Subgroups — #17060978

Order 4352.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529508	C₂	1	2	yes	maximal
2	#14529503	C₂	1	2	yes	maximal
3	#14529350	C₂	1	2	yes	maximal
4	#14529350	C₂	1	2	yes	maximal
5	#14529343	C₂	1	2	yes	maximal
6	#14529343	C₂	1	2	yes	maximal
7	#14528901	C₂	1	1	yes	maximal, characteristic
8	#14528901	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#3622	C₂²	1	2	yes	—
12	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#3622	C₂²	1	2	yes	—
14	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
16	C₂×#10587474	C₂²	1	2	yes	—
17	C₂×C₄×#3622	C₂²	1	2	yes	—
18	#11762564	C₂²	1	2	yes	—
19	#11762564	C₂²	1	2	yes	—
20	#11762564	C₂²	1	2	yes	—
21	#11762564	C₂²	1	2	yes	—
22	#11762564	C₂²	1	2	yes	—
23	#11762564	C₂²	1	2	yes	—
24	#11762564	C₂²	1	2	yes	—
25	#11762564	C₂²	1	2	yes	—
26	C₂×C₁₇×#125	C₄	1	1	yes	characteristic
27	C₁₇×#389	C₄	1	1	yes	characteristic
28	C₁₇×#389	C₄	1	1	yes	characteristic
29	C₁₇×#387	C₄	1	2	yes	—
30	C₁₇×#387	C₄	1	2	yes	—
31	C₁₇×#382	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
36	C₄³×C₁₇	C₂²	1	1	yes	characteristic
37	C₁₇×#125	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
55	C₂²×#3622	C₂³	1	2	yes	—
56	C₂²×#3622	C₂³	1	2	yes	—
57	C₂²×#3622	C₂³	1	2	yes	—
58	C₂²×#3622	C₂³	1	2	yes	—
59	C₁₇×#95	C₂×C₄	1	2	yes	—
60	C₁₇×#95	C₂×C₄	1	2	yes	—
61	C₁₇×#95	C₂×C₄	1	2	yes	—
62	C₁₇×#95	C₂×C₄	1	2	yes	—
63	C₁₇×#95	C₂×C₄	1	2	yes	—
64	C₁₇×#95	C₂×C₄	1	2	yes	—
65	C₄⋊#3622	—	4	2	no	—
66	C₄⋊#3622	—	4	2	no	—
67	C₄⋊#3622	—	4	1	no	—
68	C₄⋊#3622	—	4	1	no	—
69	C₄⋊#3622	—	4	1	no	—
70	C₄⋊#3622	—	4	1	no	—
71	#10587474	—	4	2	no	—
72	C₄×#3622	—	4	2	no	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	—	2	4	no	—
85	C₄⋊C₄×C₁₇	—	2	4	no	—
86	C₄⋊C₄×C₁₇	—	2	2	no	—
87	C₄⋊C₄×C₁₇	—	2	4	no	—
88	C₄⋊C₄×C₁₇	—	2	4	no	—
89	C₄⋊C₄×C₁₇	—	4	1	no	—
90	C₄⋊C₄×C₁₇	—	4	1	no	—
91	C₄⋊C₄×C₁₇	M₁₆	1	4	yes	—
92	C₄²×C₁₇	—	2	2	no	—
93	C₄²×C₁₇	—	2	2	no	—
94	C₄²×C₁₇	—	2	2	no	—
95	C₄²×C₁₇	—	2	2	no	—
96	C₄²×C₁₇	—	2	2	no	—
97	C₄²×C₁₇	—	2	1	no	—
98	C₄²×C₁₇	—	2	1	no	—
99	C₄²×C₁₇	—	2	1	no	—
100	C₄²×C₁₇	—	2	1	no	—
101	C₄²×C₁₇	—	2	2	no	—
102	C₄²×C₁₇	—	2	1	no	—
103	C₄²×C₁₇	—	2	1	no	—
104	C₄²×C₁₇	—	2	2	no	—
105	C₄²×C₁₇	—	4	1	no	—
106	C₂×#3622	—	2	2	no	—
107	C₂×#3622	—	2	2	no	—
108	C₂×#3622	—	2	2	no	—
109	C₂×#3622	—	2	2	no	—
110	C₂×#3622	—	2	2	no	—
111	C₂×#3622	—	2	2	no	—
112	C₂×#3622	—	2	2	no	—
113	C₂×#3622	—	2	2	no	—
114	C₂×#3622	—	2	2	no	—
115	C₂×#3622	—	2	2	no	—
116	C₂×#3622	—	2	2	no	—
117	C₂×#3622	—	2	2	no	—
118	#23371	—	17	1	no	maximal, Sylow
119	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	2	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	1	no	—
127	C₂×C₄×C₁₇	—	2	1	no	—
128	C₂×C₄×C₁₇	—	2	1	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	2	1	no	—
132	C₂×C₄×C₁₇	—	2	2	no	—
133	C₂×C₄×C₁₇	—	2	2	no	—
134	C₂×C₄×C₁₇	—	2	1	no	—
135	C₂×C₄×C₁₇	—	2	1	no	—
