Subgroups — #17053273

Order 4352.

822 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#14529542	C₂	1	3	yes	maximal
4	#14529540	C₂	1	3	yes	maximal
5	#14528911	C₂	1	3	yes	maximal
6	C₄×#10587478	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₁₇	—	2	1	no	—
9	Q₈×#3622	C₄	1	8	yes	complemented
10	C₂×#10587478	C₂²	1	3	yes	—
11	C₂×#10587478	C₂²	1	3	yes	—
12	C₂×C₄⋊#3622	C₂²	1	3	yes	—
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	1	yes	characteristic
20	C₂×C₄×Q₈×C₁₇	—	2	1	no	—
21	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#389	—	2	3	no	—
23	C₁₇×#384	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₁₇	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₇	—	4	4	no	—
27	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
28	C₄×Q₈×C₁₇	Q₈	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	3	no	—
31	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₁₇	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	—	2	3	no	—
35	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
36	Q₈×Dic₁₇	—	4	4	no	—
37	C₂×C₄⋊Dic₁₇	—	2	3	no	—
38	C₂×C₄⋊Dic₁₇	—	2	3	no	—
39	C₂×C₄⋊Dic₁₇	C₂³	1	3	yes	—
40	C₂×C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₁₇	—	2	3	no	—
42	C₂²×#3622	C₂³	1	1	yes	characteristic
43	C₂²×#3622	C₂³	1	3	yes	—
44	C₁₇×#95	—	2	3	no	—
45	#10587502	—	2	3	no	—
46	#10587478	—	2	6	no	—
47	#10587478	—	2	6	no	—
48	C₄⋊#3622	C₂×C₄	1	12	yes	—
49	C₄×#3622	C₂×C₄	1	12	yes	—
50	C₂×Q₈×C₁₇	C₄⋊C₄	1	4	yes	—
51	C₂×Q₈×C₁₇	D₈	1	4	yes	—
52	C₂×Q₈×C₁₇	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	—	2	3	no	—
55	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₁₇	—	2	1	no	—
59	C₂²×Dic₁₇	—	2	3	no	—
60	C₄⋊C₄×C₁₇	—	2	1	no	—
61	C₄⋊C₄×C₁₇	—	2	3	no	—
62	C₄⋊C₄×C₁₇	—	2	3	no	—
63	C₄⋊C₄×C₁₇	—	4	6	no	—
64	C₄⋊C₄×C₁₇	—	8	3	no	—
65	C₄⋊C₄×C₁₇	C₄⋊C₄	1	12	yes	—
66	C₄⋊C₄×C₁₇	#41	1	2	yes	—
67	C₄²×C₁₇	—	4	6	no	—
68	C₄²×C₁₇	C₂²×C₄	1	6	yes	—
69	C₄²×C₁₇	C₂×Q₈	1	6	yes	—
70	C₄⋊Dic₁₇	—	2	1	no	—
71	C₄⋊Dic₁₇	—	2	3	no	—
72	C₄⋊Dic₁₇	—	2	3	no	—
73	C₄⋊Dic₁₇	—	4	6	no	—
74	C₄⋊Dic₁₇	—	8	3	no	—
75	C₄⋊Dic₁₇	#41	1	2	yes	—
76	C₄×Dic₁₇	—	4	6	no	—
77	C₂×#3622	—	2	3	no	—
78	C₂×#3622	—	2	3	no	—
79	C₂×#3622	—	2	1	no	—
80	C₂×#3622	—	2	1	no	—
81	C₂×#3622	C₂²×C₄	1	2	yes	—
82	C₂×#3622	C₂²×C₄	1	6	yes	—
83	#22740	—	17	1	no	maximal, Sylow
84	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₇	#107	1	16	yes	complemented
86	C₂×C₄×C₁₇	—	2	3	no	—
87	C₂×C₄×C₁₇	—	2	3	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	4	1	no	—
90	C₂×C₄×C₁₇	—	4	1	no	—
91	C₂×C₄×C₁₇	—	4	3	no	—
92	C₂×C₄×C₁₇	—	4	3	no	—
93	C₂×C₄×C₁₇	—	4	3	no	—
94	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
96	C₂×C₄×C₁₇	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
98	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂×Dic₁₇	—	2	2	no	—
