Subgroups — Q₈×#10587506 #17050931

Order 4352.

1086 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14529746	C₂	1	3	yes	maximal
4	#14529733	C₂	1	3	yes	maximal
5	#14528789	C₂	1	3	yes	maximal
6	C₄×#10587506	C₂	1	3	yes	maximal
7	C₂×Q₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₁₇	—	2	1	no	—
9	Q₈×#3622	—	2	4	no	—
10	C₂×C₄⋊#3622	C₂²	1	3	yes	—
11	C₂×#10587506	C₂²	1	3	yes	—
12	C₂×#10587506	C₂²	1	1	yes	characteristic
13	C₂×C₄×#3622	C₂²	1	3	yes	—
14	#11762708	—	2	3	no	—
15	#11762679	C₂²	1	3	yes	—
16	C₄×C₄⋊Dic₁₇	C₂²	1	3	yes	—
17	#11762571	C₂²	1	6	yes	—
18	#11762571	C₂²	1	3	yes	—
19	#11762571	C₂²	1	3	yes	—
20	Q₈²×C₁₇	—	2	4	no	—
21	Q₈²×C₁₇	C₄	1	8	yes	complemented
22	C₂×C₁₇×C₄⋊Q₈	—	2	3	no	—
23	C₂×C₁₇×C₄⋊Q₈	C₂²	1	3	yes	—
24	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
25	C₂×C₄×Q₈×C₁₇	C₂²	1	3	yes	—
26	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₇	—	2	3	no	—
28	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₁₇×C₄⋊Q₈	—	2	24	no	—
31	C₁₇×C₄⋊Q₈	—	2	6	no	—
32	C₁₇×C₄⋊Q₈	C₂×C₄	1	12	yes	—
33	C₄×Q₈×C₁₇	—	2	8	no	—
34	C₄×Q₈×C₁₇	—	2	6	no	—
35	C₄×Q₈×C₁₇	C₂×C₄	1	12	yes	—
36	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₁₇	D₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
39	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
40	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
41	C₂×C₄⋊C₄×C₁₇	C₂³	1	3	yes	—
42	C₂×C₄²×C₁₇	—	2	3	no	—
43	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
44	Q₈×Dic₁₇	—	4	4	no	—
45	C₂×C₄⋊Dic₁₇	—	2	3	no	—
46	C₂×C₄⋊Dic₁₇	—	2	3	no	—
47	C₂×C₄⋊Dic₁₇	C₂³	1	3	yes	—
48	C₂×C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₁₇	—	2	3	no	—
50	C₂²×#3622	C₂³	1	3	yes	—
51	C₂²×#3622	C₂³	1	1	yes	characteristic
52	#10587506	—	2	6	no	—
53	#10587506	Q₈	1	4	yes	direct product factor
54	#10587502	—	2	3	no	—
55	C₄⋊#3622	—	2	6	no	—
56	C₄×#3622	—	2	6	no	—
57	C₂×Q₈×C₁₇	—	2	12	no	—
58	C₂×Q₈×C₁₇	—	2	1	no	—
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂×Q₈×C₁₇	—	8	3	no	—
62	C₂×Q₈×C₁₇	—	8	3	no	—
63	C₂×Q₈×C₁₇	C₂²⋊C₄	1	4	yes	—
64	C₂×Q₈×C₁₇	SD₁₆	1	4	yes	—
65	C₂×Q₈×C₁₇	Q₁₆	1	4	yes	—
66	C₂×Q₈×C₁₇	C₂²×C₄	1	2	yes	—
67	C₂×Q₈×C₁₇	C₂×Q₈	1	2	yes	—
68	C₂×Q₈×C₁₇	#41	1	2	yes	—
69	C₂²×C₄×C₁₇	—	2	1	no	—
70	C₂²×C₄×C₁₇	—	2	3	no	—
71	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₁₇	—	2	1	no	—
75	C₂²×Dic₁₇	—	2	3	no	—
76	C₄⋊C₄×C₁₇	—	2	12	no	—
77	C₄⋊C₄×C₁₇	—	2	12	no	—
78	C₄⋊C₄×C₁₇	—	2	3	no	—
