Subgroups — #17952267

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3624	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763650	C₂	1	2	yes	maximal
4	#14543606	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₄	1	2	yes	—
7	C₂×C₄×#3624	C₄	1	2	yes	—
8	C₂×C₄×#3624	C₄	1	2	yes	—
9	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3624	C₂²	1	2	yes	—
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	#11763675	C₄	1	2	yes	—
13	#11763650	—	2	4	no	—
14	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
16	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762563	C₄	1	2	yes	—
18	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
19	C₂²×#3624	C₂×C₄	1	4	yes	—
20	C₂²×#3624	C₂×C₄	1	2	yes	—
21	C₂²×#3624	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
30	C₂²×#3622	C₂×C₄	1	2	yes	—
31	C₄⋊#3624	—	2	4	no	—
32	C₄×#3624	—	2	4	no	—
33	C₄×#3624	—	2	2	no	—
34	C₄×#3624	—	2	1	no	—
35	C₄×#3624	—	2	1	no	—
36	C₄×#3624	—	2	2	no	—
37	C₄×#3624	—	2	4	no	—
38	C₄×#3624	D₄	1	2	yes	—
39	C₄×#3624	Q₈	1	2	yes	—
40	C₁₇×C₄⋊C₈	C₈	1	4	yes	complemented
41	C₈×Dic₁₇	—	2	2	no	—
42	C₈×Dic₁₇	—	2	2	no	—
43	C₄⋊#3622	C₈	1	4	yes	complemented
44	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
48	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
49	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3624	—	2	2	no	—
52	C₂×#3624	—	2	2	no	—
53	C₂×#3624	—	2	2	no	—
54	C₂×#3624	—	2	2	no	—
55	C₂×#3624	—	2	4	no	—
56	C₂×#3624	—	2	4	no	—
57	C₂×#3624	—	2	4	no	—
58	C₂×#3624	C₄⋊C₄	1	4	yes	—
59	C₂×#3624	M₁₆	1	4	yes	—
60	C₂×C₈×C₁₇	—	2	2	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
63	C₄²×C₁₇	C₂×C₈	1	2	yes	—
64	C₄²×C₁₇	C₂×C₈	1	2	yes	—
65	C₄×Dic₁₇	—	2	1	no	—
66	C₄×Dic₁₇	—	2	1	no	—
67	C₄×Dic₁₇	—	2	2	no	—
68	C₄×Dic₁₇	—	2	2	no	—
69	C₄×Dic₁₇	M₁₆	1	4	yes	—
70	C₄×Dic₁₇	M₁₆	1	4	yes	—
71	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
72	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
73	C₄×Dic₁₇	#41	1	1	yes	characteristic
74	C₄×Dic₁₇	#41	1	1	yes	characteristic
75	C₂×#3622	—	2	2	no	—
76	C₂×#3622	—	2	2	no	—
77	C₂×#3622	C₂×C₈	1	4	yes	—
78	#8261	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
84	C₂×C₄×C₁₇	#97	1	2	yes	—
85	C₂×C₄×C₁₇	#97	1	2	yes	—
86	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
91	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
92	C₂×Dic₁₇	—	2	1	no	—
93	C₂×Dic₁₇	—	2	1	no	—
94	C₂×Dic₁₇	—	2	1	no	—
95	C₂×Dic₁₇	—	2	1	no	—
96	C₂×Dic₁₇	#97	1	2	yes	—
97	C₂×Dic₁₇	#97	1	2	yes	—
98	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₇	#117	1	1	yes	characteristic
100	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
101	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
