Subgroups — #17978690

Order 4352.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	2	yes	maximal, complemented
2	#14545168	C₂	1	2	yes	maximal, complemented
3	#14545164	C₂	1	2	yes	maximal, complemented
4	#14545164	C₂	1	2	yes	maximal, complemented
5	#14545149	C₂	1	2	yes	maximal, complemented
6	#14545145	C₂	1	1	yes	maximal, characteristic, complemented
7	#14544993	C₂	1	2	yes	maximal, complemented
8	C₄×#10587538	C₂	1	1	yes	maximal, characteristic, complemented
9	#14537760	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587538	C₂²	1	2	yes	—
11	C₂×#10587538	C₂²	1	2	yes	—
12	C₂²×#63216	C₂²	1	2	yes	—
13	C₄⋊#63216	C₂²	1	2	yes	—
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	2	yes	—
16	C₂²⋊#63216	C₂²	1	1	yes	characteristic
17	C₂²⋊#63216	C₂²	1	2	yes	—
18	C₂²⋊#63216	C₂²	1	2	yes	—
19	C₂²⋊#63216	C₂²	1	1	yes	characteristic
20	#11763697	C₂²	1	2	yes	—
21	#11763697	C₂²	1	2	yes	—
22	#11763697	C₂²	1	2	yes	—
23	C₄×#63216	C₂²	1	2	yes	—
24	C₄×#63216	C₂²	1	1	yes	characteristic
25	C₄×#63216	C₂²	1	1	yes	characteristic
26	#11762940	C₄	1	1	yes	characteristic
27	Q₈×#3622	C₂²	1	1	yes	characteristic
28	#11762887	C₄	1	1	yes	characteristic
29	#11762881	C₄	1	2	yes	—
30	D₄×#3622	C₂²	1	1	yes	characteristic
31	D₄×#3622	C₂²	1	2	yes	—
32	#11762870	C₄	1	2	yes	—
33	#11762866	C₄	1	1	yes	characteristic
34	#11762765	C₂²	1	2	yes	—
35	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
36	C₁₇×#529	C₄	1	1	yes	characteristic
37	C₁₇×#126	C₈	1	2	yes	—
38	C₁₇×#125	C₈	1	1	yes	characteristic
39	C₁₇×#124	C₈	1	1	yes	characteristic
40	C₁₇×#123	C₈	1	2	yes	—
41	C₁₇×C₂²⋊Q₈	C₈	1	1	yes	characteristic
42	C₁₇×#121	C₈	1	1	yes	characteristic
43	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#117	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×#63223	C₂×C₄	1	2	yes	—
49	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
50	C₂²×#3622	C₂³	1	1	yes	characteristic
51	C₂²×#3622	C₂³	1	2	yes	—
52	#10587538	—	2	4	no	—
53	#10587538	—	2	4	no	—
54	C₂×#63216	—	2	4	no	—
55	C₂×#63216	C₂³	1	2	yes	—
56	C₂×#63216	C₂³	1	2	yes	—
57	C₂×#63216	C₂³	1	2	yes	—
58	C₂×#63216	C₂³	1	2	yes	—
59	C₂²⋊#3622	C₂×C₄	1	2	yes	—
60	C₂²⋊#3622	C₂×C₄	1	2	yes	—
61	C₂²⋊#3622	C₂×C₄	1	2	yes	—
62	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
64	C₄⋊#3622	C₂×C₄	1	2	yes	—
65	C₄⋊#3622	C₂×C₄	1	2	yes	—
66	#10587474	C₂×C₄	1	2	yes	—
67	#10587474	C₂×C₄	1	2	yes	—
68	C₄×#3622	C₂×C₄	1	1	yes	characteristic
69	C₄×#3622	C₂×C₄	1	1	yes	characteristic
70	C₄×#3622	C₂³	1	1	yes	characteristic
71	C₄×#3622	C₂³	1	2	yes	—
72	C₄×#3622	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
80	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
82	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
84	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
85	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
87	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
88	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
91	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₁₇	C₂×C₈	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	2	yes	—
