Subgroups — C₄×#11763225 #17186966

Order 4352.

834 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763225	C₂	1	2	yes	maximal
2	C₄×#10587587	C₂	1	1	yes	maximal, characteristic
3	#14530380	C₂	1	2	yes	maximal, complemented
4	#14530379	C₂	1	4	yes	maximal, complemented
5	C₄×#10587483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10587469	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₆×D₁₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10587587	C₄	1	1	yes	characteristic
10	C₂×#10587587	C₂²	1	1	yes	characteristic
11	C₄×#63252	C₄	1	1	yes	characteristic
12	#11763225	C₄	1	16	yes	direct product factor
13	C₂×#10587483	C₂²	1	2	yes	—
14	C₂×#10587469	C₂²	1	4	yes	—
15	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
16	#11762764	C₄	1	1	yes	characteristic
17	#11762764	C₂²	1	1	yes	characteristic
18	#11762763	C₄	1	2	yes	—
19	#11762763	C₂²	1	2	yes	—
20	C₄×#63223	C₄	1	1	yes	characteristic
21	C₄×#63218	C₄	1	2	yes	—
22	C₄×C₈×D₁₇	C₂²	1	2	yes	—
23	#11762585	C₂²	1	2	yes	—
24	#11762584	C₂²	1	4	yes	—
25	C₄×#63214	C₂²	1	2	yes	—
26	C₁₆×Dic₁₇	C₂²	1	4	yes	—
27	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
31	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
32	C₂×#63252	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₇	C₈	1	4	yes	—
37	#10587612	C₂×C₄	1	1	yes	characteristic
38	C₂×#63223	C₂×C₄	1	1	yes	characteristic
39	C₂×#63223	C₂×C₄	1	1	yes	characteristic
40	#10587587	C₈	1	8	yes	—
41	#10587587	C₂×C₄	1	8	yes	—
42	C₂×#63218	C₂×C₄	1	2	yes	—
43	C₂×#63218	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₇	C₂³	1	2	yes	—
46	C₄×D₆₈	C₈	1	2	yes	—
47	#10587558	C₂×C₄	1	2	yes	—
48	C₄²×D₁₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₃₄	C₈	1	2	yes	—
50	C₂×C₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
51	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
52	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
53	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
57	#10587486	C₂×C₄	1	2	yes	—
58	#10587486	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₇	C₂³	1	2	yes	—
61	#10587483	C₂×C₄	1	8	yes	complemented
62	C₂×#63214	C₂³	1	4	yes	—
63	#10587474	C₂×C₄	1	2	yes	—
64	C₄×#3622	C₂×C₄	1	2	yes	—
65	#10587469	C₂×C₄	1	16	yes	complemented
66	C₁₆×D₁₇	C₂×C₄	1	16	yes	complemented
67	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₇	C₂×C₈	1	2	yes	—
70	#63252	C₄²	1	4	yes	—
71	#63252	C₂×C₈	1	4	yes	—
72	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
82	C₄²×C₁₇	C₂×C₈	1	2	yes	—
83	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	C₂×C₈	1	2	yes	—
86	#63227	C₂×C₈	1	4	yes	—
87	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
88	#63225	C₂×C₈	1	4	yes	—
89	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
90	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
91	#63223	C₄²	1	4	yes	—
92	#63223	C₂×C₈	1	4	yes	—
93	C₂×#3622	C₂²×C₄	1	2	yes	—
94	C₂×#3622	C₂²×C₄	1	2	yes	—
95	#63218	C₄²	1	8	yes	—
96	#63218	C₂×C₈	1	8	yes	—