105	C₂×Dic₁₇	—	4	1	no	—
106	C₂×Dic₁₇	—	4	1	no	—
107	C₂×Dic₁₇	—	4	3	no	—
108	C₂×Dic₁₇	—	4	3	no	—
109	C₂×Dic₁₇	—	4	3	no	—
110	#3622	—	2	6	no	—
111	#3622	—	2	2	no	—
112	C₂×C₈×Q₈	—	17	1	no	—
113	Q₈×C₄⋊C₄	—	17	1	no	—
114	Q₈×C₄⋊C₄	D₁₇	1	1	yes	characteristic
115	#1947	—	17	3	no	—
116	#1947	—	17	3	no	—
117	#1834	—	17	3	no	—
118	C₄×#107	—	17	3	no	—
119	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₁₇	#548	1	1	yes	characteristic
121	C₂²×C₁₇	#548	1	1	yes	characteristic
122	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
123	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂²×C₁₇	#577	1	1	yes	characteristic
125	C₂²×C₁₇	#577	1	1	yes	characteristic
126	C₄×C₁₇	—	2	6	no	—
127	C₄×C₁₇	—	4	2	no	—
128	C₄×C₁₇	—	8	3	no	—
129	C₄×C₁₇	C₂×#107	1	12	yes	—
130	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
131	C₄×C₁₇	#519	1	1	yes	characteristic
132	C₄×C₁₇	#557	1	1	yes	characteristic
133	C₄×C₁₇	#557	1	1	yes	characteristic
134	Dic₁₇	—	4	2	no	—
135	Dic₁₇	—	8	3	no	—
136	C₂×C₄×Q₈	—	2	1	no	—
137	C₂×C₄×Q₈	—	34	1	no	—
138	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
139	C₈×Q₈	—	17	8	no	—
140	C₂×#107	—	17	3	no	—
141	C₂×#107	—	17	3	no	—
142	C₂×C₄⋊C₈	—	17	3	no	—
143	C₂×C₄×C₈	—	17	3	no	—
144	#389	—	2	3	no	—
145	#389	—	34	3	no	—
146	#384	—	17	3	no	—
147	#384	C₂×D₁₇	1	3	yes	—
148	C₄×C₄⋊C₄	—	17	3	no	—
149	C₄×C₄⋊C₄	C₂×D₁₇	1	3	yes	—
150	#340	—	17	3	no	—
151	#340	—	17	6	no	—
152	#340	—	17	1	no	—
153	C₂×C₁₇	#2351	1	1	yes	characteristic
154	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
155	C₂×C₁₇	#2912	1	1	yes	characteristic
156	C₂×C₁₇	#3358	1	1	yes	characteristic
157	C₂×C₁₇	#3358	1	1	yes	characteristic
158	C₂×C₁₇	#3363	1	1	yes	characteristic
159	C₂×C₁₇	#3363	1	1	yes	characteristic
160	C₂²×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
161	C₂²×C₈	—	17	1	no	—
162	C₂²×C₈	—	17	3	no	—
163	C₄×Q₈	—	4	4	no	—
164	C₄×Q₈	—	68	4	no	—
165	C₄×Q₈	Dic₃₄	1	4	yes	—
166	C₄×Q₈	C₄×D₁₇	1	4	yes	—
167	C₂×C₄⋊C₄	—	2	3	no	—
168	C₂×C₄⋊C₄	—	2	3	no	—
169	C₂×C₄⋊C₄	—	17	1	no	—
170	C₂×C₄⋊C₄	—	17	3	no	—
171	C₂×C₄⋊C₄	—	34	3	no	—
172	C₂×C₄⋊C₄	—	34	3	no	—
173	C₂×C₄⋊C₄	C₂²⋊D₁₇	1	3	yes	—
174	C₂×C₄⋊C₄	C₂²×D₁₇	1	3	yes	—
175	C₂×C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
176	C₂×C₄²	—	2	3	no	—
177	C₂×C₄²	—	34	3	no	—
178	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
179	#107	—	34	6	no	—
180	#107	—	34	6	no	—
181	C₄⋊C₈	—	17	12	no	—
182	C₄×C₈	—	17	12	no	—
183	#95	—	2	3	no	—
184	#95	—	34	3	no	—
185	C₁₇	#22740	1	1	yes	characteristic, complemented, Sylow
186	C₂×Q₈	#63225	1	4	yes	—
187	C₂×Q₈	#63228	1	4	yes	—
188	C₂×Q₈	#63231	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	3	no	—
191	C₂²×C₄	—	34	1	no	—
192	C₂²×C₄	—	34	3	no	—
