Subgroups — #16950282

Order 4352.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762630	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530412	C₂	1	2	yes	maximal
4	#14526870	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₇×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587483	C₂²	1	2	yes	—
8	#11762630	—	2	1	no	—
9	#11762630	—	2	1	no	—
10	#11762630	—	2	1	no	—
11	#11762630	—	2	1	no	—
12	#11762593	—	2	4	no	—
13	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
14	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
15	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
16	C₁₇×#405	—	2	1	no	—
17	C₁₇×#405	—	2	1	no	—
18	C₁₇×#405	C₄	1	1	yes	characteristic
19	C₁₇×#405	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₁₇	—	2	1	no	—
21	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₁₇×#131	—	2	2	no	—
26	C₁₇×#131	—	2	1	no	—
27	C₁₇×#131	—	2	1	no	—
28	C₁₇×#131	C₈	1	4	yes	—
29	C₁₇×#131	D₄	1	2	yes	—
30	C₁₇×#131	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₇	—	2	1	no	—
32	C₂×M₁₆×C₁₇	—	2	1	no	—
33	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	—	2	1	no	—
36	C₂²×C₈×C₁₇	—	2	1	no	—
37	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₁₇	—	2	1	no	—
40	C₄×D₄×C₁₇	—	2	1	no	—
41	C₄×D₄×C₁₇	—	2	2	no	—
42	C₁₇×#117	—	2	1	no	—
43	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₇	—	2	1	no	—
45	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#97	—	2	1	no	—
47	C₁₇×#97	—	2	1	no	—
48	C₁₇×#97	—	2	1	no	—
49	C₄×C₈×C₁₇	—	2	1	no	—
50	C₄×C₈×C₁₇	—	2	1	no	—
51	C₄×C₈×C₁₇	—	2	1	no	—
52	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
56	#10587483	—	2	2	no	—
57	#10587483	—	2	2	no	—
58	#10587483	—	2	2	no	—
59	#10587483	—	2	2	no	—
60	C₂×#63214	—	2	2	no	—
61	C₂×#63214	—	2	2	no	—
62	#41×C₁₇	—	2	1	no	—
63	#41×C₁₇	—	2	1	no	—
64	#41×C₁₇	C₄⋊C₄	1	2	yes	—
65	#41×C₁₇	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
67	C₂×D₄×C₁₇	—	2	1	no	—
68	C₂×D₄×C₁₇	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	—	2	1	no	—
70	C₂²×C₄×C₁₇	—	2	2	no	—
71	C₂²×C₄×C₁₇	—	2	1	no	—
72	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₁₇	—	2	1	no	—
74	M₁₆×C₁₇	—	2	1	no	—
75	M₁₆×C₁₇	—	2	4	no	—
76	M₁₆×C₁₇	—	2	2	no	—
77	M₁₆×C₁₇	—	2	2	no	—
78	M₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
79	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₇	—	2	1	no	—
81	C₂×C₈×C₁₇	—	2	1	no	—
82	C₂×C₈×C₁₇	—	2	4	no	—
83	C₂×C₈×C₁₇	—	2	2	no	—
84	C₂×C₈×C₁₇	—	2	2	no	—
85	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
89	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
94	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₁₇	—	2	2	no	—
96	C₄⋊C₄×C₁₇	—	2	1	no	—
97	C₂²⋊C₄×C₁₇	—	2	2	no	—
98	C₂²⋊C₄×C₁₇	M₁₆	1	2	yes	—
99	C₄²×C₁₇	—	2	2	no	—
100	C₄²×C₁₇	—	2	1	no	—
101	C₄²×C₁₇	—	2	1	no	—
102	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
104	#63214	—	4	2	no	—
105	#63214	—	4	2	no	—
106	#12792	—	17	1	no	maximal, Sylow
107	C₂³×C₁₇	—	2	1	no	—
108	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	C₄⋊C₈	1	2	yes	—
111	D₄×C₁₇	—	2	2	no	—
112	D₄×C₁₇	—	2	2	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	2	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	4	2	no	—
122	C₂×C₄×C₁₇	—	4	4	no	—
123	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
129	C₈×C₁₇	—	2	2	no	—
130	C₈×C₁₇	—	2	2	no	—
131	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
132	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
133	C₈×C₁₇	#121	1	1	yes	characteristic
134	C₈×C₁₇	#121	1	1	yes	characteristic
135	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₄×#131	D₁₇	1	1	yes	characteristic
138	C₂×#364	—	17	1	no	—
139	C₂×#364	—	17	1	no	—
140	#2117	—	17	2	no	—
141	#1378	—	17	2	no	—
142	C₂²×C₁₇	—	2	2	no	—
143	C₂²×C₁₇	—	4	1	no	—
144	C₂²×C₁₇	—	4	2	no	—
145	C₂²×C₁₇	#407	1	1	yes	characteristic
146	C₂²×C₁₇	#423	1	1	yes	characteristic
147	C₂²×C₁₇	#431	1	1	yes	characteristic
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	2	2	no	—
