Subgroups — #17952348

Order 4352.

446 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	C₄⋊C₄×#3624	C₂	1	2	yes	maximal
2	C₂×#11763673	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
4	#14543620	C₂	1	2	yes	maximal
5	#14528676	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3624	C₂²	1	2	yes	—
7	C₂×C₄×#3624	—	2	2	no	—
8	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
10	#11763675	—	2	2	no	—
11	#11763673	C₄	1	4	yes	complemented
12	C₈×#3624	C₄	1	4	yes	complemented
13	C₂×#10587478	C₄	1	2	yes	—
14	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₇	C₂²	1	2	yes	—
16	#11762572	C₄	1	1	yes	characteristic
17	C₁₇×#340	C₄	1	1	yes	characteristic
18	C₂²×#3624	—	2	1	no	—
19	C₂²×#3624	—	2	2	no	—
20	C₂²×#3624	D₄	1	1	yes	characteristic
21	C₂²×#3624	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₇	—	2	2	no	—
26	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
28	C₄⋊#3624	—	2	2	no	—
29	C₄⋊#3624	C₂×C₄	1	4	yes	—
30	C₄×#3624	—	2	1	no	—
31	C₄×#3624	—	4	4	no	—
32	C₄×#3624	C₂×C₄	1	2	yes	—
33	C₄×#3624	C₂×C₄	1	2	yes	—
34	C₄×#3624	Q₈	1	2	yes	—
35	#10587502	—	2	2	no	—
36	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
37	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
38	#10587478	C₈	1	8	yes	complemented
39	C₂²×C₄×C₁₇	—	2	2	no	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	2	2	no	—
42	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	#41	1	1	yes	characteristic
44	C₂×#3624	—	2	4	no	—
45	C₂×#3624	—	2	2	no	—
46	C₂×#3624	—	2	1	no	—
47	C₂×#3624	—	2	1	no	—
48	C₂×#3624	—	2	4	no	—
49	C₂×#3624	—	4	2	no	—
50	C₂×#3624	—	4	2	no	—
51	C₂×#3624	C₄⋊C₄	1	2	yes	—
52	C₂×#3624	C₄⋊C₄	1	2	yes	—
53	C₂×#3624	D₈	1	2	yes	—
54	C₂×#3624	Q₁₆	1	2	yes	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	C₄²	1	2	yes	—
57	C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
59	C₄⋊Dic₁₇	—	2	2	no	—
60	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
61	C₄×Dic₁₇	—	4	2	no	—
62	C₄×Dic₁₇	—	4	2	no	—
63	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₁₇	#41	1	1	yes	characteristic
67	C₂×#3622	C₄²	1	2	yes	—
68	C₂×#3622	C₂×C₈	1	4	yes	—
69	C₈×#107	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₁₇	#131	1	2	yes	—
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	—	2	4	no	—
80	C₂×Dic₁₇	—	4	2	no	—
81	C₂×Dic₁₇	—	4	2	no	—
82	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×Dic₁₇	#117	1	1	yes	characteristic
84	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
85	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
86	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
87	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×Dic₁₇	#135	1	1	yes	characteristic
89	C₂×Dic₁₇	#135	1	1	yes	characteristic
90	#3624	—	2	4	no	—
91	#3624	—	2	2	no	—
92	#3624	—	4	4	no	—
93	#3624	#107	1	4	yes	complemented
94	C₈×C₁₇	—	2	2	no	—
95	#3622	C₄×C₈	1	4	yes	complemented
96	C₄×#107	—	17	1	no	—
97	C₈×C₄⋊C₄	—	17	2	no	—
98	C₂×#335	—	17	1	no	—
99	C₂×C₈²	—	17	1	no	—
100	#1279	—	17	2	no	—
101	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₇	C₈×D₄	1	2	yes	—
103	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
104	C₂²×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
105	C₂²×C₁₇	#443	1	2	yes	—
106	C₄×C₁₇	—	4	4	no	—
107	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₁₇	#405	1	1	yes	characteristic
109	C₄×C₁₇	C₈×Q₈	1	2	yes	—
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	—	4	4	no	—
