Subgroups — #17982857

Order 4352.

734 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14546028	C₂	1	2	yes	maximal
2	#14545909	C₂	1	6	yes	maximal, complemented
3	#14545908	C₂	1	3	yes	maximal, complemented
4	C₂×#11763898	C₂	1	3	yes	maximal, complemented
5	#131×#595	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3633	—	2	2	no	—
7	D₄×#3633	—	2	6	no	—
8	#11763922	C₂²	1	6	yes	—
9	C₂×C₄⋊#3633	C₂²	1	6	yes	—
10	#11763899	C₂²	1	6	yes	—
11	#11763898	C₂²	1	3	yes	—
12	#11763898	C₂²	1	6	yes	—
13	#11763898	C₂²	1	1	yes	characteristic
14	D₁₇⋊#131	C₄	1	1	yes	characteristic
15	#41×#595	C₂²	1	1	yes	characteristic
16	#11763775	C₄	1	3	yes	—
17	#11763772	C₄	1	3	yes	—
18	M₁₆×#595	C₂²	1	3	yes	—
19	C₂×C₈×#595	C₂²	1	3	yes	—
20	#131×D₁₇	C₄	1	1	yes	characteristic
21	C₂²×#3633	—	2	6	no	—
22	#41×D₁₇	C₂×C₄	1	1	yes	characteristic
23	Q₈×#595	D₄	1	1	yes	characteristic
24	Q₈×#595	Q₈	1	1	yes	characteristic
25	#10587680	—	2	1	no	—
26	D₄×#595	D₄	1	3	yes	—
27	D₄×#595	Q₈	1	3	yes	—
28	#10587678	—	2	3	no	—
29	#10587677	C₂×C₄	1	3	yes	—
30	C₂×C₄×#595	C₂³	1	3	yes	—
31	D₁₇⋊M₁₆	C₂×C₄	1	3	yes	—
32	C₂×#63242	C₂×C₄	1	3	yes	—
33	C₂²⋊#3633	—	2	6	no	—
34	C₄⋊#3633	—	2	6	no	—
35	C₄⋊#3633	C₂³	1	6	yes	—
36	C₄⋊#3633	C₂³	1	2	yes	—
37	C₄×#3633	—	2	6	no	—
38	C₁₇×#131	C₈	1	1	yes	characteristic, complemented
39	#10587634	C₈	1	1	yes	characteristic, complemented
40	#10587593	C₈	1	3	yes	complemented
41	M₁₆×D₁₇	C₂×C₄	1	3	yes	—
42	#10587587	C₈	1	3	yes	complemented
43	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
44	#10587532	C₂×C₄	1	6	yes	—
45	C₈×#595	C₂×C₄	1	3	yes	—
46	C₈×#595	C₂×C₄	1	3	yes	—
47	C₈×#595	C₂³	1	1	yes	characteristic
48	C₈×#595	C₂³	1	3	yes	—
49	C₂²×#595	C₂×D₄	1	3	yes	—
50	C₂²×#595	C₂×Q₈	1	3	yes	—
51	C₂×#3633	—	2	2	no	—
52	C₂×#3633	—	2	6	no	—
53	C₂×#3633	—	8	6	no	—
54	#41×C₁₇	C₂×C₈	1	1	yes	characteristic
55	#63256	M₁₆	1	1	yes	characteristic
56	Q₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
57	#63254	M₁₆	1	3	yes	—
58	D₄×D₁₇	C₄⋊C₄	1	3	yes	—
59	#63252	C₂×C₈	1	3	yes	—
60	C₂×C₄×D₁₇	C₂²×C₄	1	3	yes	—
61	C₂²⋊#595	C₄⋊C₄	1	6	yes	—
62	#63247	—	2	3	no	—
63	C₂×#3624	—	2	3	no	—
64	C₄⋊#595	C₄⋊C₄	1	3	yes	—
65	C₄⋊#595	C₄⋊C₄	1	3	yes	—
66	C₄×#595	C₂²×C₄	1	3	yes	—
67	C₄×#595	C₂×D₄	1	3	yes	—
68	C₄×#595	C₂×Q₈	1	3	yes	—
69	C₄×#595	C₂⁴	1	1	yes	characteristic
70	#63243	—	2	3	no	—
71	#63242	—	2	3	no	—
72	#63242	C₂²×C₄	1	1	yes	characteristic
73	#63242	C₂²×C₄	1	3	yes	—
74	M₁₆×C₁₇	C₂×C₈	1	3	yes	complemented
75	C₂×C₈×C₁₇	C₂×C₈	1	3	yes	complemented
76	#63223	C₂×C₈	1	3	yes	complemented
77	C₂×#3622	C₂×C₈	1	3	yes	complemented
78	#63218	C₂×C₈	1	6	yes	complemented
79	C₈×D₁₇	C₂×C₈	1	3	yes	complemented
80	C₈×D₁₇	C₂×C₈	1	3	yes	complemented
81	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₈×D₁₇	C₂²×C₄	1	3	yes	—
