Subgroups — #17948933

Order 4352.

262 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763730	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763673	C₂	1	1	yes	maximal, characteristic
3	#14528942	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	—	2	1	no	—
5	C₂×C₄⋊#3624	—	2	1	no	—
6	#11763730	—	2	1	no	—
7	#11763730	C₄	1	2	yes	complemented
8	#11763673	—	2	1	no	—
9	#11763673	C₄	1	2	yes	complemented
10	C₂×#10587489	C₄	1	1	yes	characteristic
11	C₂×#10587478	—	2	1	no	—
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762678	—	2	1	no	—
14	C₂×C₁₇×#107	C₄	1	1	yes	characteristic
15	C₂²×#3624	—	4	1	no	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
18	C₂×C₄⋊Dic₁₇	—	2	1	no	—
19	C₂×#63225	—	4	1	no	—
20	C₂×#63225	—	4	1	no	—
21	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	C₂²×#3622	D₄	1	1	yes	characteristic
23	C₄⋊#3624	—	2	2	no	—
24	C₄⋊#3624	—	4	1	no	—
25	C₂×#63216	—	2	2	no	—
26	C₂×#63216	—	4	1	no	—
27	C₂×#63216	—	4	1	no	—
28	C₁₇×#107	—	4	1	no	—
29	#10587489	—	4	1	no	—
30	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	D₄	1	1	yes	characteristic
33	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
34	#10587478	—	4	1	no	—
35	#10587478	—	4	1	no	—
36	C₂²×C₄×C₁₇	—	4	1	no	—
37	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	—	4	1	no	—
39	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
40	C₂²×Dic₁₇	SD₁₆	1	1	yes	characteristic
41	C₂×#3624	—	4	2	no	—
42	C₂×#3624	—	8	1	no	—
43	C₂×#3624	—	8	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	—	2	1	no	—
46	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₇	—	4	1	no	—
49	C₄⋊C₄×C₁₇	—	4	1	no	—
50	C₄⋊Dic₁₇	—	4	1	no	—
51	C₄⋊Dic₁₇	—	4	1	no	—
52	#63225	—	8	2	no	—
53	#63225	—	8	1	no	—
54	#63225	—	8	1	no	—
55	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
58	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
59	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#3622	D₈	1	1	yes	characteristic
62	C₂×#3622	SD₁₆	1	1	yes	characteristic
63	C₂×#3622	SD₁₆	1	1	yes	characteristic
64	C₂×#3622	Q₁₆	1	1	yes	characteristic
65	#63216	—	4	2	no	—
66	#7274	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	#102	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	8	1	no	—
69	C₂×C₄×C₁₇	—	8	1	no	—
70	C₂×C₄×C₁₇	—	8	1	no	—
71	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#100	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#101	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	8	1	no	—
80	C₂×Dic₁₇	—	8	1	no	—
81	C₂×Dic₁₇	—	8	1	no	—
82	C₂×Dic₁₇	#102	1	1	yes	characteristic
83	C₂×Dic₁₇	#103	1	1	yes	characteristic
84	C₂×Dic₁₇	#106	1	1	yes	characteristic
85	C₂×Dic₁₇	#107	1	1	yes	characteristic
86	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
87	C₂×Dic₁₇	SD₃₂	1	1	yes	characteristic
88	C₂×Dic₁₇	SD₃₂	1	1	yes	characteristic
89	C₂×Dic₁₇	Q₃₂	1	1	yes	characteristic
90	#3624	—	8	2	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	—	2	1	no	—
93	#3622	#102	1	1	yes	characteristic
94	#3622	#103	1	1	yes	characteristic
95	#3622	#106	1	1	yes	characteristic
96	#3622	#107	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	17	1	no	—
98	#1845	—	17	1	no	—
99	C₂×#335	—	17	1	no	—
100	C₂²×C₁₇	#331	1	1	yes	characteristic
101	C₂²×C₁₇	#332	1	1	yes	characteristic
102	C₂²×C₁₇	#340	1	1	yes	characteristic
103	C₂²×C₁₇	#357	1	1	yes	characteristic
104	C₂²×C₁₇	#358	1	1	yes	characteristic
105	C₂²×C₁₇	#361	1	1	yes	characteristic
106	C₂²×C₁₇	#362	1	1	yes	characteristic
107	C₄×C₁₇	—	16	1	no	—
108	C₄×C₁₇	#332	1	1	yes	characteristic
109	C₄×C₁₇	#333	1	1	yes	characteristic
110	C₄×C₁₇	#340	1	1	yes	characteristic
111	C₄×C₁₇	#343	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	—	2	1	no	—