136	C₂×C₄×C₁₇	—	2	2	no	—
137	C₂×C₄×C₁₇	—	2	1	no	—
138	C₂×C₄×C₁₇	—	2	1	no	—
139	C₂×C₄×C₁₇	—	2	1	no	—
140	C₂×C₄×C₁₇	—	2	1	no	—
141	C₂×C₄×C₁₇	—	2	2	no	—
142	C₂×C₄×C₁₇	—	2	2	no	—
143	C₂×C₄×C₁₇	—	2	2	no	—
144	C₂×C₄×C₁₇	—	2	2	no	—
145	C₂×C₄×C₁₇	—	2	2	no	—
146	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₁₇	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₁₇	#131	1	2	yes	—
149	C₂×C₄×C₁₇	#131	1	2	yes	—
150	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
155	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
156	#3622	—	4	2	no	—
157	#3622	—	4	2	no	—
158	#3622	—	4	2	no	—
159	#3622	—	4	2	no	—
160	C₄×#125	D₁₇	1	1	yes	characteristic
161	#1940	—	17	2	no	—
162	#1939	—	17	2	no	—
163	#1917	—	17	2	no	—
164	#1917	—	17	2	no	—
165	#1916	—	17	2	no	—
166	#1916	—	17	2	no	—
167	#1831	—	17	1	no	—
168	#1831	—	17	1	no	—
169	C₂²×C₁₇	#519	1	1	yes	characteristic
170	C₂²×C₁₇	#520	1	1	yes	characteristic
171	C₂²×C₁₇	#520	1	1	yes	characteristic
172	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
173	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
174	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
175	C₂²×C₁₇	#568	1	1	yes	characteristic
176	C₄×C₁₇	—	2	2	no	—
177	C₄×C₁₇	—	2	4	no	—
178	C₄×C₁₇	—	2	2	no	—
179	C₄×C₁₇	—	2	2	no	—
180	C₄×C₁₇	—	4	1	no	—
181	C₄×C₁₇	—	4	1	no	—
182	C₄×C₁₇	—	4	1	no	—
183	C₄×C₁₇	—	4	2	no	—
184	C₄×C₁₇	—	4	2	no	—
185	C₄×C₁₇	—	4	2	no	—
186	C₄×C₁₇	#562	1	1	yes	characteristic
187	C₄×C₁₇	#563	1	1	yes	characteristic
188	C₄×C₁₇	#563	1	1	yes	characteristic
189	C₄×C₁₇	#564	1	1	yes	characteristic
190	C₂×#125	Dic₁₇	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	17	1	no	—
192	C₂×C₄⋊C₈	—	17	1	no	—
193	C₂×C₄⋊C₈	—	17	1	no	—
194	C₂×C₄⋊C₈	—	17	1	no	—
195	C₂×C₄⋊C₈	—	17	2	no	—
196	C₂×C₄⋊C₈	—	17	2	no	—
197	C₂×#97	—	17	2	no	—
198	C₂×C₄×C₈	—	17	2	no	—
199	#389	Dic₁₇	1	1	yes	characteristic
200	#389	Dic₁₇	1	1	yes	characteristic
201	#387	Dic₁₇	1	2	yes	—
202	#387	Dic₁₇	1	2	yes	—
203	#382	Dic₁₇	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
205	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
206	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
208	C₄³	C₂×D₁₇	1	1	yes	characteristic
209	#336	—	17	2	no	—
210	#336	—	17	2	no	—
211	#336	—	17	2	no	—
212	#336	—	17	2	no	—
213	#336	—	17	2	no	—
214	#336	—	17	2	no	—
215	#336	—	17	2	no	—
216	#336	—	17	2	no	—
217	C₂×C₁₇	#2382	1	1	yes	characteristic
218	C₂×C₁₇	#2983	1	1	yes	characteristic
219	C₂×C₁₇	#2986	1	1	yes	characteristic
220	C₂×C₁₇	#2990	1	1	yes	characteristic
221	C₂×C₁₇	#2991	1	1	yes	characteristic
222	C₂×C₁₇	#2992	1	1	yes	characteristic
223	C₂×C₁₇	#2992	1	1	yes	characteristic
224	C₂²×C₈	—	17	2	no	—
225	C₂²×C₈	—	17	2	no	—
226	C₂²×C₈	—	17	2	no	—
227	C₂²×C₈	—	17	2	no	—
228	#125	—	2	4	no	—
229	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