97	C₈×D₁₇	C₂×C₈	1	8	yes	—
98	C₈×D₁₇	C₂²×C₄	1	8	yes	—
99	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
100	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
101	#63214	C₂²×C₄	1	8	yes	—
102	C₄×#504	—	17	1	no	maximal, Sylow
103	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₇	—	2	4	no	—
106	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
112	C₂²⋊D₁₇	C₄×C₈	1	8	yes	—
113	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
114	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
115	D₆₈	C₄×C₈	1	4	yes	—
116	C₄×D₁₇	—	2	8	no	—
117	C₄×D₁₇	C₂×C₄²	1	4	yes	—
118	C₄×D₁₇	C₂²×C₈	1	4	yes	—
119	Dic₃₄	C₄×C₈	1	4	yes	—
120	C₈×C₁₇	C₂×C₄²	1	2	yes	—
121	C₈×C₁₇	C₂²×C₈	1	2	yes	—
122	C₈×C₁₇	C₂²×C₈	1	2	yes	—
123	C₈×C₁₇	C₂³×C₄	1	2	yes	—
124	#3622	C₂×C₄²	1	4	yes	—
125	#3622	C₂²×C₈	1	4	yes	—
126	C₂×#504	—	17	2	no	—
127	C₄×#131	—	17	1	no	—
128	#2108	—	17	2	no	—
129	#2108	—	17	4	no	—
130	C₄×M₃₂	—	17	1	no	—
131	C₄×M₃₂	—	17	2	no	—
132	C₂×C₄×C₁₆	—	17	2	no	—
133	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic, complemented
134	C₂²×C₁₇	—	2	2	no	—
135	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
136	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₁₇	—	2	4	no	—
138	C₂×D₁₇	C₂×C₄×C₈	1	4	yes	—
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
141	C₄×C₁₇	#504	1	4	yes	complemented
142	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
144	Dic₁₇	—	2	4	no	—
145	Dic₁₇	C₂×C₄×C₈	1	4	yes	—
146	C₂×#131	—	17	1	no	—
147	C₂×#131	—	17	1	no	—
148	C₄×#41	—	17	1	no	—
149	#504	—	17	16	no	—
150	C₂×M₃₂	—	17	2	no	—
151	C₂×M₃₂	—	17	4	no	—
152	C₂²×C₁₆	—	17	4	no	—
153	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
154	#405	—	17	1	no	—
155	#405	—	17	1	no	—
156	#405	—	17	2	no	—
157	#405	—	17	2	no	—
158	C₄×M₁₆	—	17	1	no	—
159	C₄×M₁₆	—	17	2	no	—
160	C₂×C₄×C₈	—	17	2	no	—
161	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
162	#346	—	17	4	no	—
163	#346	—	17	2	no	—
164	C₄×C₁₆	—	17	4	no	—
165	C₄×C₁₆	—	17	2	no	—
166	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
167	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
168	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
169	C₂×C₁₇	—	2	2	no	—
170	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
171	C₂×C₁₇	C₂×#504	1	2	yes	—
172	D₁₇	—	2	4	no	—
173	C₂×#41	—	17	1	no	—
174	#131	—	17	8	no	—
175	#131	—	17	8	no	—
176	C₂×M₁₆	—	17	1	no	—
177	C₂×M₁₆	—	17	1	no	—
178	C₂×M₁₆	—	17	2	no	—
179	C₂×M₁₆	—	17	2	no	—
180	C₂²×C₈	—	17	2	no	—
181	C₂²×C₈	—	17	2	no	—
182	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
184	C₄×Q₈	—	17	2	no	—
185	C₄×D₄	—	17	4	no	—
186	C₄×D₄	—	17	2	no	—
187	#117	—	17	1	no	—
188	#117	—	17	2	no	—
189	C₂×C₄²	—	17	2	no	—
190	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
191	M₃₂	—	17	8	no	—
192	M₃₂	—	17	16	no	—
193	C₂×C₁₆	—	17	4	no	—
194	C₂×C₁₆	—	17	16	no	—
195	C₂×C₁₆	C₄×D₁₇	1	8	yes	complemented
196	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
