Subgroups — #17039011

Order 4352.

814 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	#14529721	C₂	1	2	yes	maximal
2	#14529503	C₂	1	2	yes	maximal
3	#14529343	C₂	1	4	yes	maximal
4	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#3622	C₂	1	2	yes	maximal
6	#14528530	C₂	1	2	yes	maximal
7	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#3622	C₂²	1	4	yes	—
10	C₂×C₄⋊#3622	C₂²	1	2	yes	—
11	C₂×C₄×#3622	C₂²	1	2	yes	—
12	C₂×C₄×#3622	C₂²	1	2	yes	—
13	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	2	yes	—
15	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
16	#11762564	C₂²	1	4	yes	—
17	#11762564	C₂²	1	4	yes	—
18	#11762564	C₂²	1	2	yes	—
19	#11762564	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
29	C₄³×C₁₇	C₂²	1	2	yes	—
30	C₄³×C₁₇	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	—	2	4	no	—
33	C₄×Q₈×C₁₇	—	2	4	no	—
34	C₄×Q₈×C₁₇	C₈	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
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×C₄²×C₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
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	C₂²×#3622	C₂³	1	4	yes	—
55	C₂²×#3622	C₂³	1	2	yes	—
56	C₂²×#3622	C₂³	1	2	yes	—
57	C₁₇×#95	C₂×C₄	1	2	yes	—
58	C₁₇×#95	C₂×C₄	1	2	yes	—
59	C₁₇×#95	C₂×C₄	1	2	yes	—
60	C₄⋊#3622	—	2	8	no	—
61	C₄⋊#3622	—	2	4	no	—
62	C₄×#3622	—	2	4	no	—
63	C₄×#3622	—	2	4	no	—
64	C₄×#3622	—	2	2	no	—
65	C₄×#3622	—	2	4	no	—
66	C₄×#3622	Q₈	1	4	yes	complemented
67	C₂×Q₈×C₁₇	—	2	2	no	—
68	C₂×Q₈×C₁₇	—	2	2	no	—
69	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	—	2	4	no	—
82	C₄⋊C₄×C₁₇	—	2	2	no	—
83	C₄⋊C₄×C₁₇	—	2	4	no	—
84	C₄⋊C₄×C₁₇	—	2	2	no	—
85	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
86	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
87	C₄²×C₁₇	—	2	4	no	—
88	C₄²×C₁₇	—	2	2	no	—
89	C₄²×C₁₇	—	2	2	no	—
90	C₄²×C₁₇	—	2	4	no	—
91	C₄²×C₁₇	—	2	4	no	—
92	C₄²×C₁₇	—	2	2	no	—
93	C₄²×C₁₇	—	2	4	no	—
94	C₄²×C₁₇	—	2	4	no	—
95	C₄²×C₁₇	C₂×C₈	1	8	yes	—
96	C₄²×C₁₇	C₂×C₈	1	4	yes	—
97	C₄²×C₁₇	M₁₆	1	4	yes	—
98	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
99	C₄²×C₁₇	#41	1	4	yes	—
100	C₄²×C₁₇	#41	1	2	yes	—
101	C₂×#3622	—	2	2	no	—
102	C₂×#3622	—	2	2	no	—
103	C₂×#3622	—	2	2	no	—
104	C₂×#3622	—	2	2	no	—
105	C₂×#3622	—	2	4	no	—
106	C₂×#3622	—	2	4	no	—
107	C₂×#3622	—	2	4	no	—
108	C₂×#3622	C₂×Q₈	1	4	yes	—
109	C₂×#3622	#41	1	4	yes	—
110	#21339	—	17	1	no	maximal, Sylow
111	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
112	Q₈×C₁₇	—	2	8	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	2	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
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	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	#117	1	2	yes	—
130	C₂×C₄×C₁₇	#117	1	2	yes	—
131	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
133	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
134	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₁₇	#131	1	2	yes	—
142	C₂×C₄×C₁₇	#131	1	2	yes	—
143	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
147	#3622	—	2	4	no	—
148	#3622	—	2	4	no	—
149	#3622	—	4	4	no	—
150	C₄²×Q₈	D₁₇	1	1	yes	characteristic
151	#1984	—	17	2	no	—
152	#1939	—	17	2	no	—
153	#1916	—	17	4	no	—
154	C₄×C₄⋊C₈	—	17	1	no	—
155	C₄×C₄⋊C₈	—	17	2	no	—
156	#1744	—	17	2	no	—
157	C₄²×C₈	—	17	1	no	—
158	C₂²×C₁₇	C₂×#117	1	1	yes	characteristic
159	C₂²×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₂²×C₁₇	C₄×#41	1	1	yes	characteristic
161	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
162	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
164	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
165	C₄×C₁₇	—	2	4	no	—
166	C₄×C₁₇	—	2	2	no	—
167	C₄×C₁₇	—	2	2	no	—
168	C₄×C₁₇	—	2	4	no	—
169	C₄×C₁₇	—	2	2	no	—
170	C₄×C₁₇	—	4	2	no	—
171	C₄×C₁₇	—	4	2	no	—
172	C₄×C₁₇	#431	1	4	yes	—
173	C₄×C₁₇	C₈×Q₈	1	4	yes	—
174	C₄×C₁₇	#446	1	4	yes	—
175	C₄×C₁₇	#529	1	2	yes	—
176	C₄×C₁₇	#533	1	2	yes	—
177	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
178	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	17	4	no	—
181	C₂×C₄⋊C₈	—	17	2	no	—
182	C₂×C₄×C₈	—	17	1	no	—
183	C₂×C₄×C₈	—	17	2	no	—
184	C₂×C₄×C₈	—	17	2	no	—
185	C₂×C₄×C₈	—	17	1	no	—
186	C₂×C₄×C₈	—	17	2	no	—
187	C₄×C₄⋊C₄	Dic₁₇	1	2	yes	—
188	C₄×C₄⋊C₄	Dic₁₇	1	1	yes	characteristic
189	C₄×C₄⋊C₄	Dic₁₇	1	2	yes	—