96	#63223	—	2	4	no	—
97	C₂×#3622	—	2	2	no	—
98	C₂×#3622	—	2	2	no	—
99	C₂×#3622	—	2	2	no	—
100	C₂×#3622	—	2	2	no	—
101	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
102	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
103	C₂×#3622	C₂²×C₄	1	2	yes	—
104	C₂×#3622	C₂²×C₄	1	2	yes	—
105	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
106	C₂×#3622	C₂⁴	1	1	yes	characteristic
107	#63216	—	2	2	no	—
108	#63216	—	2	2	no	—
109	#63216	—	2	2	no	—
110	#63216	#41	1	4	yes	—
111	#33481	—	17	1	no	maximal, Sylow
112	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
113	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
114	Q₈×C₁₇	—	4	1	no	—
115	D₄×C₁₇	—	4	1	no	—
116	D₄×C₁₇	—	4	2	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	4	1	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
132	#3622	—	2	2	no	—
133	#3622	—	2	1	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	2	no	—
136	#3622	#131	1	2	yes	—
137	#3622	C₂×#41	1	2	yes	—
138	#2966	—	17	1	no	—
139	#2183	—	17	2	no	—
140	#2169	—	17	2	no	—
141	#2168	—	17	2	no	—
142	#2168	—	17	2	no	—
143	#2164	—	17	2	no	—
144	#2163	—	17	1	no	—
145	#2108	—	17	2	no	—
146	C₄×M₃₂	—	17	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	2	no	—
150	C₂²×C₁₇	—	2	2	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	2	no	—
153	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	2	no	—
159	C₄×C₁₇	—	2	2	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	2	no	—
162	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
164	#529	#595	1	1	yes	characteristic
165	C₂×M₃₂	—	17	2	no	—
166	C₂×M₃₂	—	17	2	no	—
167	C₂²×C₁₆	—	17	2	no	—
168	#446	—	17	1	no	—
169	C₈×Q₈	—	17	1	no	—
170	#436	—	17	1	no	—
171	#435	—	17	2	no	—
172	C₈×D₄	—	17	1	no	—
173	C₈×D₄	—	17	2	no	—
174	#433	—	17	2	no	—
175	#432	—	17	1	no	—
176	#405	—	17	2	no	—
177	C₂×C₄×C₈	—	17	1	no	—
178	C₄⋊C₁₆	—	17	2	no	—
179	C₄⋊C₁₆	—	17	2	no	—
180	C₄⋊C₁₆	—	17	2	no	—
181	C₂²⋊C₁₆	—	17	1	no	—
182	C₂²⋊C₁₆	—	17	1	no	—
183	C₂²⋊C₁₆	—	17	2	no	—
184	C₂²⋊C₁₆	—	17	2	no	—
185	#346	—	17	2	no	—
186	#346	—	17	2	no	—
187	#346	—	17	2	no	—
188	C₄×C₁₆	—	17	1	no	—
189	C₄×C₁₆	—	17	1	no	—
190	C₄×C₁₆	—	17	2	no	—
191	C₂×C₁₇	—	4	1	no	—
192	C₂×C₁₇	—	4	2	no	—
193	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
194	C₂×C₁₇	#3407	1	1	yes	characteristic
195	C₂×C₁₇	#3407	1	1	yes	characteristic
196	C₂×M₁₆	—	17	2	no	—
197	C₂²×C₈	—	17	1	no	—
198	C₂²×C₈	—	17	2	no	—
199	C₂²×C₈	—	17	1	no	—
200	#126	#3624	1	2	yes	—
201	#125	#3624	1	1	yes	characteristic
202	#124	#3624	1	1	yes	characteristic
203	#123	#3624	1	2	yes	—
204	C₂²⋊Q₈	#3624	1	1	yes	characteristic
205	#121	#3624	1	1	yes	characteristic
206	C₄×Q₈	C₂×#595	1	1	yes	characteristic
207	C₄×D₄	C₂×#595	1	2	yes	—
208	C₄×D₄	C₂×#595	1	1	yes	characteristic
209	#117	C₂×#595	1	2	yes	—
210	C₂×C₄²	C₂×#595	1	1	yes	characteristic
211	M₃₂	—	34	4	no	—
212	M₃₂	—	34	4	no	—
213	C₂×C₁₆	—	17	2	no	—
214	C₂×C₁₆	—	17	2	no	—
215	C₂×C₁₆	—	17	2	no	—
216	C₂×C₁₆	—	17	2	no	—
217	C₂×C₁₆	—	34	4	no	—