102	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
103	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
104	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
105	C₂×Dic₁₇	#131	1	2	yes	—
106	#3624	—	2	4	no	—
107	#3624	—	2	4	no	—
108	#3624	—	4	4	no	—
109	C₈×C₁₇	—	2	4	no	—
110	#3622	—	2	4	no	—
111	C₄×C₄⋊C₈	—	17	1	no	—
112	C₄×C₄⋊C₈	—	17	1	no	—
113	C₄²×C₈	—	17	1	no	—
114	C₂×#322	—	17	2	no	—
115	#1273	—	17	2	no	—
116	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
119	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
120	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
121	C₂²×C₁₇	#405	1	1	yes	characteristic
122	C₂²×C₁₇	#405	1	1	yes	characteristic
123	C₄×C₁₇	—	2	2	no	—
124	C₄×C₁₇	#322	1	4	yes	complemented
125	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
126	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
127	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
128	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
129	Dic₁₇	—	2	2	no	—
130	Dic₁₇	—	2	2	no	—
131	Dic₁₇	#423	1	2	yes	—
132	Dic₁₇	#432	1	2	yes	—
133	Dic₁₇	#436	1	2	yes	—
134	Dic₁₇	#446	1	2	yes	—
135	C₂×C₄⋊C₈	—	17	1	no	—
136	C₂×C₄⋊C₈	—	17	2	no	—
137	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
138	C₂×C₄×C₈	—	17	1	no	—
139	C₂×C₄×C₈	—	17	2	no	—
140	C₂×C₄×C₈	—	17	2	no	—
141	C₂×C₄×C₈	—	17	2	no	—
142	C₂×C₄×C₈	—	17	1	no	—
143	C₂×C₄×C₈	—	17	2	no	—
144	C₄³	—	17	1	no	—
145	#336	—	17	2	no	—
146	#336	—	17	2	no	—
147	#322	—	34	4	no	—
148	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
149	C₂×C₁₇	C₈×M₁₆	1	1	yes	characteristic
150	C₂×C₁₇	#1451	1	1	yes	characteristic
151	C₂×C₁₇	#1452	1	1	yes	characteristic
152	C₂×C₁₇	#1767	1	1	yes	characteristic
153	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₁₇	#1770	1	1	yes	characteristic
155	C₂²×C₈	—	17	2	no	—
156	C₂²×C₈	—	17	2	no	—
157	C₂²×C₈	—	17	2	no	—
158	C₂²×C₈	—	17	4	no	—
159	C₂²×C₈	C₂×#595	1	2	yes	—
160	C₂×C₄²	—	17	1	no	—
161	C₂×C₄²	—	17	1	no	—
162	C₂×C₄²	—	17	1	no	—
163	C₂×C₄²	—	17	1	no	—
164	C₂×C₄²	—	17	1	no	—
165	C₂×C₄²	—	17	1	no	—
166	C₂×C₄²	C₂×#595	1	1	yes	characteristic
167	C₄⋊C₈	—	17	4	no	—
168	C₄⋊C₈	—	34	4	no	—
169	C₄⋊C₈	#3624	1	4	yes	complemented
170	C₄×C₈	—	17	2	no	—
171	C₄×C₈	—	17	2	no	—
172	C₄×C₈	—	34	2	no	—
173	C₄×C₈	—	34	2	no	—
174	C₄×C₈	—	34	2	no	—
175	C₄×C₈	—	34	1	no	—
176	C₄×C₈	—	34	1	no	—
177	C₄×C₈	—	34	2	no	—
178	C₄×C₈	—	34	4	no	—
179	C₄×C₈	—	34	4	no	—
180	C₁₇	#8261	1	1	yes	characteristic, complemented, Sylow
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	—	17	1	no	—
184	C₂²×C₄	—	17	1	no	—
185	C₂²×C₄	C₄×#595	1	1	yes	characteristic
186	C₂²×C₄	C₄×#595	1	1	yes	characteristic
187	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	17	4	no	—