113	Dic₁₇	C₂×#107	1	1	yes	characteristic
114	Dic₁₇	#427	1	1	yes	characteristic
115	Dic₁₇	C₄×D₈	1	1	yes	characteristic
116	Dic₁₇	C₄×Q₁₆	1	1	yes	characteristic
117	C₂×#107	—	17	2	no	—
118	C₂×C₄⋊C₈	—	17	2	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₂×C₄×C₈	—	34	2	no	—
123	C₄×C₄⋊C₄	—	17	2	no	—
124	#340	—	17	1	no	—
125	#340	#595	1	1	yes	characteristic
126	#336	—	34	2	no	—
127	#335	—	17	4	no	—
128	C₈²	—	17	4	no	—
129	C₂×C₁₇	C₈×D₈	1	2	yes	—
130	C₂×C₁₇	C₈×Q₁₆	1	2	yes	—
131	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₁₇	#1772	1	1	yes	characteristic
133	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	34	1	no	—
138	C₂²×C₈	—	34	2	no	—
139	C₂²×C₈	C₂×#595	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	17	2	no	—
141	C₂×C₄⋊C₄	C₂×#595	1	2	yes	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	34	2	no	—
144	#107	—	17	8	no	—
145	C₄⋊C₈	—	17	4	no	—
146	C₄⋊C₈	—	34	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	17	2	no	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	2	no	—
152	C₄×C₈	—	34	1	no	—
153	C₄×C₈	—	68	4	no	—
154	#95	—	34	2	no	—
155	C₁₇	C₈×#107	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	34	2	no	—
160	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	17	2	no	—
165	C₂×C₈	—	17	4	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	34	2	no	—
169	C₂×C₈	—	34	1	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	4	no	—
172	C₂×C₈	—	34	4	no	—
173	C₂×C₈	—	68	2	no	—
174	C₂×C₈	—	68	2	no	—
175	C₂×C₈	C₄×#595	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	17	4	no	—
178	C₄⋊C₄	—	34	2	no	—
179	C₄⋊C₄	#63242	1	4	yes	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	17	1	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	68	2	no	—
185	C₄²	—	68	2	no	—
186	C₂³	D₄×#595	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	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	—	34	2	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	—	68	2	no	—
201	C₂×C₄	—	68	2	no	—
202	C₂×C₄	C₂×#63242	1	2	yes	—
203	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
204	C₂×C₄	#10587678	1	2	yes	—
205	C₂×C₄	Q₈×#595	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	17	4	no	—
208	C₈	—	17	4	no	—
209	C₈	—	34	2	no	—
210	C₈	—	34	4	no	—
211	C₈	—	68	4	no	—
212	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
213	C₂²	#11763836	1	2	yes	—
214	C₂²	D₈×#595	1	1	yes	characteristic
215	C₂²	Q₁₆×#595	1	1	yes	characteristic
216	C₂²	#11763852	1	2	yes	—
217	C₄	—	4	4	no	—
218	C₄	—	17	1	no	—
219	C₄	—	17	1	no	—
220	C₄	—	17	1	no	—
221	C₄	—	17	1	no	—
222	C₄	—	34	1	no	—
223	C₄	—	34	1	no	—
224	C₄	—	68	4	no	—
225	C₄	C₄×#63242	1	1	yes	characteristic
226	C₄	#11763773	1	1	yes	characteristic
227	C₄	#11763858	1	2	yes	—
228	C₂	#14544364	1	2	yes	—
229	C₂	#14544371	1	2	yes	—
230	C₂	#14544576	1	1	yes	characteristic
231	C₂	#14544587	1	1	yes	characteristic
232	C₂	#107×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#340	C₄
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#340	#595
hypocenter	C₁₇	C₈×#107
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
11. #11763673	C₄	semidirect
12. C₈×#3624	C₄	semidirect
38. #10587478	C₈	semidirect
93. #3624	#107	semidirect
95. #3622	C₄×C₈	semidirect
155. C₁₇	C₈×#107	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#3624	2
C₂×#11763673	2
C₂×C₈×#3624	2
#14543620	2
#14528676	2
C₈×#107	17