197	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
198	#97	—	17	2	no	—
199	#97	—	17	2	no	—
200	#97	—	17	2	no	—
201	C₄×C₈	—	17	2	no	—
202	C₄×C₈	—	17	2	no	—
203	C₄×C₈	—	17	2	no	—
204	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
205	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
208	C₁₇	C₄×#504	1	1	yes	characteristic, complemented, Sylow
209	#41	—	17	4	no	—
210	#41	—	17	4	no	—
211	C₂×Q₈	—	17	1	no	—
212	C₂×D₄	—	17	2	no	—
213	C₂×D₄	—	17	1	no	—
214	C₂²×C₄	—	17	2	no	—
215	C₂²×C₄	—	17	4	no	—
216	C₂²×C₄	C₈×D₁₇	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
218	M₁₆	—	17	4	no	—
219	M₁₆	—	17	4	no	—
220	M₁₆	—	17	8	no	—
221	M₁₆	—	17	8	no	—
222	C₂×C₈	—	17	2	no	—
223	C₂×C₈	—	17	2	no	—
224	C₂×C₈	—	17	8	no	—
225	C₂×C₈	—	17	8	no	—
226	C₂×C₈	C₈×D₁₇	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
232	C₄⋊C₄	—	17	4	no	—
233	C₄⋊C₄	—	17	2	no	—
234	C₂²⋊C₄	—	17	2	no	—
235	C₂²⋊C₄	—	17	4	no	—
236	C₄²	—	17	2	no	—
237	C₄²	—	17	4	no	—
238	C₄²	C₈×D₁₇	1	2	yes	—
239	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
241	C₁₆	—	17	8	no	—
242	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
243	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
244	C₂³	—	17	2	no	—
245	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
246	Q₈	—	17	4	no	—
247	D₄	—	17	8	no	—
248	D₄	—	17	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	4	no	—
252	C₂×C₄	—	17	4	no	—
253	C₂×C₄	—	17	4	no	—
254	C₂×C₄	—	34	8	no	—
255	C₂×C₄	C₄²×D₁₇	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
261	C₈	—	17	4	no	—
262	C₈	—	17	4	no	—
263	C₈	C₄²×D₁₇	1	2	yes	—
264	C₈	C₂×C₈×D₁₇	1	2	yes	—
265	C₈	C₂×C₈×D₁₇	1	2	yes	—
266	C₈	C₂²×C₄×D₁₇	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	17	4	no	—
269	C₂²	—	34	4	no	—
270	C₂²	C₄×C₈×D₁₇	1	2	yes	—
271	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	17	4	no	—
274	C₄	—	34	4	no	—
275	C₄	C₄×C₈×D₁₇	1	2	yes	—
276	C₄	#11763225	1	4	yes	direct product factor
277	C₄	C₂×C₄²×D₁₇	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	34	4	no	—
281	C₂	C₂×C₄×C₈×D₁₇	1	1	yes	characteristic
282	C₂	C₂×#11763225	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	C₄×#504
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14530380	C₂	semidirect
4. #14530379	C₂	semidirect
5. C₄×#10587483	C₂	semidirect
6. C₄×#10587469	C₂	semidirect
7. C₄×C₁₆×D₁₇	C₂	semidirect
8. C₂×C₄×C₁₆×C₁₇	C₂	semidirect
12. #11763225	C₄	direct (N × H)
50. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
61. #10587483	C₂×C₄	semidirect
65. #10587469	C₂×C₄	semidirect
66. C₁₆×D₁₇	C₂×C₄	semidirect
133. C₂×C₄×C₁₆	D₁₇	semidirect
141. C₄×C₁₇	#504	semidirect
195. C₂×C₁₆	C₄×D₁₇	semidirect
208. C₁₇	C₄×#504	semidirect
276. C₄	#11763225	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763225	2
C₄×#10587587	2
#14530380	2
#14530379	2
C₄×#10587483	2
C₄×#10587469	2
C₄×C₁₆×D₁₇	2
C₂×C₄×C₁₆×C₁₇	2
C₄×#504	17

Subgroups — C4×#11763225 #17186966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11763225 #17186966