Subgroups — #17978672

Order 4352.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545201	C₂	1	4	yes	maximal, complemented
2	#14545200	C₂	1	2	yes	maximal, complemented
3	#14545198	C₂	1	2	yes	maximal, complemented
4	#14545118	C₂	1	2	yes	maximal
5	#14545116	C₂	1	1	yes	maximal, characteristic
6	#14544990	C₂	1	2	yes	maximal
7	C₄×#10587538	C₂	1	1	yes	maximal, characteristic
8	C₂×#11762933	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	1	yes	characteristic
10	C₂×#10587538	C₂²	1	2	yes	—
11	C₂×#10587538	C₂²	1	1	yes	characteristic
12	#11763730	C₂²	1	4	yes	—
13	#11763730	C₂²	1	4	yes	—
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	2	yes	—
16	#11763698	C₂²	1	4	yes	—
17	#11763697	C₂²	1	2	yes	—
18	#11763695	C₂²	1	4	yes	—
19	C₄×#63216	C₂²	1	2	yes	—
20	C₂×C₄×Dic₃₄	C₄	1	1	yes	characteristic
21	#11762933	C₄	1	2	yes	—
22	#11762933	C₄	1	1	yes	characteristic
23	#11762933	C₄	1	1	yes	characteristic
24	#11762933	C₂²	1	1	yes	characteristic
25	#11762933	C₂²	1	1	yes	characteristic
26	#11762933	C₂²	1	2	yes	—
27	C₂×#10587485	C₄	1	2	yes	—
28	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
30	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
31	C₂²×Dic₃₄	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
35	C₂×#63225	C₈	1	2	yes	—
36	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
37	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
38	C₂²×#3622	C₂³	1	1	yes	characteristic
39	C₄×Dic₃₄	C₈	1	2	yes	—
40	C₄×Dic₃₄	C₈	1	1	yes	characteristic
41	C₄×Dic₃₄	C₈	1	1	yes	characteristic
42	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
44	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
45	#10587538	—	2	2	no	—
46	#10587538	—	2	4	no	—
47	#10587538	Q₈	1	4	yes	complemented
48	C₂×#63216	C₂³	1	4	yes	—
49	C₂×#63216	C₂³	1	2	yes	—
50	C₂×#63216	C₂³	1	2	yes	—
51	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	#10587485	C₂×C₄	1	2	yes	—
55	#10587485	C₂×C₄	1	2	yes	—
56	#10587485	C₂×C₄	1	2	yes	—
57	#10587485	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₇	C₂³	1	2	yes	—
61	C₈×Dic₁₇	C₂³	1	2	yes	—
62	C₄×#3622	C₂×C₄	1	1	yes	characteristic
63	C₄×#3622	C₂×C₄	1	1	yes	characteristic
64	C₄×#3622	C₂³	1	1	yes	characteristic
65	C₄×#3622	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
70	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
71	C₂×Dic₃₄	—	2	4	no	—
72	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
74	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₁₇	—	2	4	no	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
83	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
85	#63225	C₂×C₈	1	2	yes	—
86	#63225	C₂×C₈	1	2	yes	—
87	#63225	C₂×C₈	1	2	yes	—
88	#63225	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
91	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
93	C₂×#3622	—	2	2	no	—
94	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
