Subgroups — C₄×#11762594 #16949375

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587587	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762594	C₂	1	2	yes	maximal
3	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526874	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587587	C₄	1	1	yes	characteristic
7	C₂×#10587587	C₂²	1	1	yes	characteristic
8	C₄×#63252	C₄	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	2	yes	—
10	#11762764	C₄	1	1	yes	characteristic
11	#11762764	C₂²	1	1	yes	characteristic
12	#11762763	—	2	1	no	—
13	#11762763	—	2	1	no	—
14	C₄×#63223	C₄	1	1	yes	characteristic
15	C₄×#63218	—	2	1	no	—
16	C₄×C₈×D₁₇	—	2	1	no	—
17	#11762594	C₄	1	8	yes	direct product factor
18	C₄×#63214	—	2	1	no	—
19	C₄×#63214	—	2	1	no	—
20	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#346	—	2	1	no	—
23	C₄×C₁₆×C₁₇	—	2	1	no	—
24	C₂×#63252	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₇	—	2	2	no	—
29	#10587612	C₂×C₄	1	1	yes	characteristic
30	C₂×#63223	C₂×C₄	1	1	yes	characteristic
31	C₂×#63223	C₂×C₄	1	1	yes	characteristic
32	#10587587	—	2	2	no	—
33	#10587587	—	2	2	no	—
34	#10587587	C₈	1	4	yes	—
35	#10587587	C₂×C₄	1	4	yes	—
36	C₂×#63218	—	2	1	no	—
37	C₂×#63218	—	2	1	no	—
38	C₂×C₈×D₁₇	—	2	1	no	—
39	C₂×C₈×D₁₇	—	2	1	no	—
40	C₄×D₆₈	C₈	1	2	yes	—
41	#10587558	—	2	1	no	—
42	C₄²×D₁₇	—	2	1	no	—
43	C₄×Dic₃₄	C₈	1	2	yes	—
44	C₁₇×M₃₂	—	2	2	no	—
45	C₁₇×M₃₂	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₆×C₁₇	—	2	2	no	—
47	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
51	#10587486	—	2	1	no	—
52	#10587486	—	2	1	no	—
53	C₈×Dic₁₇	—	2	1	no	—
54	C₈×Dic₁₇	—	2	1	no	—
55	C₂×#63214	—	2	2	no	—
56	C₂×#63214	—	2	2	no	—
57	C₂×#63214	C₂×C₄	1	4	yes	complemented
58	#10587474	—	2	1	no	—
59	C₄×#3622	—	2	1	no	—
60	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	#63252	—	2	1	no	—
64	#63252	—	2	1	no	—
65	#63252	C₄²	1	2	yes	—
66	#63252	C₂×C₈	1	2	yes	—
67	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	2	no	—
70	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
75	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	#41	1	2	yes	—
79	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂×C₈	1	2	yes	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63227	—	2	2	no	—
84	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
85	#63225	—	2	2	no	—
86	C₄×Dic₁₇	—	2	2	no	—
87	C₄×Dic₁₇	—	2	1	no	—
88	#63223	—	2	1	no	—
89	#63223	—	2	1	no	—
90	#63223	C₄²	1	2	yes	—
91	#63223	C₂×C₈	1	2	yes	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	1	no	—
94	#63218	—	2	4	no	—
95	#63218	—	2	4	no	—
96	C₈×D₁₇	—	2	4	no	—
97	C₈×D₁₇	—	2	4	no	—
98	C₁₆×C₁₇	—	2	4	no	—
99	#63214	—	2	4	no	—
100	C₄×#350	—	17	1	no	maximal, Sylow
101	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂²×D₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	4	no	—