83	#17788	—	17	1	no	maximal, Sylow
84	C₂²×D₁₇	C₂×C₄⋊C₄	1	3	yes	—
85	C₂×#595	—	2	6	no	—
86	C₂×#595	—	4	3	no	—
87	C₂×#595	C₂×C₄⋊C₄	1	3	yes	—
88	C₂×#595	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×#595	C₂²×D₄	1	1	yes	characteristic
90	C₂×#595	C₂²×Q₈	1	1	yes	characteristic
91	#3633	—	4	2	no	—
92	#3633	—	4	6	no	—
93	Q₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
94	D₄×C₁₇	C₄⋊C₈	1	3	yes	—
95	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
96	C₂²⋊D₁₇	C₄⋊C₈	1	6	yes	—
97	C₂×Dic₁₇	C₂×M₁₆	1	3	yes	—
98	D₆₈	C₄⋊C₈	1	3	yes	—
99	C₄×D₁₇	C₂×C₄⋊C₄	1	3	yes	—
100	C₄×D₁₇	C₂²×C₈	1	3	yes	—
101	C₄×D₁₇	C₂×M₁₆	1	3	yes	—
102	C₄×D₁₇	C₂³×C₄	1	1	yes	characteristic
103	Dic₃₄	C₄⋊C₈	1	3	yes	—
104	#3624	—	2	1	no	—
105	#3624	—	2	3	no	—
106	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₁₇	C₂²×C₈	1	3	yes	—
108	#3622	C₂²×C₈	1	1	yes	characteristic
109	#3622	C₂²×C₈	1	3	yes	—
110	#2988	—	17	2	no	—
111	C₄×#131	—	17	1	no	—
112	#1565	—	17	6	no	—
113	#1564	—	17	3	no	—
114	C₂×#334	—	17	3	no	—
115	C₂²×C₁₇	C₂×C₄⋊C₈	1	3	yes	—
116	C₂×D₁₇	—	2	3	no	—
117	C₂×D₁₇	—	2	3	no	—
118	C₂×D₁₇	C₂×C₄⋊C₈	1	3	yes	—
119	C₂×D₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	#595	—	2	3	no	—
121	#595	—	2	1	no	—
122	#595	—	2	3	no	—
123	#595	#577	1	2	yes	—
124	C₄×C₁₇	C₂×C₄⋊C₈	1	3	yes	—
125	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
126	Dic₁₇	C₂×C₄⋊C₈	1	3	yes	—
127	Dic₁₇	C₂²×M₁₆	1	1	yes	characteristic
128	C₂×#131	—	17	1	no	—
129	C₂×#131	—	17	1	no	—
130	C₄×#41	—	17	1	no	—
131	C₈×Q₈	—	34	2	no	—
132	C₈×D₄	—	34	6	no	—
133	#431	—	17	6	no	—
134	C₂×C₄⋊C₈	—	17	6	no	—
135	#405	—	17	3	no	—
136	#405	—	17	3	no	—
137	C₄×M₁₆	—	17	3	no	—
138	C₂×C₄×C₈	—	17	3	no	—
139	#335	—	17	6	no	—
140	#334	—	17	3	no	—
141	#334	—	17	6	no	—
142	#334	—	17	1	no	—
143	C₂×C₁₇	—	2	3	no	—
144	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
145	D₁₇	—	2	3	no	—
146	D₁₇	#2912	1	1	yes	characteristic
147	D₁₇	#2916	1	1	yes	characteristic
148	C₂×#41	—	17	1	no	—
149	#131	—	17	1	no	—
150	#131	—	17	3	no	—
151	#131	—	17	3	no	—
152	#131	—	34	3	no	—
153	#131	—	34	1	no	—
154	#131	#3633	1	1	yes	characteristic, complemented
155	C₂×M₁₆	—	17	3	no	—
156	C₂×M₁₆	—	17	3	no	—
157	C₂²×C₈	—	17	3	no	—
158	C₂²×C₈	—	17	3	no	—
159	C₂²×C₈	—	34	6	no	—
160	C₄×Q₈	—	17	1	no	—
161	C₄×Q₈	—	17	1	no	—
162	C₄×D₄	—	17	3	no	—
163	C₄×D₄	—	17	3	no	—
164	#117	—	17	3	no	—
165	C₂×C₄²	—	17	3	no	—
166	C₄⋊C₈	—	17	6	no	—
167	C₄⋊C₈	—	17	2	no	—
168	C₄⋊C₈	—	34	6	no	—
169	C₂²⋊C₈	—	34	6	no	—
170	#97	—	17	6	no	—
171	C₄×C₈	—	17	1	no	—
172	C₄×C₈	—	17	3	no	—
173	C₄×C₈	—	17	3	no	—
174	C₄×C₈	—	17	3	no	—
175	C₄×C₈	—	34	6	no	—
176	C₁₇	#17788	1	1	yes	characteristic, complemented, Sylow
177	#41	—	17	1	no	—
178	#41	—	17	3	no	—
179	#41	—	17	3	no	—
180	#41	C₂×#3633	1	1	yes	characteristic