218	C₄⋊C₈	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	C₄⋊C₈	—	17	2	no	—
221	C₄⋊C₈	—	17	2	no	—
222	C₂²⋊C₈	—	17	2	no	—
223	C₂²⋊C₈	—	17	2	no	—
224	C₂²⋊C₈	—	17	2	no	—
225	#97	—	17	2	no	—
226	#97	—	17	2	no	—
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	2	no	—
231	C₄×C₈	—	17	1	no	—
232	C₁₇	#33481	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
234	C₂×D₄	C₂×#3624	1	2	yes	—
235	C₂×D₄	C₂×#3624	1	1	yes	characteristic
236	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
237	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
238	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
239	C₂²×C₄	C₂²×#595	1	2	yes	—
240	M₁₆	—	34	4	no	—
241	C₂×C₈	—	17	1	no	—
242	C₂×C₈	—	17	1	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	17	2	no	—
246	C₂×C₈	—	17	2	no	—
247	C₂×C₈	—	34	2	no	—
248	C₂×C₈	—	34	2	no	—
249	C₂×C₈	—	34	2	no	—
250	C₂×C₈	—	34	2	no	—
251	C₄⋊C₄	C₂×#3624	1	2	yes	—
252	C₄⋊C₄	C₂×#3624	1	2	yes	—
253	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
254	C₄⋊C₄	C₂×#3624	1	2	yes	—
255	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
256	C₄⋊C₄	C₂×#3624	1	2	yes	—
257	C₂²⋊C₄	C₂×#3624	1	2	yes	—
258	C₂²⋊C₄	C₂×#3624	1	2	yes	—
259	C₂²⋊C₄	C₂×#3624	1	2	yes	—
260	C₂²⋊C₄	C₂×#3624	1	2	yes	—
261	C₂²⋊C₄	C₂×#3624	1	2	yes	—
262	C₄²	C₂×#3624	1	1	yes	characteristic
263	C₄²	C₂×#3624	1	1	yes	characteristic
264	C₄²	C₂²×#595	1	1	yes	characteristic
265	C₄²	C₂²×#595	1	1	yes	characteristic
266	C₄²	C₂²×#595	1	2	yes	—
267	C₁₆	—	17	4	no	—
268	C₁₆	—	34	2	no	—
269	C₁₆	—	34	2	no	—
270	C₁₆	—	34	2	no	—
271	C₂³	C₂²×#3624	1	2	yes	—
272	C₂³	C₂²×#3624	1	1	yes	characteristic
273	Q₈	—	4	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
283	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
284	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
285	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
286	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
287	C₂×C₄	C₂²×#3624	1	2	yes	—
288	C₂×C₄	C₂²×#3624	1	2	yes	—
289	C₂×C₄	C₂²×#3624	1	2	yes	—
290	C₂×C₄	C₂³×#595	1	1	yes	characteristic
291	C₈	—	17	2	no	—
292	C₈	—	17	2	no	—
293	C₈	—	34	1	no	—
294	C₈	—	34	2	no	—
295	C₈	—	34	1	no	—
296	C₈	—	34	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂³×#3624	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	#41×#595	1	1	yes	characteristic
313	C₄	D₁₇⋊#131	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#41×#3624	1	1	yes	characteristic
317	C₂	#14545751	1	1	yes	characteristic
318	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#529	C₄
prime radical	C₁₇×#121	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#529	#595
hypocenter	C₁₇	#33481
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. #14545168	C₂	semidirect
3. #14545164	C₂	semidirect
4. #14545164	C₂	semidirect
5. #14545149	C₂	semidirect
6. #14545145	C₂	semidirect
7. #14544993	C₂	semidirect
8. C₄×#10587538	C₂	semidirect
232. C₁₇	#33481	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545168	2
#14545164	2
#14545149	2
#14545145	2
#14544993	2
C₄×#10587538	2
#14537760	2
#33481	17