191	C₂×C₈	—	17	4	no	—
192	C₂×C₈	—	17	4	no	—
193	C₂×C₈	—	34	2	no	—
194	C₂×C₈	—	34	2	no	—
195	C₂×C₈	—	34	2	no	—
196	C₂×C₈	—	34	2	no	—
197	C₂×C₈	—	34	2	no	—
198	C₂×C₈	—	34	2	no	—
199	C₂×C₈	—	34	4	no	—
200	C₂×C₈	—	34	4	no	—
201	C₂×C₈	—	34	4	no	—
202	C₂×C₈	C₂×#3624	1	4	yes	—
203	C₄²	—	17	1	no	—
204	C₄²	—	17	1	no	—
205	C₄²	—	17	1	no	—
206	C₄²	—	17	1	no	—
207	C₄²	—	17	4	no	—
208	C₄²	—	17	4	no	—
209	C₄²	—	34	2	no	—
210	C₄²	—	34	2	no	—
211	C₄²	—	34	1	no	—
212	C₄²	—	34	1	no	—
213	C₄²	#63242	1	2	yes	—
214	C₄²	C₂×#3624	1	2	yes	—
215	C₂³	C₂×C₄×#595	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	1	no	—
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	2	no	—
223	C₂×C₄	—	17	2	no	—
224	C₂×C₄	—	17	2	no	—
225	C₂×C₄	—	17	2	no	—
226	C₂×C₄	—	17	2	no	—
227	C₂×C₄	—	17	2	no	—
228	C₂×C₄	—	34	1	no	—
229	C₂×C₄	—	34	1	no	—
230	C₂×C₄	—	34	1	no	—
231	C₂×C₄	—	34	1	no	—
232	C₂×C₄	#10587532	1	2	yes	—
233	C₂×C₄	#10587532	1	2	yes	—
234	C₂×C₄	C₄×#3624	1	2	yes	—
235	C₂×C₄	C₄×#3624	1	2	yes	—
236	C₂×C₄	C₂×#63242	1	1	yes	characteristic
237	C₂×C₄	#10587678	1	1	yes	characteristic
238	C₂×C₄	D₄×#595	1	1	yes	characteristic
239	C₂×C₄	#10587680	1	1	yes	characteristic
240	C₂×C₄	Q₈×#595	1	1	yes	characteristic
241	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	34	4	no	—
244	C₈	—	34	4	no	—
245	C₈	—	34	4	no	—
246	C₈	—	68	4	no	—
247	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
248	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
249	C₂²	C₂×#10587532	1	1	yes	characteristic
250	C₂²	C₂×#10587532	1	1	yes	characteristic
251	C₂²	M₁₆×#595	1	1	yes	characteristic
252	C₂²	#11763774	1	1	yes	characteristic
253	C₂²	#11763775	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	17	2	no	—
256	C₄	—	17	2	no	—
257	C₄	—	17	2	no	—
258	C₄	—	17	2	no	—
259	C₄	—	34	2	no	—
260	C₄	—	34	2	no	—
261	C₄	#11763650	1	4	yes	complemented
262	C₄	D₄×#3624	1	1	yes	characteristic
263	C₄	#11763837	1	1	yes	characteristic
264	C₄	Q₈×#3624	1	1	yes	characteristic
265	C₄	#11763858	1	1	yes	characteristic
266	C₂	C₂×#11763650	1	1	yes	characteristic
267	C₂	M₁₆×#3624	1	1	yes	characteristic
268	C₂	#14543996	1	1	yes	characteristic
269	C₂	#14544000	1	1	yes	characteristic
270	C₂	#14544571	1	1	yes	characteristic
271	C₂	C₄⋊C₄×#3624	1	1	yes	characteristic
272	C₂	#14544581	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	#8261
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
40. C₁₇×C₄⋊C₈	C₈	semidirect
43. C₄⋊#3622	C₈	semidirect
124. C₄×C₁₇	#322	semidirect
169. C₄⋊C₈	#3624	semidirect
180. C₁₇	#8261	semidirect
261. C₄	#11763650	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₄²×#3624	2
C₂×#11763650	2
#14543606	2
C₄⋊C₈×Dic₁₇	2
#8261	17