181	C₂×Q₈	—	17	1	no	—
182	C₂×D₄	—	17	3	no	—
183	C₂²×C₄	—	17	3	no	—
184	C₂²×C₄	—	17	3	no	—
185	C₂²×C₄	—	17	3	no	—
186	M₁₆	—	17	3	no	—
187	M₁₆	—	17	6	no	—
188	M₁₆	—	34	3	no	—
189	M₁₆	—	34	3	no	—
190	M₁₆	C₂×#3633	1	3	yes	complemented
191	C₂×C₈	—	17	3	no	—
192	C₂×C₈	—	17	1	no	—
193	C₂×C₈	—	17	3	no	—
194	C₂×C₈	—	17	3	no	—
195	C₂×C₈	—	17	3	no	—
196	C₂×C₈	—	17	1	no	—
197	C₂×C₈	—	17	3	no	—
198	C₂×C₈	—	34	3	no	—
199	C₂×C₈	—	34	3	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	34	6	no	—
202	C₂×C₈	—	136	6	no	—
203	C₂×C₈	C₂×#3633	1	3	yes	complemented
204	C₄⋊C₄	—	17	3	no	—
205	C₄⋊C₄	—	17	3	no	—
206	C₂²⋊C₄	—	17	6	no	—
207	C₄²	—	17	3	no	—
208	C₄²	—	17	3	no	—
209	C₄²	—	17	3	no	—
210	C₄²	—	17	1	no	—
211	C₂³	—	17	3	no	—
212	Q₈	—	17	3	no	—
213	Q₈	C₄⋊#3633	1	1	yes	characteristic
214	D₄	—	17	6	no	—
215	D₄	—	17	3	no	—
216	D₄	C₄⋊#3633	1	3	yes	—
217	C₂×C₄	—	17	1	no	—
218	C₂×C₄	—	17	3	no	—
219	C₂×C₄	—	17	3	no	—
220	C₂×C₄	—	17	3	no	—
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	3	no	—
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	3	no	—
225	C₂×C₄	—	17	3	no	—
226	C₂×C₄	—	34	6	no	—
227	C₂×C₄	—	68	3	no	—
228	C₂×C₄	C₂²×#3633	1	3	yes	—
229	C₈	—	17	1	no	—
230	C₈	—	17	3	no	—
231	C₈	—	34	1	no	—
232	C₈	—	34	3	no	—
233	C₈	—	68	6	no	—
234	C₈	—	68	2	no	—
235	C₈	C₂²×#3633	1	1	yes	characteristic
236	C₈	C₂²×#3633	1	3	yes	—
237	C₂²	—	17	1	no	—
238	C₂²	—	17	3	no	—
239	C₂²	—	34	3	no	—
240	C₂²	—	34	3	no	—
241	C₂²	C₂×C₄⋊#3633	1	3	yes	—
242	C₄	—	17	2	no	—
243	C₄	—	17	1	no	—
244	C₄	—	17	3	no	—
245	C₄	—	34	3	no	—
246	C₄	—	34	1	no	—
247	C₄	—	34	3	no	—
248	C₄	C₂×C₄⋊#3633	1	3	yes	—
249	C₄	C₂³×#3633	1	1	yes	characteristic
250	C₂	—	2	3	no	—
251	C₂	—	17	1	no	—
252	C₂	—	17	1	no	—
253	C₂	—	34	3	no	—
254	C₂	C₂²×C₄⋊#3633	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#3633
commutator	C₄×C₁₇	C₂³×C₈
Frattini	C₄	C₂³×#3633
Fitting	C₁₇×#131	C₈
socle	C₂×C₁₇	C₂²×C₄⋊C₈
prime radical	#41×D₁₇	C₂×C₄
cosocle	C₄×#595	C₂⁴
hypercenter	#131	#3633
hypocenter	C₁₇	#17788
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
2. #14545909	C₂	semidirect
3. #14545908	C₂	semidirect
4. C₂×#11763898	C₂	semidirect
38. C₁₇×#131	C₈	semidirect
39. #10587634	C₈	semidirect
40. #10587593	C₈	semidirect
42. #10587587	C₈	semidirect
74. M₁₆×C₁₇	C₂×C₈	semidirect
75. C₂×C₈×C₁₇	C₂×C₈	semidirect
76. #63223	C₂×C₈	semidirect
77. C₂×#3622	C₂×C₈	semidirect
78. #63218	C₂×C₈	semidirect
79. C₈×D₁₇	C₂×C₈	semidirect
80. C₈×D₁₇	C₂×C₈	semidirect
154. #131	#3633	semidirect
176. C₁₇	#17788	semidirect
190. M₁₆	C₂×#3633	semidirect
203. C₂×C₈	C₂×#3633	semidirect

Maximal subgroups

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

subgroup	index
#14546028	2
#14545909	2
#14545908	2
C₂×#11763898	2
#131×#595	2
#17788	17