Subgroups — #17948620

Order 4352.

274 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763698	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763670	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	—	2	1	no	—
5	C₂×C₄⋊#3624	—	2	1	no	—
6	#11763698	—	2	1	no	—
7	#11763698	C₄	1	2	yes	complemented
8	#11763670	—	2	1	no	—
9	#11763670	C₄	1	2	yes	complemented
10	C₂×#10587489	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₁₇	—	2	1	no	—
13	#11762571	—	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×C₄×Dic₁₇	—	4	1	no	—
20	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
21	C₂²×#3622	D₄	1	1	yes	characteristic
22	C₄⋊#3624	—	2	2	no	—
23	C₄⋊#3624	—	4	1	no	—
24	C₂×#63216	—	2	2	no	—
25	C₂×#63216	—	2	2	no	—
26	C₂×#63216	—	2	2	no	—
27	C₁₇×#107	—	2	1	no	—
28	C₁₇×#107	C₈	1	2	yes	complemented
29	#10587502	—	4	1	no	—
30	#10587489	—	2	1	no	—
31	#10587489	C₈	1	2	yes	complemented
32	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₇	D₄	1	1	yes	characteristic
35	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
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	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	—	2	2	no	—
51	C₄⋊C₄×C₁₇	—	4	1	no	—
52	C₄⋊Dic₁₇	—	2	2	no	—
53	C₄⋊Dic₁₇	—	4	1	no	—
54	C₄×Dic₁₇	—	8	2	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	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
63	#63216	—	2	4	no	—
64	#7178	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	#102	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	—	8	1	no	—
68	C₂×C₄×C₁₇	—	8	1	no	—
69	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	1	no	—
76	C₂×Dic₁₇	—	2	1	no	—
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	4	2	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	#3624	—	8	2	no	—
87	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
88	C₈×C₁₇	#97	1	1	yes	characteristic
89	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
90	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
91	#3622	—	2	1	no	—
92	#3622	—	2	1	no	—
93	C₂×#346	—	17	1	no	—
94	C₄×#107	—	17	1	no	—
95	C₂×#334	—	17	1	no	—
96	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₁₇	#340	1	1	yes	characteristic
99	C₂²×C₁₇	#360	1	1	yes	characteristic
100	C₂²×C₁₇	#360	1	1	yes	characteristic
101	C₂²×C₁₇	#361	1	1	yes	characteristic
102	C₂²×C₁₇	#362	1	1	yes	characteristic
103	C₄×C₁₇	—	8	2	no	—
104	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
105	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
106	C₄×C₁₇	#336	1	1	yes	characteristic
107	C₄×C₁₇	#339	1	1	yes	characteristic
108	Dic₁₇	—	2	1	no	—
109	Dic₁₇	—	2	1	no	—
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	—	8	2	no	—
113	C₂²×C₁₆	—	34	1	no	—
114	C₂×#107	—	17	1	no	—
115	C₂×#107	#595	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	34	1	no	—
117	C₂×C₄×C₈	—	17	1	no	—
118	C₄×C₄⋊C₄	—	34	1	no	—
119	#346	—	17	2	no	—
120	#346	—	34	1	no	—
121	#340	—	34	1	no	—
122	#334	—	17	2	no	—
123	#334	—	34	1	no	—
124	C₂×C₁₇	#1279	1	1	yes	characteristic
125	C₂×C₁₇	#1333	1	1	yes	characteristic
126	C₂×C₁₇	#1333	1	1	yes	characteristic
127	C₂×C₁₇	#1334	1	1	yes	characteristic
128	C₂×C₁₇	#1334	1	1	yes	characteristic
129	C₂×C₁₇	#1386	1	1	yes	characteristic
130	C₂×C₁₇	#1387	1	1	yes	characteristic
131	C₂²×C₈	—	17	1	no	—
132	C₂²×C₈	—	68	1	no	—
