₇	—	2	4	no	—
68	C₂²×C₄×C₁₇	—	4	2	no	—
69	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
87	M₁₆×C₁₇	—	4	4	no	—
88	C₂×C₈×C₁₇	—	2	4	no	—
89	C₄²×C₁₇	C₄²	1	1	yes	characteristic
90	C₄²×C₁₇	C₄²	1	1	yes	characteristic
91	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	#63223	—	2	8	no	—
106	#63223	—	2	8	no	—
107	#63223	—	2	8	no	—
108	#63223	—	2	4	no	—
109	#63223	M₁₆	1	8	yes	complemented
110	C₂×#3622	—	2	8	no	—
111	C₂×#3622	—	2	8	no	—
112	C₂×#3622	—	2	4	no	—
113	C₂×#3622	—	2	4	no	—
114	#7887	—	17	1	no	maximal, Sylow
115	C₂³×C₁₇	—	2	2	no	—
116	C₂³×C₁₇	—	2	2	no	—
117	C₂³×C₁₇	—	4	1	no	—
118	C₂³×C₁₇	#95	1	1	yes	characteristic
119	C₂³×C₁₇	#95	1	1	yes	characteristic
120	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	2	2	no	—
124	C₂×C₄×C₁₇	—	2	2	no	—
125	C₂×C₄×C₁₇	—	2	2	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	2	2	no	—
129	C₂×C₄×C₁₇	—	2	2	no	—
130	C₂×C₄×C₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	2	2	no	—
132	C₂×C₄×C₁₇	—	2	2	no	—
133	C₂×C₄×C₁₇	—	2	2	no	—
134	C₂×C₄×C₁₇	—	4	1	no	—
135	C₂×C₄×C₁₇	—	4	1	no	—
136	C₂×C₄×C₁₇	—	4	1	no	—
137	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
138	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
139	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
140	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
141	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
142	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
143	C₂×C₄×C₁₇	#104	1	4	yes	—
144	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₇	C₂×M₁₆	1	4	yes	—
151	C₈×C₁₇	—	4	4	no	—
152	#3622	—	2	8	no	—
153	#3622	—	2	8	no	—
154	#2920	D₁₇	1	1	yes	characteristic
155	C₂×C₄×M₁₆	—	17	1	no	—
156	C₂×C₄×M₁₆	—	17	1	no	—
157	#1278	—	17	4	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	—	2	1	no	—
161	C₂²×C₁₇	—	2	1	no	—
162	C₂²×C₁₇	—	2	1	no	—
163	C₂²×C₁₇	—	2	1	no	—
164	C₂²×C₁₇	—	2	1	no	—
165	C₂²×C₁₇	—	4	4	no	—
166	C₂²×C₁₇	#339	1	2	yes	—
167	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
168	C₂²×C₁₇	#423	1	2	yes	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	2	1	no	—
171	C₄×C₁₇	—	2	1	no	—
172	C₄×C₁₇	—	2	1	no	—
173	C₄×C₁₇	—	2	1	no	—
174	C₄×C₁₇	—	2	1	no	—
175	C₄×C₁₇	—	2	1	no	—
176	C₄×C₁₇	#339	1	2	yes	—
177	C₄×C₁₇	C₄×M₁₆	1	2	yes	—
178	C₄×C₁₇	#407	1	2	yes	—
179	C₄×C₁₇	C₂×#104	1	2	yes	—
180	C₄×C₁₇	#430	1	2	yes	—
181	C₂²×M₁₆	—	17	2	no	—
182	C₂²×M₁₆	—	34	1	no	—
183	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
184	#432	—	2	1	no	—
185	#432	Dic₁₇	1	2	yes	—
186	#431	—	2	1	no	—
187	#431	C₂×D₁₇	1	2	yes	—
188	#423	Dic₁₇	1	2	yes	—
189	#407	—	2	1	no	—
190	C₄×M₁₆	—	2	1	no	—
191	C₄×M₁₆	—	17	2	no	—
192	C₄×M₁₆	—	17	2	no	—
193	C₄×M₁₆	—	17	2	no	—
194	C₄×M₁₆	—	17	2	no	—
195	C₄×M₁₆	—	34	1	no	—
196	C₄×M₁₆	—	34	1	no	—
197	C₄×M₁₆	—	34	1	no	—
198	C₄×M₁₆	—	34	1	no	—
199	C₂×#97	—	34	1	no	—
200	C₂×#97	—	34	1	no	—
201	C₂×C₄×C₈	—	17	2	no	—
202	C₂×C₄×C₈	—	34	1	no	—
203	C₂×C₁₇	—	2	2	no	—
204	C₂×C₁₇	—	2	2	no	—
205	C₂×C₁₇	—	4	1	no	—
206	C₂×C₁₇	#1728	1	1	yes	characteristic
207	C₂×C₁₇	C₂×#339	1	1	yes	characteristic
208	C₂×C₁₇	#1738	1	1	yes	characteristic
209	C₂³×C₄	Dic₃₄	1	1	yes	characteristic
210	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
211	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	17	4	no	—
214	C₂×M₁₆	—	34	8	no	—
215	C₂×M₁₆	—	34	8	no	—
216	C₂×M₁₆	—	34	2	no	—
217	C₂×M₁₆	—	34	2	no	—
218	C₂×M₁₆	—	34	2	no	—
219	C₂²×C₈	—	34	2	no	—
220	C₂²×C₈	—	34	2	no	—
221	C₂×C₄²	Dic₃₄	1	1	yes	characteristic
222	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
225	C₂×C₄²	D₆₈	1	1	yes	characteristic
226	C₂×C₄²	D₆₈	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