79	C₄⋊C₄×C₁₇	C₂²⋊C₄	1	12	yes	—
80	C₄⋊C₄×C₁₇	C₂²×C₄	1	6	yes	—
81	C₄²×C₁₇	—	2	12	no	—
82	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
83	C₄²×C₁₇	C₂×D₄	1	6	yes	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	3	no	—
86	C₄⋊Dic₁₇	—	2	3	no	—
87	C₄⋊Dic₁₇	—	4	6	no	—
88	C₄⋊Dic₁₇	—	8	3	no	—
89	C₄⋊Dic₁₇	C₂×Q₈	1	2	yes	—
90	C₄×Dic₁₇	—	4	6	no	—
91	C₂×#3622	—	2	1	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	3	no	—
94	C₂×#3622	—	2	3	no	—
95	C₂×#3622	—	2	3	no	—
96	C₂×#3622	C₂×Q₈	1	2	yes	—
97	Q₈×#103	—	17	1	no	maximal, Sylow
98	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₁₇	—	2	2	no	—
100	Q₈×C₁₇	—	2	2	no	—
101	Q₈×C₁₇	—	2	2	no	—
102	Q₈×C₁₇	—	8	12	no	—
103	Q₈×C₁₇	#103	1	16	yes	complemented
104	Q₈×C₁₇	C₄×Q₈	1	4	yes	complemented
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	2	3	no	—
109	C₂×C₄×C₁₇	—	2	3	no	—
110	C₂×C₄×C₁₇	—	2	6	no	—
111	C₂×C₄×C₁₇	—	4	3	no	—
112	C₂×C₄×C₁₇	—	4	3	no	—
113	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	6	yes	—
114	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₄×C₁₇	C₂×SD₁₆	1	6	yes	—
116	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
117	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₁₇	—	2	2	no	—
124	C₂×Dic₁₇	—	4	1	no	—
125	C₂×Dic₁₇	—	4	1	no	—
126	C₂×Dic₁₇	—	4	3	no	—
127	C₂×Dic₁₇	—	4	3	no	—
128	C₂×Dic₁₇	—	4	3	no	—
129	#3622	—	2	2	no	—
130	#3622	—	4	3	no	—
131	C₂×Q₈²	D₁₇	1	1	yes	characteristic
132	C₂×C₈×Q₈	—	17	1	no	—
133	Q₈×C₄⋊C₄	—	17	1	no	—
134	#1990	—	17	3	no	—
135	#1987	—	17	3	no	—
136	#1803	—	17	3	no	—
137	C₄×#103	—	17	3	no	—
138	C₂²×C₁₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₇	#548	1	1	yes	characteristic
140	C₂²×C₁₇	D₄×Q₈	1	1	yes	characteristic
141	C₂²×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
142	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
143	C₂²×C₁₇	#576	1	1	yes	characteristic
144	C₂²×C₁₇	#577	1	1	yes	characteristic
145	C₄×C₁₇	—	2	4	no	—
146	C₄×C₁₇	—	2	6	no	—
147	C₄×C₁₇	—	4	6	no	—
148	C₄×C₁₇	C₂×#103	1	12	yes	—
149	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₁₇	#519	1	1	yes	characteristic
151	C₄×C₁₇	#548	1	1	yes	characteristic
152	C₄×C₁₇	D₄×Q₈	1	1	yes	characteristic
153	Dic₁₇	—	4	2	no	—
154	Dic₁₇	—	8	3	no	—
155	Q₈²	—	2	4	no	—
156	Q₈²	Dic₁₇	1	8	yes	complemented
157	C₂×C₄⋊Q₈	—	2	3	no	—
158	C₂×C₄⋊Q₈	C₂×D₁₇	1	3	yes	—
159	C₂×C₄×Q₈	—	2	1	no	—
160	C₂×C₄×Q₈	—	34	1	no	—
161	C₂×C₄×Q₈	C₂×D₁₇	1	3	yes	—
