Subgroups — #16923752

Order 4352.

552 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537417	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762862	C₂	1	1	yes	maximal, characteristic
3	#14526615	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#432	C₂	1	1	yes	maximal, characteristic
5	#11762866	—	2	1	no	—
6	#11762866	—	2	1	no	—
7	#11762862	—	2	1	no	—
8	#11762862	—	2	1	no	—
9	#11762862	C₄	1	2	yes	—
10	#11762862	C₄	1	2	yes	—
11	#11762862	C₂²	1	2	yes	—
12	#11762862	C₂²	1	2	yes	—
13	#11762833	—	2	1	no	—
14	C₂×C₄⋊#3622	—	2	1	no	—
15	#11762773	—	2	1	no	—
16	C₂×C₂²⋊#3622	—	2	1	no	—
17	C₄×#63223	C₄	1	2	yes	—
18	C₂×C₄×#3622	C₄	1	2	yes	—
19	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#432	—	2	1	no	—
21	C₁₇×#432	—	2	1	no	—
22	C₁₇×#432	—	2	1	no	—
23	C₁₇×#432	—	2	1	no	—
24	C₂×C₁₇×C₄⋊C₈	C₄	1	2	yes	—
25	C₂×C₁₇×C₂²⋊C₈	C₄	1	2	yes	—
26	C₂×C₁₇×#97	C₂²	1	2	yes	—
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	C₂²×C₈×C₁₇	C₂×C₄	1	4	yes	—
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	2	no	—
44	C₂²×#3622	—	2	2	no	—
45	C₁₇×C₄⋊C₈	—	2	2	no	—
46	C₁₇×C₄⋊C₈	—	2	2	no	—
47	C₁₇×C₂²⋊C₈	—	2	2	no	—
48	C₁₇×C₂²⋊C₈	—	2	2	no	—
49	C₁₇×#97	—	2	2	no	—
50	C₁₇×#97	—	2	2	no	—
51	C₂²⋊#3622	—	2	2	no	—
52	C₂²⋊#3622	—	2	2	no	—
53	C₂²⋊#3622	—	2	2	no	—
54	C₂²⋊#3622	—	2	2	no	—
55	C₄⋊#3622	—	2	2	no	—
56	C₄⋊#3622	—	2	2	no	—
57	C₄⋊#3622	—	2	2	no	—
58	C₄⋊#3622	—	2	2	no	—
59	#10587474	—	2	2	no	—
60	C₄×#3622	—	2	2	no	—
61	C₄×#3622	C₂×C₄	1	4	yes	—
62	C₄×#3622	C₂×C₄	1	4	yes	—
63	C₂⁴×C₁₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	4	no	—
65	C₂²×C₄×C₁₇	—	4	2	no	—
66	C₂²×C₄×C₁₇	—	4	2	no	—
67	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₁₇	—	2	4	no	—
86	C₂×C₈×C₁₇	—	2	8	no	—
87	C₄²×C₁₇	C₄²	1	1	yes	characteristic
88	C₄²×C₁₇	C₄²	1	1	yes	characteristic
89	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₇	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₇	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
103	#63223	—	4	4	no	—
104	C₂×#3622	—	2	4	no	—
105	C₂×#3622	—	2	4	no	—
106	C₂×#3622	—	4	4	no	—
107	#7863	—	17	1	no	maximal, Sylow
108	C₂³×C₁₇	—	2	2	no	—
109	C₂³×C₁₇	—	2	2	no	—
110	C₂³×C₁₇	—	4	1	no	—
111	C₂³×C₁₇	#95	1	1	yes	characteristic
112	C₂³×C₁₇	#95	1	1	yes	characteristic
113	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	—	2	2	no	—
115	C₂×C₄×C₁₇	—	2	2	no	—
116	C₂×C₄×C₁₇	—	2	2	no	—
117	C₂×C₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	2	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
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	—	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	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	1	no	—
131	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
132	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
133	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
134	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
135	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
136	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₇	#131	1	4	yes	—
144	C₈×C₁₇	—	4	4	no	—
145	#3622	—	4	4	no	—
146	#3622	—	4	4	no	—
147	#2920	—	17	1	no	—
148	C₂×#432	D₁₇	1	1	yes	characteristic
149	C₂×#431	—	17	1	no	—
150	#1275	—	17	4	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	4	4	no	—
159	C₂²×C₁₇	H(3,ℤ/4)	1	2	yes	—
160	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
161	C₂²×C₁₇	#424	1	2	yes	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	2	1	no	—
171	C₄×C₁₇	#338	1	2	yes	—
172	C₄×C₁₇	#405	1	2	yes	—
173	C₄×C₁₇	#408	1	2	yes	—
174	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
175	#432	—	2	1	no	—
176	#432	—	2	1	no	—
177	#432	—	2	1	no	—
178	#432	—	2	1	no	—
179	#432	—	34	1	no	—
180	#432	—	34	1	no	—
181	#431	—	17	2	no	—
182	#431	—	17	2	no	—
183	#431	—	17	2	no	—
184	#431	—	17	2	no	—
185	#431	—	34	1	no	—
186	#431	—	34	1	no	—
187	#423	—	34	1	no	—
188	C₂×C₄⋊C₈	—	34	1	no	—
189	C₂×C₄⋊C₈	Dic₁₇	1	2	yes	—
190	#407	—	34	1	no	—
191	C₂×C₂²⋊C₈	—	34	1	no	—
192	C₂×C₂²⋊C₈	Dic₁₇	1	2	yes	—
193	C₄×M₁₆	—	17	2	no	—
194	C₂×#97	C₂×D₁₇	1	2	yes	—