193	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
195	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
196	C₂×C₈	—	17	2	no	—
197	C₂×C₈	—	17	6	no	—
198	C₂×C₈	—	34	3	no	—
199	C₂×C₈	—	34	1	no	—
200	C₂×C₈	—	34	3	no	—
201	C₂×C₈	—	34	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	3	no	—
204	C₄⋊C₄	—	2	3	no	—
205	C₄⋊C₄	—	4	6	no	—
206	C₄⋊C₄	—	8	3	no	—
207	C₄⋊C₄	—	17	2	no	—
208	C₄⋊C₄	—	34	1	no	—
209	C₄⋊C₄	—	34	3	no	—
210	C₄⋊C₄	—	34	3	no	—
211	C₄⋊C₄	—	68	6	no	—
212	C₄⋊C₄	—	136	3	no	—
213	C₄⋊C₄	#63225	1	12	yes	—
214	C₄⋊C₄	#63256	1	2	yes	—
215	C₄²	—	4	6	no	—
216	C₄²	—	68	6	no	—
217	C₄²	C₂×Dic₃₄	1	6	yes	—
218	C₄²	C₂×C₄×D₁₇	1	6	yes	—
219	C₂³	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
220	Q₈	#10587478	1	16	yes	complemented
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	3	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	34	2	no	—
230	C₂×C₄	—	68	1	no	—
231	C₂×C₄	—	68	1	no	—
232	C₂×C₄	—	68	3	no	—
233	C₂×C₄	—	68	3	no	—
234	C₂×C₄	—	68	3	no	—
235	C₂×C₄	C₂×#63225	1	6	yes	—
236	C₂×C₄	C₂×#63225	1	6	yes	—
237	C₂×C₄	C₂×#63228	1	6	yes	—
238	C₂×C₄	C₂×#63231	1	6	yes	—
239	C₂×C₄	C₂²×Dic₃₄	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
241	C₂×C₄	C₂×#63256	1	1	yes	characteristic
242	C₂×C₄	#10587694	1	1	yes	characteristic
243	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
244	C₂×C₄	#10587697	1	1	yes	characteristic
245	C₈	—	34	6	no	—
246	C₈	—	34	2	no	—
247	C₂²	C₂²×#63225	1	1	yes	characteristic
248	C₂²	#11763466	1	1	yes	characteristic
249	C₂²	#11763470	1	1	yes	characteristic
250	C₂²	C₂²×#63228	1	1	yes	characteristic
251	C₂²	C₂²×#63231	1	1	yes	characteristic
252	C₂²	#11763617	1	1	yes	characteristic
253	C₂²	#11763620	1	1	yes	characteristic
254	C₄	—	2	6	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	3	no	—
257	C₄	—	68	2	no	—
258	C₄	—	136	3	no	—
259	C₄	C₂×#10587478	1	12	yes	—
260	C₄	C₄×#63256	1	1	yes	characteristic
261	C₄	#11763285	1	1	yes	characteristic
262	C₄	#11763535	1	1	yes	characteristic
263	C₄	#11763538	1	1	yes	characteristic
264	C₂	#14531702	1	1	yes	characteristic
265	C₂	C₂²×#10587478	1	1	yes	characteristic
266	C₂	#14537377	1	1	yes	characteristic
267	C₂	#14541415	1	1	yes	characteristic
268	C₂	#14541419	1	1	yes	characteristic
269	C₂	#14541458	1	1	yes	characteristic
270	C₂	#14541462	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	Q₈×C₄⋊C₄×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂²×D₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₁₇
hypocenter	C₁₇	#22740
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×#3622	C₄	semidirect
85. Q₈×C₁₇	#107	semidirect
185. C₁₇	#22740	semidirect
220. Q₈	#10587478	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
#14529542	2
#14529540	2
#14528911	2
C₄×#10587478	2
Q₈×C₄⋊C₄×C₁₇	2
#22740	17