104	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#117	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	C₂×M₁₆	1	1	yes	characteristic
112	C₂²⋊D₁₇	—	2	2	no	—
113	C₂²⋊D₁₇	—	2	2	no	—
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	2	no	—
116	D₆₈	—	2	1	no	—
117	D₆₈	C₄×C₈	1	2	yes	—
118	C₄×D₁₇	—	2	2	no	—
119	C₄×D₁₇	—	2	2	no	—
120	C₄×D₁₇	—	4	4	no	—
121	Dic₃₄	—	2	1	no	—
122	Dic₃₄	C₄×C₈	1	2	yes	—
123	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
124	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
125	C₈×C₁₇	C₄×D₄	1	2	yes	—
126	C₈×C₁₇	C₄×D₄	1	2	yes	—
127	#3622	—	2	2	no	—
128	#3622	—	2	2	no	—
129	C₄×#131	—	17	1	no	—
130	C₂×#350	—	17	2	no	—
131	C₄×M₃₂	D₁₇	1	1	yes	characteristic, complemented
132	C₂×C₄×C₁₆	—	17	1	no	—
133	#1379	—	17	2	no	—
134	C₂²×C₁₇	—	2	2	no	—
135	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
136	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₁₇	#431	1	1	yes	characteristic
138	C₂×D₁₇	—	2	2	no	—
139	C₂×D₁₇	—	4	2	no	—
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	#350	1	4	yes	complemented
142	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
144	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₁₇	#431	1	1	yes	characteristic
146	Dic₁₇	—	2	2	no	—
147	Dic₁₇	—	4	2	no	—
148	C₂×#131	—	17	1	no	—
149	C₂×#131	—	17	1	no	—
150	C₄×#41	—	17	1	no	—
151	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
152	C₂²×C₁₆	—	17	2	no	—
153	#405	—	17	1	no	—
154	#405	—	17	1	no	—
155	#405	—	34	1	no	—
156	#405	—	34	1	no	—
157	C₄×M₁₆	—	17	1	no	—
158	C₄×M₁₆	—	34	1	no	—
159	C₂×C₄×C₈	—	34	1	no	—
160	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
161	#350	—	17	8	no	—
162	#346	—	2	1	no	—
163	C₄×C₁₆	—	2	1	no	—
164	C₄×C₁₆	—	34	1	no	—
165	C₄×C₁₆	—	34	1	no	—
166	C₂×C₁₇	—	2	1	no	—
167	C₂×C₁₇	—	2	1	no	—
168	C₂×C₁₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
169	C₂×C₁₇	C₂×#350	1	2	yes	—
170	D₁₇	—	4	2	no	—
171	C₂×#41	—	17	1	no	—
172	#131	—	17	4	no	—
173	#131	—	17	4	no	—
174	#131	—	34	2	no	—
175	#131	—	34	2	no	—
176	C₂×M₁₆	—	17	1	no	—
177	C₂×M₁₆	—	17	1	no	—
178	C₂×M₁₆	—	34	1	no	—
179	C₂×M₁₆	—	34	1	no	—
180	C₂²×C₈	—	34	1	no	—
181	C₂²×C₈	—	34	1	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	2	no	—
186	C₄×D₄	—	34	2	no	—
187	#117	—	17	1	no	—
188	#117	—	34	1	no	—
189	C₂×C₄²	—	34	1	no	—
190	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	C₄×D₁₇	1	4	yes	complemented
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	17	4	no	—
195	C₂×C₁₆	—	34	2	no	—
196	C₂×C₁₆	—	34	2	no	—
197	#97	—	34	1	no	—
198	#97	—	34	1	no	—
199	#97	—	34	1	no	—
200	C₄×C₈	—	34	1	no	—
201	C₄×C₈	—	34	1	no	—
202	C₄×C₈	—	34	1	no	—
203	C₄×C₈	D₆₈	1	1	yes	characteristic
204	C₄×C₈	D₆₈	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
207	C₁₇	C₄×#350	1	1	yes	characteristic, complemented, Sylow
208	#41	—	17	2	no	—
209	#41	—	17	2	no	—
210	#41	—	34	1	no	—
211	#41	—	34	1	no	—
212	C₂×Q₈	—	17	1	no	—
213	C₂×D₄	—	17	1	no	—
214	C₂×D₄	—	34	1	no	—