95	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
96	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
97	C₂×#3622	C₂×Q₈	1	2	yes	—
98	C₂×#3622	#41	1	2	yes	—
99	C₂×#3622	C₂⁴	1	1	yes	characteristic
100	#63216	—	2	4	no	—
101	#63216	—	2	2	no	—
102	#63216	C₂×Q₈	1	4	yes	complemented
103	Q₈×M₃₂	—	17	1	no	maximal, Sylow
104	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	4	1	no	—
107	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#131	1	2	yes	—
114	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₁₇	—	2	4	no	—
116	C₂×Dic₁₇	—	4	2	no	—
117	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
118	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
119	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
120	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
121	Dic₃₄	—	2	2	no	—
122	Dic₃₄	—	2	4	no	—
123	Dic₃₄	M₃₂	1	4	yes	complemented
124	C₈×C₁₇	—	2	2	no	—
125	C₈×C₁₇	—	2	2	no	—
126	#3622	—	2	1	no	—
127	#3622	—	2	1	no	—
128	#3622	C₄×Q₈	1	2	yes	—
129	#3622	C₂²×Q₈	1	1	yes	characteristic
130	#3622	C₂×#41	1	1	yes	characteristic
131	C₂×C₈×Q₈	—	17	1	no	—
132	#2183	—	17	2	no	—
133	#2183	—	17	4	no	—
134	Q₈×C₁₆	—	17	2	no	—
135	#2150	—	17	1	no	—
136	#2150	—	17	2	no	—
137	C₄×M₃₂	—	17	1	no	—
138	C₄×M₃₂	—	17	2	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
142	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	C₈×Q₈	1	2	yes	—
147	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
148	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
150	Dic₁₇	—	2	2	no	—
151	Dic₁₇	—	2	2	no	—
152	Dic₁₇	—	2	2	no	—
153	Dic₁₇	C₂×M₃₂	1	4	yes	—
154	C₂×C₄×Q₈	—	17	1	no	—
155	C₂×M₃₂	—	17	1	no	—
156	C₂×M₃₂	—	17	1	no	—
157	C₂×M₃₂	—	17	2	no	—
158	C₈×Q₈	—	17	1	no	—
159	C₈×Q₈	—	17	2	no	—
160	C₈×Q₈	—	17	1	no	—
161	C₈×Q₈	—	17	2	no	—
162	C₈×Q₈	—	17	1	no	—
163	C₈×Q₈	—	17	1	no	—
164	C₂×C₄⋊C₈	—	17	2	no	—
165	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
166	C₂×C₄×C₈	—	17	2	no	—
167	C₂×C₄×C₈	—	17	1	no	—
168	C₄⋊C₁₆	—	17	2	no	—
169	C₄⋊C₁₆	—	17	2	no	—
170	C₄⋊C₁₆	—	17	4	no	—
171	C₄⋊C₁₆	—	17	4	no	—
172	#346	—	17	2	no	—
173	#346	—	17	4	no	—
174	C₄×C₁₆	—	17	2	no	—
175	C₄×C₁₆	—	17	4	no	—
176	C₂×C₁₇	—	2	2	no	—
177	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
178	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
179	C₂×C₁₇	#3407	1	1	yes	characteristic
180	C₂²×Q₈	—	17	1	no	—
181	C₂²×C₈	—	17	1	no	—
182	C₂²×C₈	—	17	1	no	—
183	C₂²×C₈	C₂×#595	1	2	yes	—
184	C₄×Q₈	—	17	1	no	—
185	C₄×Q₈	—	17	2	no	—
186	C₄×Q₈	—	17	1	no	—
187	C₄×Q₈	—	17	1	no	—
188	C₄×Q₈	—	17	2	no	—
189	C₄×Q₈	—	17	1	no	—
190	C₂×C₄⋊C₄	—	17	2	no	—
191	C₂×C₄⋊C₄	—	17	1	no	—
192	C₂×C₄²	—	17	2	no	—
193	C₂×C₄²	C₂×#595	1	1	yes	characteristic
194	M₃₂	—	17	4	no	—
195	M₃₂	—	34	2	no	—
196	M₃₂	—	34	4	no	—
197	C₂×C₁₆	—	17	2	no	—
198	C₂×C₁₆	—	17	2	no	—
199	C₂×C₁₆	—	17	4	no	—
200	C₄⋊C₈	—	17	2	no	—
201	C₄⋊C₈	—	17	2	no	—