162	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
163	C₈×Q₈	—	34	4	no	—
164	C₂×C₄⋊C₈	—	17	3	no	—
165	C₂×#103	—	17	1	no	—
166	C₂×#103	—	17	3	no	—
167	C₂×C₄×C₈	—	17	3	no	—
168	#389	—	34	3	no	—
169	#384	—	17	3	no	—
170	C₄×C₄⋊C₄	—	17	3	no	—
171	#340	—	17	6	no	—
172	#340	—	17	3	no	—
173	#340	—	17	3	no	—
174	C₂×C₁₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₁₇	C₂²×#103	1	1	yes	characteristic
176	C₂×C₁₇	#2897	1	1	yes	characteristic
177	C₂×C₁₇	#3362	1	1	yes	characteristic
178	C₂×C₁₇	#3363	1	1	yes	characteristic
179	C₂×C₁₇	Q₈×SD₁₆	1	1	yes	characteristic
180	C₂×C₁₇	Q₈×Q₁₆	1	1	yes	characteristic
181	C₂²×Q₈	—	2	3	no	—
182	C₂²×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
183	C₂²×Q₈	C₂²×D₁₇	1	1	yes	characteristic
184	C₂²×C₈	—	17	1	no	—
185	C₂²×C₈	—	17	3	no	—
186	C₄⋊Q₈	—	2	24	no	—
187	C₄⋊Q₈	—	2	6	no	—
188	C₄⋊Q₈	C₂×Dic₁₇	1	12	yes	—
189	C₄×Q₈	—	2	8	no	—
190	C₄×Q₈	—	2	6	no	—
191	C₄×Q₈	—	68	4	no	—
192	C₄×Q₈	C₂×Dic₁₇	1	12	yes	—
193	C₄×Q₈	C₂×Dic₁₇	1	4	yes	—
194	C₄×Q₈	C₂²⋊D₁₇	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	17	1	no	—
198	C₂×C₄⋊C₄	—	17	3	no	—
199	C₂×C₄⋊C₄	—	34	3	no	—
200	C₂×C₄⋊C₄	—	34	3	no	—
201	C₂×C₄⋊C₄	C₂²⋊D₁₇	1	3	yes	—
202	C₂×C₄⋊C₄	C₂²×D₁₇	1	3	yes	—
203	C₂×C₄²	—	2	3	no	—
204	C₂×C₄²	—	34	3	no	—
205	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
206	C₄⋊C₈	—	34	6	no	—
207	#103	—	17	4	no	—
208	#103	—	34	6	no	—
209	C₄×C₈	—	34	6	no	—
210	#95	—	34	3	no	—
211	C₁₇	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	12	no	—
216	C₂×Q₈	—	8	3	no	—
217	C₂×Q₈	—	8	3	no	—
218	C₂×Q₈	#63230	1	4	yes	—
219	C₂×Q₈	#63231	1	4	yes	—
220	C₂×Q₈	C₂²⋊Dic₁₇	1	4	yes	—
221	C₂×Q₈	Q₈×D₁₇	1	2	yes	—
222	C₂×Q₈	#63256	1	2	yes	—
223	C₂×Q₈	C₂²×Dic₁₇	1	2	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	—	34	3	no	—
228	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
230	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
231	C₂×C₈	—	17	2	no	—
232	C₂×C₈	—	34	1	no	—
233	C₂×C₈	—	34	1	no	—
234	C₂×C₈	—	34	3	no	—
235	C₂×C₈	—	34	3	no	—
236	C₂×C₈	—	34	3	no	—
237	C₄⋊C₄	—	2	3	no	—
238	C₄⋊C₄	—	2	12	no	—
239	C₄⋊C₄	—	2	12	no	—
240	C₄⋊C₄	—	17	2	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	3	no	—
243	C₄⋊C₄	—	34	3	no	—
244	C₄⋊C₄	—	68	6	no	—
245	C₄⋊C₄	—	136	3	no	—
246	C₄⋊C₄	C₂²⋊Dic₁₇	1	12	yes	—
247	C₄⋊C₄	C₂²×Dic₁₇	1	6	yes	—
248	C₄²	—	2	12	no	—
249	C₄²	—	68	6	no	—
250	C₄²	C₂²×Dic₁₇	1	6	yes	—
251	C₄²	C₂×C₂²⋊D₁₇	1	6	yes	—
