Subgroups — #17961023

Order 4352.

334 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763732	C₂	1	1	yes	maximal, characteristic
2	#14544995	C₂	1	1	yes	maximal, characteristic
3	#14544993	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
5	#14543839	C₂	1	1	yes	maximal, characteristic
6	#14543838	C₂	1	1	yes	maximal, characteristic
7	#14528662	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587538	C₂²	1	1	yes	characteristic
9	C₂×#10587538	C₂²	1	1	yes	characteristic
10	C₂²×#63216	C₂²	1	1	yes	characteristic
11	C₂²×#63216	C₂²	1	1	yes	characteristic
12	#11763732	C₄	1	4	yes	complemented
13	#11763698	—	2	1	no	—
14	#11763697	—	2	1	no	—
15	#11763695	—	2	1	no	—
16	C₄×#63216	—	2	1	no	—
17	C₄×#63216	C₄	1	4	yes	complemented
18	C₂×#10587493	C₄	1	1	yes	characteristic
19	#11762845	C₄	1	1	yes	characteristic
20	#11762767	C₂²	1	1	yes	characteristic
21	#11762765	C₂²	1	1	yes	characteristic
22	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
23	#11762569	C₄	1	1	yes	characteristic
24	C₁₇×#339	C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	#10587612	C₂×C₄	1	1	yes	characteristic
29	C₂×#63223	C₂×C₄	1	1	yes	characteristic
30	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
31	C₂²×#3622	C₂³	1	1	yes	characteristic
32	#10587538	—	2	1	no	—
33	#10587538	—	2	1	no	—
34	#10587538	C₂×C₄	1	2	yes	—
35	#10587538	C₂×C₄	1	2	yes	—
36	C₂×#63216	—	2	1	no	—
37	C₂×#63216	—	2	1	no	—
38	C₂×#63216	—	2	1	no	—
39	C₂×#63216	—	2	1	no	—
40	C₂×#63216	C₂×C₄	1	2	yes	—
41	C₂×#63216	C₂×C₄	1	2	yes	—
42	C₂×#63216	D₄	1	1	yes	characteristic
43	C₂×#63216	D₄	1	1	yes	characteristic
44	C₂×#63216	Q₈	1	2	yes	—
45	#10587493	C₈	1	4	yes	—
46	#10587486	—	2	1	no	—
47	C₈×Dic₁₇	—	2	1	no	—
48	#10587479	C₈	1	2	yes	—
49	#10587478	C₈	1	2	yes	—
50	#10587474	—	2	1	no	—
51	C₄×#3622	—	2	1	no	—
52	C₄×#3622	C₂×C₄	1	2	yes	—
53	C₄×#3622	C₂×C₄	1	2	yes	—
54	C₂²×C₄×C₁₇	—	2	1	no	—
55	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₇	—	2	1	no	—
57	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
60	C₂²⋊C₄×C₁₇	—	2	1	no	—
61	C₄²×C₁₇	—	2	1	no	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	C₄²	1	2	yes	—
64	C₂²⋊Dic₁₇	—	2	1	no	—
65	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₇	—	2	1	no	—
67	#63223	—	2	1	no	—
68	#63223	C₂×C₈	1	2	yes	—
69	C₂×#3622	—	2	1	no	—
70	C₂×#3622	C₄²	1	2	yes	—
71	C₂×#3622	C₂×C₈	1	1	yes	characteristic
72	C₂×#3622	C₂×C₈	1	1	yes	characteristic
73	C₂×#3622	C₂×C₈	1	2	yes	—
74	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
75	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
76	C₂×#3622	C₂×D₄	1	1	yes	characteristic
77	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
78	C₂×#3622	#41	1	1	yes	characteristic
79	C₂×#3622	#41	1	1	yes	characteristic
80	#63216	—	2	2	no	—
81	#63216	—	2	2	no	—
82	#63216	—	2	2	no	—
83	#63216	C₄⋊C₄	1	2	yes	—
84	#63216	C₄⋊C₄	1	2	yes	—
85	#12664	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	#131	1	1	yes	characteristic
87	C₂×C₄×C₁₇	—	2	4	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
96	C₂×Dic₁₇	—	2	2	no	—
97	C₈×C₁₇	—	2	2	no	—
98	#3622	—	2	2	no	—
99	#3622	C₄×C₈	1	2	yes	—