228	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	C₂×Dic₁₇	1	4	yes	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	#97	—	2	2	no	—
238	#97	—	34	2	no	—
239	#97	—	34	2	no	—
240	#97	—	34	2	no	—
241	#97	—	34	2	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	—	17	4	no	—
244	C₄×C₈	—	34	2	no	—
245	C₄×C₈	—	34	2	no	—
246	C₄×C₈	—	34	2	no	—
247	C₁₇	#7887	1	1	yes	characteristic, complemented, Sylow
248	C₂⁴	#63225	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
253	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
254	C₂²×C₄	#63225	1	1	yes	characteristic
255	C₂²×C₄	#63225	1	1	yes	characteristic
256	C₂²×C₄	#63225	1	1	yes	characteristic
257	C₂²×C₄	#63225	1	1	yes	characteristic
258	C₂²×C₄	#63225	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
261	C₂²×C₄	#63227	1	1	yes	characteristic
262	C₂²×C₄	#63227	1	1	yes	characteristic
263	C₂²×C₄	#63227	1	1	yes	characteristic
264	C₂²×C₄	#63227	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
270	M₁₆	—	4	4	no	—
271	M₁₆	—	17	8	no	—
272	M₁₆	—	34	8	no	—
273	M₁₆	—	34	8	no	—
274	M₁₆	—	34	8	no	—
275	M₁₆	—	34	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	34	8	no	—
278	C₂×C₈	—	34	8	no	—
279	C₂×C₈	—	34	4	no	—
280	C₂×C₈	—	34	4	no	—
281	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
282	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
283	C₄²	#63225	1	1	yes	characteristic
284	C₄²	#63225	1	1	yes	characteristic
285	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
286	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
287	C₄²	#63227	1	1	yes	characteristic
288	C₄²	#63227	1	1	yes	characteristic
289	C₄²	#63227	1	1	yes	characteristic
290	C₄²	#63227	1	1	yes	characteristic
291	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
292	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
294	C₄²	C₂×D₆₈	1	1	yes	characteristic
295	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
296	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	#10587502	1	1	yes	characteristic
301	C₂³	#10587502	1	1	yes	characteristic
302	C₂³	C₂×#63225	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	#10587502	1	1	yes	characteristic
320	C₂×C₄	#10587502	1	1	yes	characteristic
321	C₂×C₄	#10587502	1	1	yes	characteristic
322	C₂×C₄	#10587502	1	1	yes	characteristic
323	C₂×C₄	#10587502	1	1	yes	characteristic
324	C₂×C₄	#10587502	1	1	yes	characteristic
325	C₂×C₄	#10587508	1	4	yes	—
326	C₂×C₄	M₁₆×D₁₇	1	4	yes	—
327	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
328	C₂×C₄	C₂×#63225	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
330	C₂×C₄	C₂×#63227	1	1	yes	characteristic
331	C₂×C₄	C₂×#63227	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	34	8	no	—
335	C₈	—	34	8	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	4	no	—
344	C₂²	#11762568	1	2	yes	—
345	C₂²	C₂×#10587502	1	1	yes	characteristic
346	C₂²	#11762832	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	#11762568	1	2	yes	—
355	C₄	M₁₆×Dic₁₇	1	2	yes	—
356	C₄	#11762772	1	2	yes	—
357	C₄	C₂×#10587508	1	2	yes	—
358	C₄	#11762857	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	2	2	no	—
361	C₂	—	4	1	no	—
362	C₂	#14528477	1	1	yes	characteristic
363	C₂	C₂×#11762568	1	1	yes	characteristic
364	C₂	#14528510	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2920	C₂
socle	C₂²×C₁₇	C₂×#95
prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2920	D₁₇
hypocenter	C₁₇	#7887
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #14526620	C₂	semidirect
109. #63223	M₁₆	semidirect
247. C₁₇	#7887	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#63223	2
#14526620	2
C₁₇×#2920	2
#7887	17