202	C₄⋊C₈	—	17	2	no	—
203	C₄⋊C₈	—	17	2	no	—
204	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
205	C₄⋊C₈	C₂×#595	1	2	yes	—
206	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
207	C₄×C₈	—	17	2	no	—
208	C₄×C₈	—	17	2	no	—
209	C₄×C₈	—	17	2	no	—
210	C₄×C₈	—	17	2	no	—
211	C₄×C₈	—	17	1	no	—
212	C₄×C₈	—	17	1	no	—
213	C₄×C₈	—	17	1	no	—
214	C₄×C₈	—	17	1	no	—
215	C₁₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
216	C₂×Q₈	—	17	1	no	—
217	C₂×Q₈	—	17	2	no	—
218	C₂×Q₈	—	17	1	no	—
219	C₂×Q₈	—	34	4	no	—
220	C₂²×C₄	—	17	2	no	—
221	C₂²×C₄	—	17	2	no	—
222	C₂²×C₄	#63242	1	1	yes	characteristic
223	C₂²×C₄	#63242	1	1	yes	characteristic
224	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	17	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	34	2	no	—
233	C₂×C₈	C₂²×#595	1	2	yes	—
234	C₂×C₈	C₂²×#595	1	2	yes	—
235	C₄⋊C₄	—	17	2	no	—
236	C₄⋊C₄	—	17	2	no	—
237	C₄⋊C₄	—	17	2	no	—
238	C₄⋊C₄	—	17	2	no	—
239	C₄⋊C₄	—	17	2	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄²	—	17	2	no	—
243	C₄²	—	17	2	no	—
244	C₄²	—	17	2	no	—
245	C₄²	—	17	2	no	—
246	C₄²	#63242	1	1	yes	characteristic
247	C₄²	#63242	1	1	yes	characteristic
248	C₄²	C₂²×#595	1	1	yes	characteristic
249	C₄²	C₂²×#595	1	1	yes	characteristic
250	C₁₆	—	17	4	no	—
251	C₁₆	—	34	2	no	—
252	C₁₆	—	34	4	no	—
253	C₂³	C₂×#63242	1	1	yes	characteristic
254	Q₈	—	17	4	no	—
255	Q₈	—	34	4	no	—
256	Q₈	—	34	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	17	2	no	—
260	C₂×C₄	—	17	2	no	—
261	C₂×C₄	—	17	2	no	—
262	C₂×C₄	—	17	2	no	—
263	C₂×C₄	—	34	4	no	—
264	C₂×C₄	—	68	2	no	—
265	C₂×C₄	C₂×#63242	1	1	yes	characteristic
266	C₂×C₄	C₂×#63242	1	1	yes	characteristic
267	C₂×C₄	C₂×#63242	1	1	yes	characteristic
268	C₂×C₄	C₂×#63242	1	1	yes	characteristic
269	C₂×C₄	C₂×#63242	1	1	yes	characteristic
270	C₂×C₄	#10587678	1	2	yes	—
271	C₂×C₄	Q₈×#595	1	2	yes	—
272	C₂×C₄	C₂³×#595	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	17	1	no	—
276	C₈	—	17	1	no	—
277	C₈	—	17	2	no	—
278	C₈	—	34	1	no	—
279	C₈	—	34	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#11763858	1	2	yes	—
283	C₂²	C₂²×#63242	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	17	4	no	—
288	C₄	—	34	2	no	—
289	C₄	—	34	2	no	—
290	C₄	—	34	2	no	—
291	C₄	#11763858	1	2	yes	—
292	C₄	#11763865	1	2	yes	—
293	C₄	C₂×Q₈×#595	1	1	yes	characteristic
294	C₄	C₂×#10587678	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#11763858	1	1	yes	characteristic
297	C₂	C₂×#11763865	1	1	yes	characteristic
298	C₂	#14545753	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #14545201	C₂	semidirect
2. #14545200	C₂	semidirect
3. #14545198	C₂	semidirect
47. #10587538	Q₈	semidirect
102. #63216	C₂×Q₈	semidirect
123. Dic₃₄	M₃₂	semidirect
215. C₁₇	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#14545201	2
#14545200	2
#14545198	2
#14545118	2
#14545116	2
#14544990	2
C₄×#10587538	2
C₂×#11762933	2
Q₈×M₃₂	17