252	C₂³	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	8	12	no	—
257	Q₈	#10587506	1	16	yes	direct product factor
258	Q₈	Q₈×Dic₁₇	1	4	yes	complemented
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	3	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	4	3	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	34	2	no	—
268	C₂×C₄	—	68	1	no	—
269	C₂×C₄	—	68	1	no	—
270	C₂×C₄	—	68	3	no	—
271	C₂×C₄	—	68	3	no	—
272	C₂×C₄	—	68	3	no	—
273	C₂×C₄	C₂×#63230	1	6	yes	—
274	C₂×C₄	C₂×#63231	1	6	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	6	yes	—
276	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	6	yes	—
277	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#63256	1	1	yes	characteristic
279	C₂×C₄	#10587694	1	1	yes	characteristic
280	C₂×C₄	#10587694	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
283	C₈	—	34	2	no	—
284	C₈	—	68	3	no	—
285	C₂²	C₂²×C₂²⋊Dic₁₇	1	1	yes	characteristic
286	C₂²	#11763470	1	1	yes	characteristic
287	C₂²	Q₈×C₂²⋊D₁₇	1	1	yes	characteristic
288	C₂²	C₂²×#63230	1	1	yes	characteristic
289	C₂²	C₂²×#63231	1	1	yes	characteristic
290	C₂²	#11763612	1	1	yes	characteristic
291	C₂²	#11763620	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	6	no	—
294	C₄	—	4	6	no	—
295	C₄	—	68	2	no	—
296	C₄	—	136	3	no	—
297	C₄	C₂×#10587506	1	12	yes	—
298	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
299	C₄	#11763287	1	1	yes	characteristic
300	C₄	#11763470	1	1	yes	characteristic
301	C₄	Q₈×C₂²⋊D₁₇	1	1	yes	characteristic
302	C₂	Q₈×C₂²⋊Dic₁₇	1	1	yes	characteristic
303	C₂	C₂²×#10587506	1	1	yes	characteristic
304	C₂	#14537310	1	1	yes	characteristic
305	C₂	#14541455	1	1	yes	characteristic
306	C₂	#14541462	1	1	yes	characteristic
307	C₂	Q₈×#63230	1	1	yes	characteristic
308	C₂	Q₈×#63231	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₂×Q₈²×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×Q₈²	D₁₇
hypocenter	C₁₇	Q₈×#103
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
21. Q₈²×C₁₇	C₄	semidirect
53. #10587506	Q₈	direct (N × H)
103. Q₈×C₁₇	#103	semidirect
104. Q₈×C₁₇	C₄×Q₈	semidirect
156. Q₈²	Dic₁₇	semidirect
211. C₁₇	Q₈×#103	semidirect
257. Q₈	#10587506	direct (N × H)
258. Q₈	Q₈×Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#3622	2
Q₈×C₄⋊Dic₁₇	2
#14529746	2
#14529733	2
#14528789	2
C₄×#10587506	2
C₂×Q₈²×C₁₇	2
Q₈×#103	17

Subgroups — Q8×#10587506 #17050931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587506 #17050931