215	C₂²×C₄	—	34	1	no	—
216	C₂²×C₄	—	34	2	no	—
217	C₂²×C₄	#63218	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
219	M₁₆	—	17	2	no	—
220	M₁₆	—	17	2	no	—
221	M₁₆	—	34	1	no	—
222	M₁₆	—	34	1	no	—
223	M₁₆	—	34	4	no	—
224	M₁₆	—	34	4	no	—
225	C₂×C₈	—	34	1	no	—
226	C₂×C₈	—	34	1	no	—
227	C₂×C₈	—	34	4	no	—
228	C₂×C₈	—	34	4	no	—
229	C₂×C₈	C₈×D₁₇	1	2	yes	—
230	C₂×C₈	#63218	1	2	yes	—
231	C₂×C₈	#63227	1	1	yes	characteristic
232	C₂×C₈	#63227	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
234	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
235	C₂×C₈	#63252	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
237	C₄⋊C₄	—	17	2	no	—
238	C₄⋊C₄	—	34	2	no	—
239	C₂²⋊C₄	—	34	1	no	—
240	C₂²⋊C₄	—	34	2	no	—
241	C₄²	—	34	2	no	—
242	C₄²	—	34	1	no	—
243	C₄²	C₈×D₁₇	1	2	yes	—
244	C₄²	#63227	1	1	yes	characteristic
245	C₄²	#63227	1	1	yes	characteristic
246	C₁₆	—	2	4	no	—
247	C₁₆	—	34	4	no	—
248	C₂³	—	34	1	no	—
249	C₂³	C₂×#63218	1	1	yes	characteristic
250	Q₈	—	17	2	no	—
251	Q₈	—	34	1	no	—
252	D₄	—	17	2	no	—
253	D₄	—	34	2	no	—
254	D₄	—	34	2	no	—
255	D₄	—	34	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	34	1	no	—
258	C₂×C₄	—	34	2	no	—
259	C₂×C₄	—	34	2	no	—
260	C₂×C₄	—	34	2	no	—
261	C₂×C₄	—	68	4	no	—
262	C₂×C₄	#10587491	1	2	yes	—
263	C₂×C₄	#10587491	1	2	yes	—
264	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
265	C₂×C₄	#10587558	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
268	C₂×C₄	C₂×#63218	1	1	yes	characteristic
269	C₂×C₄	C₂×#63227	1	1	yes	characteristic
270	C₈	—	34	2	no	—
271	C₈	—	34	2	no	—
272	C₈	#10587491	1	2	yes	—
273	C₈	#10587491	1	2	yes	—
274	C₈	C₄×D₆₈	1	2	yes	—
275	C₈	C₄×C₂²⋊D₁₇	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	34	2	no	—
278	C₂²	—	68	2	no	—
279	C₂²	C₄×#63218	1	1	yes	characteristic
280	C₂²	C₂×#10587491	1	1	yes	characteristic
281	C₂²	#11762860	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	34	2	no	—
284	C₄	—	68	2	no	—
285	C₄	#11762594	1	4	yes	direct product factor
286	C₄	C₄×#63227	1	1	yes	characteristic
287	C₄	C₄×C₈×D₁₇	1	1	yes	characteristic
288	C₄	C₂×#10587491	1	1	yes	characteristic
289	C₄	#11762860	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	68	2	no	—
293	C₂	C₄×#10587491	1	1	yes	characteristic
294	C₂	C₂×#11762594	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	C₄×#350
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#63214	C₂	semidirect
4. #14526874	C₂	semidirect
5. C₄×C₁₇×M₃₂	C₂	semidirect
17. #11762594	C₄	direct (N × H)
45. C₁₇×M₃₂	C₂×C₄	semidirect
57. C₂×#63214	C₂×C₄	semidirect
131. C₄×M₃₂	D₁₇	semidirect
141. C₄×C₁₇	#350	semidirect
192. M₃₂	C₄×D₁₇	semidirect
207. C₁₇	C₄×#350	semidirect
285. C₄	#11762594	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#10587587	2
C₂×#11762594	2
C₂×C₄×#63214	2
#14526874	2
C₄×C₁₇×M₃₂	2
C₄×#350	17

Subgroups — C4×#11762594 #16949375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11762594 #16949375