150	C₄×C₁₇	—	4	2	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	#380	1	1	yes	characteristic
153	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₄×C₁₇	#432	1	1	yes	characteristic
156	C₂×#131	Dic₁₇	1	1	yes	characteristic
157	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
158	C₄×#41	Dic₁₇	1	1	yes	characteristic
159	C₂×M₃₂	—	17	2	no	—
160	C₂×M₃₂	—	17	2	no	—
161	#405	—	2	1	no	—
162	#405	—	2	1	no	—
163	#405	Dic₁₇	1	1	yes	characteristic
164	#405	C₂×D₁₇	1	1	yes	characteristic
165	C₄×M₁₆	—	2	1	no	—
166	C₄×M₁₆	Dic₁₇	1	1	yes	characteristic
167	C₂×C₄×C₈	—	2	1	no	—
168	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
169	#364	—	34	1	no	—
170	#364	—	34	1	no	—
171	#364	—	34	1	no	—
172	#364	—	34	1	no	—
173	#350	—	34	4	no	—
174	C₂×C₁₇	—	2	1	no	—
175	C₂×C₁₇	—	2	1	no	—
176	C₂×C₁₇	—	2	1	no	—
177	C₂×C₁₇	—	4	2	no	—
178	C₂×C₁₇	#1797	1	1	yes	characteristic
179	C₂×#41	C₂×Dic₁₇	1	1	yes	characteristic
180	#131	—	2	1	no	—
181	#131	—	2	1	no	—
182	#131	—	2	2	no	—
183	#131	#3622	1	4	yes	—
184	#131	Dic₃₄	1	2	yes	—
185	#131	D₆₈	1	2	yes	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
189	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
194	C₄×Q₈	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	—	2	2	no	—
197	#117	—	2	1	no	—
198	#117	C₂×Dic₁₇	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
201	M₃₂	—	34	2	no	—
202	M₃₂	—	34	2	no	—
203	M₃₂	—	34	2	no	—
204	M₃₂	—	34	2	no	—
205	C₂×C₁₆	—	34	2	no	—
206	C₂×C₁₆	—	34	2	no	—
207	#97	—	2	1	no	—
208	#97	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
217	C₁₇	#12792	1	1	yes	characteristic, complemented, Sylow
218	#41	—	2	1	no	—
219	#41	—	2	1	no	—
220	#41	C₂×#3622	1	2	yes	—
221	#41	C₄⋊Dic₁₇	1	2	yes	—
222	C₂×Q₈	#63223	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	#63223	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
229	M₁₆	—	2	1	no	—
230	M₁₆	—	2	1	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	C₂×#3622	1	2	yes	—
235	M₁₆	C₄⋊Dic₁₇	1	2	yes	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×#3622	1	2	yes	—
242	C₂×C₈	#63223	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
245	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
246	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
247	C₂×C₈	#63252	1	1	yes	characteristic
248	C₂×C₈	#63252	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	#63223	1	2	yes	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
259	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
260	C₁₆	—	68	2	no	—
261	C₁₆	—	68	2	no	—
262	C₂³	—	2	1	no	—
263	C₂³	C₂×#63223	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₄⋊#3622	1	2	yes	—
266	D₄	—	2	2	no	—
267	D₄	—	2	2	no	—
268	D₄	—	2	1	no	—
269	D₄	C₄⋊#3622	1	2	yes	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
279	C₂×C₄	C₂×#63223	1	1	yes	characteristic
280	C₂×C₄	C₂×#63223	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
282	C₂×C₄	#10587612	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂²⋊#3622	1	2	yes	—
287	C₈	C₂²⋊#3622	1	2	yes	—
288	C₈	#10587610	1	1	yes	characteristic
289	C₈	#10587610	1	1	yes	characteristic
290	C₈	#10587616	1	1	yes	characteristic
291	C₈	#10587616	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#11762773	1	1	yes	characteristic
296	C₂²	#11762833	1	1	yes	characteristic
297	C₂²	#11762862	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	#11762669	1	1	yes	characteristic
303	C₄	C₂×C₂²⋊#3622	1	1	yes	characteristic
304	C₄	C₂×C₄⋊#3622	1	1	yes	characteristic
305	C₄	#11762866	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	#14528764	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587616
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×#131	C₂
socle	C₂×C₁₇	#1797
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×#131	D₁₇
hypocenter	C₁₇	#12792
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#11762630	C₂	semidirect
2. C₂×#11762630	C₂	semidirect
4. #14526870	C₂	semidirect
217. C₁₇	#12792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762630	2
#14530412	2
#14526870	2
C₄×C₁₇×#131	2
#12792	17