133	C₂²×C₈	C₂×#595	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	34	1	no	—
136	C₂×C₄²	—	17	1	no	—
137	C₂×C₄²	—	68	1	no	—
138	C₂×C₁₆	—	34	2	no	—
139	C₂×C₁₆	—	34	2	no	—
140	C₂×C₁₆	—	34	2	no	—
141	#107	—	2	1	no	—
142	#107	—	17	2	no	—
143	#107	—	34	1	no	—
144	#107	#3624	1	2	yes	complemented
145	C₄⋊C₈	—	34	2	no	—
146	C₄⋊C₈	—	68	1	no	—
147	C₄×C₈	—	17	1	no	—
148	C₄×C₈	—	17	1	no	—
149	C₄×C₈	—	17	1	no	—
150	C₄×C₈	—	17	1	no	—
151	#95	—	68	1	no	—
152	C₁₇	#7178	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	17	1	no	—
155	C₂²×C₄	—	17	1	no	—
156	C₂²×C₄	—	68	1	no	—
157	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	1	no	—
160	C₂×C₈	—	34	1	no	—
161	C₂×C₈	—	34	1	no	—
162	C₂×C₈	—	68	2	no	—
163	C₂×C₈	—	136	1	no	—
164	C₂×C₈	—	136	1	no	—
165	C₂×C₈	#63242	1	1	yes	characteristic
166	C₂×C₈	#63243	1	1	yes	characteristic
167	C₂×C₈	C₄×#595	1	1	yes	characteristic
168	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
169	C₂×C₈	C₂×#3624	1	1	yes	characteristic
170	C₂×C₈	#63247	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	34	2	no	—
174	C₄⋊C₄	—	68	1	no	—
175	C₄²	—	17	1	no	—
176	C₄²	—	17	1	no	—
177	C₄²	—	17	1	no	—
178	C₄²	—	17	1	no	—
179	C₄²	—	136	2	no	—
180	C₁₆	—	34	4	no	—
181	C₂³	#10587547	1	1	yes	characteristic
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	8	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	17	1	no	—
188	C₂×C₄	—	17	1	no	—
189	C₂×C₄	—	34	1	no	—
190	C₂×C₄	—	34	1	no	—
191	C₂×C₄	—	34	1	no	—
192	C₂×C₄	—	34	1	no	—
193	C₂×C₄	—	68	2	no	—
194	C₂×C₄	—	136	1	no	—
195	C₂×C₄	—	136	1	no	—
196	C₂×C₄	#10587543	1	1	yes	characteristic
197	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#10587548	1	1	yes	characteristic
199	C₂×C₄	#10587549	1	1	yes	characteristic
200	C₂×C₄	#10587550	1	1	yes	characteristic
201	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
202	C₈	—	34	1	no	—
203	C₈	—	34	1	no	—
204	C₈	—	136	2	no	—
205	C₈	C₄×#3624	1	1	yes	characteristic
206	C₈	C₄⋊#3624	1	1	yes	characteristic
207	C₈	#10587542	1	1	yes	characteristic
208	C₈	#10587544	1	1	yes	characteristic
209	C₂²	D₄⋊#3624	1	1	yes	characteristic
210	C₂²	#11763657	1	1	yes	characteristic
211	C₂²	#11763682	1	1	yes	characteristic
212	C₂²	#11763725	1	1	yes	characteristic
213	C₂²	#11763726	1	1	yes	characteristic
214	C₂²	#11763727	1	1	yes	characteristic
215	C₂²	#11763728	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	34	1	no	—
218	C₄	—	34	1	no	—
219	C₄	—	34	1	no	—
220	C₄	—	34	1	no	—
221	C₄	—	136	2	no	—
222	C₄	Q₈⋊#3624	1	1	yes	characteristic
223	C₄	#11763659	1	1	yes	characteristic
224	C₄	#11763675	1	1	yes	characteristic
225	C₄	#11763679	1	1	yes	characteristic
226	C₂	#14543620	1	1	yes	characteristic
227	C₂	#14543742	1	1	yes	characteristic
228	C₂	#14543743	1	1	yes	characteristic
229	C₂	#14543744	1	1	yes	characteristic
230	C₂	#14543745	1	1	yes	characteristic
231	C₂	#14543855	1	1	yes	characteristic
232	C₂	#14543858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587547
commutator	C₈×C₁₇	C₄×C₈
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	#7178
LCS	G ▷ C₈×C₁₇ ▷ 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. #11763698	C₄	semidirect
9. #11763670	C₄	semidirect
28. C₁₇×#107	C₈	semidirect
31. #10587489	C₈	semidirect
144. #107	#3624	semidirect
152. C₁₇	#7178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763698	2
C₂×#11763670	2
#107×Dic₁₇	2
#7178	17