114	Dic₁₇	—	16	1	no	—
115	Dic₁₇	#357	1	1	yes	characteristic
116	Dic₁₇	#358	1	1	yes	characteristic
117	Dic₁₇	#366	1	1	yes	characteristic
118	Dic₁₇	#367	1	1	yes	characteristic
119	C₂²×C₁₆	—	34	1	no	—
120	C₂×#107	—	17	1	no	—
121	C₂×#107	—	34	1	no	—
122	C₂×#107	#595	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	34	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	#384	—	34	1	no	—
126	C₄⋊C₁₆	—	17	2	no	—
127	C₄⋊C₁₆	—	34	1	no	—
128	#335	—	17	2	no	—
129	#335	—	34	1	no	—
130	C₂×C₁₇	#1296	1	1	yes	characteristic
131	C₂×C₁₇	#1361	1	1	yes	characteristic
132	C₂×C₁₇	#1362	1	1	yes	characteristic
133	C₂×C₁₇	#1364	1	1	yes	characteristic
134	C₂×C₁₇	#1365	1	1	yes	characteristic
135	C₂×C₁₇	#1380	1	1	yes	characteristic
136	C₂×C₁₇	#1387	1	1	yes	characteristic
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	68	1	no	—
139	C₂²×C₈	C₂×#595	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	34	1	no	—
142	C₂×C₄⋊C₄	—	68	1	no	—
143	C₂×C₄⋊C₄	—	68	1	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₁₆	—	34	2	no	—
146	C₂×C₁₆	—	68	1	no	—
147	C₂×C₁₆	—	68	1	no	—
148	#107	—	4	1	no	—
149	#107	—	68	1	no	—
150	#107	—	68	1	no	—
151	#107	—	68	1	no	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄⋊C₈	—	68	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	17	1	no	—
158	C₁₇	#7274	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	68	1	no	—
163	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	68	2	no	—
173	C₂×C₈	—	136	1	no	—
174	C₂×C₈	—	136	1	no	—
175	C₂×C₈	C₄×#595	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	68	1	no	—
180	C₄⋊C₄	—	68	1	no	—
181	C₄⋊C₄	—	136	1	no	—
182	C₄⋊C₄	—	136	1	no	—
183	C₄⋊C₄	—	136	2	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	17	1	no	—
187	C₄²	—	17	1	no	—
188	C₁₆	—	68	2	no	—
189	C₂³	#10587547	1	1	yes	characteristic
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	1	no	—
203	C₂×C₄	—	136	1	no	—
204	C₂×C₄	—	136	1	no	—
205	C₂×C₄	—	136	1	no	—
206	C₂×C₄	#10587545	1	1	yes	characteristic
207	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	#10587547	1	1	yes	characteristic
209	C₂×C₄	#10587549	1	1	yes	characteristic
210	C₂×C₄	#10587549	1	1	yes	characteristic
211	C₂×C₄	#10587553	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	17	1	no	—
215	C₈	—	17	1	no	—
216	C₈	—	17	1	no	—
217	C₈	—	17	1	no	—
218	C₈	—	136	2	no	—
219	C₂²	#11763667	1	1	yes	characteristic
220	C₂²	#11763668	1	1	yes	characteristic
221	C₂²	#11763682	1	1	yes	characteristic
222	C₂²	#11763721	1	1	yes	characteristic
223	C₂²	#11763722	1	1	yes	characteristic
224	C₂²	#11763727	1	1	yes	characteristic
225	C₂²	#11763728	1	1	yes	characteristic
226	C₄	—	16	1	no	—
227	C₄	—	272	1	no	—
228	C₄	—	17	1	no	—
229	C₄	—	17	1	no	—
230	C₄	—	17	1	no	—
231	C₄	—	17	1	no	—
232	C₄	—	34	1	no	—
233	C₄	—	34	1	no	—
234	C₄	#11763668	1	1	yes	characteristic
235	C₄	#11763669	1	1	yes	characteristic
236	C₄	#11763682	1	1	yes	characteristic
237	C₄	#11763689	1	1	yes	characteristic
238	C₂	#14543662	1	1	yes	characteristic
239	C₂	#14543798	1	1	yes	characteristic
240	C₂	#14543799	1	1	yes	characteristic
241	C₂	#14543802	1	1	yes	characteristic
242	C₂	#14543803	1	1	yes	characteristic
243	C₂	#14543841	1	1	yes	characteristic
244	C₂	#14543858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587547
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂×C₁₇×#107	C₄
socle, prime radical	C₂³×C₁₇	#102
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂×#107	#595
hypocenter	C₁₇	#7274
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
7. #11763730	C₄	semidirect
9. #11763673	C₄	semidirect
158. C₁₇	#7274	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763730	2
C₂×#11763673	2
#14528942	2
#7274	17