195	C₂×C₄×C₈	—	17	2	no	—
196	C₂×C₁₇	—	2	2	no	—
197	C₂×C₁₇	—	2	2	no	—
198	C₂×C₁₇	—	4	1	no	—
199	C₂×C₁₇	#1729	1	1	yes	characteristic
200	C₂×C₁₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
201	C₂×C₁₇	C₂×#338	1	1	yes	characteristic
202	C₂³×C₄	Dic₃₄	1	1	yes	characteristic
203	C₂³×C₄	D₆₈	1	1	yes	characteristic
204	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
205	C₂×M₁₆	—	34	2	no	—
206	C₂²×C₈	—	34	2	no	—
207	C₂²×C₈	C₂×Dic₁₇	1	4	yes	—
208	C₂×C₄²	Dic₃₄	1	1	yes	characteristic
209	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
213	C₂×C₄²	D₆₈	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	34	2	no	—
223	C₄⋊C₈	—	34	2	no	—
224	C₄⋊C₈	—	34	2	no	—
225	C₄⋊C₈	—	34	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	34	2	no	—
229	C₂²⋊C₈	—	34	2	no	—
230	C₂²⋊C₈	—	34	2	no	—
231	C₂²⋊C₈	—	34	2	no	—
232	#97	—	2	2	no	—
233	#97	—	2	2	no	—
234	#97	—	34	2	no	—
235	C₄×C₈	—	17	4	no	—
236	C₄×C₈	—	17	4	no	—
237	C₄×C₈	—	34	2	no	—
238	C₁₇	#7863	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	C₄⋊Dic₁₇	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
244	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
245	C₂²×C₄	#63225	1	1	yes	characteristic
246	C₂²×C₄	#63225	1	1	yes	characteristic
247	C₂²×C₄	#63225	1	1	yes	characteristic
248	C₂²×C₄	#63225	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
250	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
251	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
252	C₂²×C₄	#63227	1	1	yes	characteristic
253	C₂²×C₄	#63227	1	1	yes	characteristic
254	C₂²×C₄	#63227	1	1	yes	characteristic
255	C₂²×C₄	#63227	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
261	M₁₆	—	68	4	no	—
262	C₂×C₈	—	2	8	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	34	4	no	—
265	C₂×C₈	—	34	4	no	—
266	C₂×C₈	—	68	4	no	—
267	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
268	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
269	C₄²	#63225	1	1	yes	characteristic
270	C₄²	#63225	1	1	yes	characteristic
271	C₄²	#63225	1	1	yes	characteristic
272	C₄²	#63225	1	1	yes	characteristic
273	C₄²	#63227	1	1	yes	characteristic
274	C₄²	#63227	1	1	yes	characteristic
275	C₄²	#63227	1	1	yes	characteristic
276	C₄²	#63227	1	1	yes	characteristic
277	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
278	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
280	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	#10587502	1	1	yes	characteristic
287	C₂³	#10587502	1	1	yes	characteristic
288	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	#10587502	1	1	yes	characteristic
307	C₂×C₄	#10587502	1	1	yes	characteristic
308	C₂×C₄	#10587502	1	1	yes	characteristic
309	C₂×C₄	#10587502	1	1	yes	characteristic
310	C₂×C₄	#10587502	1	1	yes	characteristic
311	C₂×C₄	#10587502	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#63225	1	1	yes	characteristic
314	C₂×C₄	C₂×#63225	1	1	yes	characteristic
315	C₂×C₄	C₂×#63227	1	1	yes	characteristic
316	C₂×C₄	C₂×#63227	1	1	yes	characteristic
317	C₂×C₄	#10587634	1	4	yes	—
318	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
319	C₈	—	4	4	no	—
320	C₈	—	68	4	no	—
321	C₈	—	68	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	4	no	—
330	C₂²	#11762566	1	2	yes	—
331	C₂²	C₂×#10587502	1	1	yes	characteristic
332	C₂²	#11762836	1	2	yes	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	#11762567	1	2	yes	—
343	C₄	#11762765	1	2	yes	—
344	C₄	#11762776	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	2	2	no	—
347	C₂	—	4	1	no	—
348	C₂	#14528481	1	1	yes	characteristic
349	C₂	C₂×#11762566	1	1	yes	characteristic
350	C₂	C₂×#11762567	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#432	C₂
socle	C₂²×C₁₇	C₂×#95
prime radical	C₂⁴×C₁₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#432	D₁₇
hypocenter	C₁₇	#7863
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #14526615	C₂	semidirect
238. C₁₇	#7863	semidirect

Maximal subgroups

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

subgroup	index
#14537417	2
C₂×#11762862	2
#14526615	2
C₂×C₁₇×#432	2
#7863	17