237	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
245	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
246	C₄⋊C₈	—	68	2	no	—
247	C₄⋊C₈	—	68	2	no	—
248	C₄⋊C₈	—	68	1	no	—
249	C₄⋊C₈	—	68	1	no	—
250	C₄⋊C₈	—	68	1	no	—
251	C₄⋊C₈	—	68	1	no	—
252	#97	—	68	2	no	—
253	C₄×C₈	—	68	2	no	—
254	#95	C₂×Dic₁₇	1	2	yes	—
255	#95	C₂×Dic₁₇	1	2	yes	—
256	#95	C₂×Dic₁₇	1	2	yes	—
257	#95	C₂×Dic₁₇	1	2	yes	—
258	#95	C₂×Dic₁₇	1	2	yes	—
259	#95	C₂×Dic₁₇	1	2	yes	—
260	C₁₇	#23371	1	1	yes	characteristic, complemented, Sylow
261	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
267	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
268	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
269	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
272	C₂×C₈	—	34	2	no	—
273	C₂×C₈	—	34	2	no	—
274	C₂×C₈	—	34	2	no	—
275	C₂×C₈	—	34	2	no	—
276	C₂×C₈	—	34	2	no	—
277	C₂×C₈	—	34	2	no	—
278	C₂×C₈	—	34	2	no	—
279	C₂×C₈	—	34	2	no	—
280	C₂×C₈	—	34	2	no	—
281	C₂×C₈	—	34	2	no	—
282	C₂×C₈	—	34	2	no	—
283	C₂×C₈	—	34	2	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	#63223	1	4	yes	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	4	1	no	—
306	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	C₂×#63223	1	2	yes	—
334	C₂×C₄	C₂×#63223	1	4	yes	—
335	C₂×C₄	#10587634	1	2	yes	—
336	C₂×C₄	#10587634	1	2	yes	—
337	C₂×C₄	#10587694	1	1	yes	characteristic
338	C₂×C₄	#10587694	1	1	yes	characteristic
339	C₂×C₄	#10587696	1	1	yes	characteristic
340	C₂×C₄	#10587696	1	1	yes	characteristic
341	C₂×C₄	#10587697	1	1	yes	characteristic
342	C₂×C₄	#10587697	1	1	yes	characteristic
343	C₈	—	68	2	no	—
344	C₈	—	68	2	no	—
345	C₈	—	68	2	no	—
346	C₈	—	68	2	no	—
347	C₂²	#11763287	1	1	yes	characteristic
348	C₂²	#11763292	1	1	yes	characteristic
349	C₂²	#11763292	1	1	yes	characteristic
350	C₂²	C₂²×#63223	1	1	yes	characteristic
351	C₂²	C₂×#10587634	1	1	yes	characteristic
352	C₂²	C₂×#10587634	1	1	yes	characteristic
353	C₂²	#11763576	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	#11763557	1	1	yes	characteristic
365	C₄	#11763563	1	1	yes	characteristic
366	C₄	#11763563	1	1	yes	characteristic
367	C₄	C₁₇⋊#564	1	1	yes	characteristic
368	C₂	#14531970	1	1	yes	characteristic
369	C₂	#14537869	1	1	yes	characteristic
370	C₂	#14537886	1	1	yes	characteristic
371	C₂	#14537909	1	1	yes	characteristic
372	C₂	#14537916	1	1	yes	characteristic
373	C₂	#14537923	1	1	yes	characteristic
374	C₂	#14537923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₇
commutator, socle, prime radical	C₂³×C₁₇	C₂³×C₄
Fitting	C₄×C₁₇×#125	C₂
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#23371
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
260. C₁₇	#23371	semidirect

Maximal subgroups

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

subgroup	index
#14529508	2
#14529503	2
#14529350	2
#14529343	2
#14528901	2
C₄×C₁₇×#125	2
#23371	17