190	C₄×C₄⋊C₄	Dic₁₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
192	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
193	C₄³	C₂×D₁₇	1	1	yes	characteristic
194	C₄³	C₂×D₁₇	1	2	yes	—
195	#336	—	17	2	no	—
196	#336	—	17	4	no	—
197	#336	—	17	4	no	—
198	#336	—	17	4	no	—
199	C₂×C₁₇	#2295	1	1	yes	characteristic
200	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
201	C₂×C₁₇	#2922	1	1	yes	characteristic
202	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
203	C₂×C₁₇	C₂×#446	1	1	yes	characteristic
204	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
205	C₂×C₁₇	#2965	1	1	yes	characteristic
206	C₂²×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
207	C₂²×C₈	—	17	2	no	—
208	C₂²×C₈	—	17	2	no	—
209	C₂²×C₈	—	17	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	#3622	1	8	yes	complemented
213	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
220	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
221	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
222	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
225	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
226	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
228	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
230	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
232	C₄⋊C₈	—	34	8	no	—
233	C₄⋊C₈	—	34	4	no	—
234	C₄×C₈	—	17	4	no	—
235	C₄×C₈	—	34	4	no	—
236	C₄×C₈	—	34	4	no	—
237	C₄×C₈	—	34	2	no	—
238	C₄×C₈	—	34	4	no	—
239	#95	C₂×Dic₁₇	1	2	yes	—
240	#95	C₂×Dic₁₇	1	2	yes	—
241	#95	C₂×Dic₁₇	1	2	yes	—
242	C₁₇	#21339	1	1	yes	characteristic, complemented, Sylow
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	C₂×#3622	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
257	C₂×C₈	—	17	4	no	—
258	C₂×C₈	—	17	4	no	—
259	C₂×C₈	—	34	2	no	—
260	C₂×C₈	—	34	2	no	—
261	C₂×C₈	—	34	2	no	—
262	C₂×C₈	—	34	2	no	—
263	C₂×C₈	—	34	4	no	—
264	C₂×C₈	—	34	4	no	—
265	C₂×C₈	—	34	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂×#3622	1	8	yes	—
271	C₄⋊C₄	C₂×#3622	1	4	yes	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	C₂×#3622	1	8	yes	—
281	C₄²	C₂×#3622	1	4	yes	—
282	C₄²	#63223	1	4	yes	—
283	C₄²	#63252	1	4	yes	—
284	C₄²	Q₈×D₁₇	1	2	yes	—
285	C₄²	#63256	1	2	yes	—
286	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	C₂²×#3622	1	4	yes	—
305	C₂×C₄	C₂²×#3622	1	2	yes	—
306	C₂×C₄	C₂²×#3622	1	4	yes	—
307	C₂×C₄	C₂²×#3622	1	2	yes	—
308	C₂×C₄	C₂²×#3622	1	2	yes	—
309	C₂×C₄	C₂×#63223	1	2	yes	—
310	C₂×C₄	C₂×#63223	1	2	yes	—
311	C₂×C₄	C₂×#63223	1	2	yes	—
312	C₂×C₄	#10587612	1	2	yes	—
313	C₂×C₄	#10587612	1	2	yes	—
314	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
315	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
316	C₂×C₄	#10587634	1	2	yes	—
317	C₂×C₄	#10587634	1	2	yes	—
318	C₂×C₄	C₂×#63252	1	2	yes	—
319	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#63256	1	1	yes	characteristic
321	C₂×C₄	#41×D₁₇	1	2	yes	—
322	C₈	—	34	4	no	—
323	C₈	—	34	4	no	—
324	C₈	—	68	4	no	—
325	C₂²	C₂×#10587612	1	1	yes	characteristic
326	C₂²	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
327	C₂²	#41×Dic₁₇	1	1	yes	characteristic
328	C₂²	C₂³×#3622	1	1	yes	characteristic
329	C₂²	C₂²×#63223	1	1	yes	characteristic
330	C₂²	C₂×#10587634	1	1	yes	characteristic
331	C₂²	C₂×#10587634	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	#11762862	1	4	yes	—
340	C₄	Q₈×#3622	1	4	yes	—
341	C₄	#11762940	1	4	yes	—
342	C₄	#11763346	1	2	yes	—
343	C₄	#11763363	1	2	yes	—
344	C₂	#14531262	1	1	yes	characteristic
345	C₂	C₂×#11762862	1	1	yes	characteristic
346	C₂	#14537428	1	1	yes	characteristic
347	C₂	C₂×Q₈×#3622	1	1	yes	characteristic
348	C₂	C₂×#11762940	1	1	yes	characteristic
349	C₂	#41×#3622	1	1	yes	characteristic
350	C₂	#14537751	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₄²×Q₈×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂³×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄²×Q₈	D₁₇
hypocenter	C₁₇	#21339
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
34. C₄×Q₈×C₁₇	C₈	semidirect
66. C₄×#3622	Q₈	semidirect
212. C₄×Q₈	#3622	semidirect
242. C₁₇	#21339	semidirect

Maximal subgroups

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

subgroup	index
#14529721	2
#14529503	2
#14529343	2
C₄×C₄⋊#3622	2
#14528530	2
C₄²×#3622	2
C₄²×Q₈×C₁₇	2
#21339	17