100	#3622	C₄×C₈	1	2	yes	—
101	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
102	#3622	#117	1	1	yes	characteristic
103	#3622	C₄×D₄	1	1	yes	characteristic
104	#3622	C₄×Q₈	1	1	yes	characteristic
105	C₂×#364	—	17	1	no	—
106	#2108	—	17	1	no	—
107	#2108	—	17	1	no	—
108	C₂×C₄×C₁₆	—	17	1	no	—
109	#1772	—	17	1	no	—
110	#1379	—	17	1	no	—
111	#1379	—	17	1	no	—
112	C₂²×C₁₇	—	2	2	no	—
113	C₂²×C₁₇	#405	1	1	yes	characteristic
114	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
115	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
116	C₄×C₁₇	—	2	4	no	—
117	C₄×C₁₇	—	4	2	no	—
118	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
120	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
121	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
122	Dic₁₇	—	4	2	no	—
123	C₂×M₃₂	—	17	1	no	—
124	C₂×M₃₂	—	17	1	no	—
125	C₂²×C₁₆	—	17	1	no	—
126	C₂²×C₁₆	—	17	1	no	—
127	C₂×#108	—	17	1	no	—
128	#427	—	17	1	no	—
129	#405	—	17	1	no	—
130	#405	—	17	1	no	—
131	C₂×C₄×C₈	—	17	1	no	—
132	#364	—	17	4	no	—
133	#346	—	34	1	no	—
134	#346	—	34	1	no	—
135	C₄×C₁₆	—	17	4	no	—
136	C₄×C₁₆	—	34	1	no	—
137	C₄×C₁₆	—	34	1	no	—
138	#339	—	17	1	no	—
139	#339	#595	1	1	yes	characteristic
140	C₂×C₁₇	—	2	1	no	—
141	C₂×C₁₇	—	2	1	no	—
142	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₁₇	#2170	1	2	yes	—
144	C₂×M₁₆	—	17	1	no	—
145	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
146	C₂²×C₈	—	17	1	no	—
147	C₂²×C₈	—	17	1	no	—
148	#117	—	17	1	no	—
149	#117	C₂×#595	1	1	yes	characteristic
150	C₂×C₄²	C₂×#595	1	1	yes	characteristic
151	M₃₂	—	17	2	no	—
152	M₃₂	—	17	2	no	—
153	M₃₂	—	34	1	no	—
154	M₃₂	—	34	1	no	—
155	C₂×C₁₆	—	17	2	no	—
156	C₂×C₁₆	—	17	1	no	—
157	C₂×C₁₆	—	17	1	no	—
158	C₂×C₁₆	—	17	2	no	—
159	C₂×C₁₆	—	17	2	no	—
160	C₂×C₁₆	—	34	1	no	—
161	C₂×C₁₆	—	34	1	no	—
162	C₂×C₁₆	—	34	1	no	—
163	C₂×C₁₆	—	34	1	no	—
164	#108	—	17	4	no	—
165	#107	—	17	2	no	—
166	#106	—	17	2	no	—
167	#97	—	34	1	no	—
168	#97	—	34	1	no	—
169	C₄×C₈	—	17	2	no	—
170	C₄×C₈	—	17	2	no	—
171	C₄×C₈	—	34	1	no	—
172	C₄×C₈	—	34	1	no	—
173	C₁₇	#12664	1	1	yes	characteristic, complemented, Sylow
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
176	M₁₆	—	2	1	no	—
177	M₁₆	—	17	2	no	—
178	M₁₆	—	34	1	no	—
179	M₁₆	#63242	1	2	yes	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	1	no	—
183	C₂×C₈	—	17	2	no	—
184	C₂×C₈	—	17	1	no	—
185	C₂×C₈	—	17	1	no	—
186	C₂×C₈	—	17	2	no	—
187	C₂×C₈	—	17	1	no	—
188	C₂×C₈	—	17	1	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	1	no	—
191	C₂×C₈	—	34	1	no	—
192	C₄⋊C₄	—	17	2	no	—
193	C₄⋊C₄	#63242	1	2	yes	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	34	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	34	1	no	—
199	C₄²	C₄×#595	1	2	yes	—
200	C₁₆	—	17	2	no	—
201	C₁₆	—	17	2	no	—
202	C₁₆	—	34	2	no	—
203	C₁₆	—	34	2	no	—
204	C₁₆	—	34	2	no	—
205	C₂³	#10587678	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	34	2	no	—
210	C₂×C₄	C₂×#63242	1	1	yes	characteristic
211	C₂×C₄	C₂×#63242	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
213	C₂×C₄	#10587678	1	1	yes	characteristic
214	C₂×C₄	D₄×#595	1	1	yes	characteristic
215	C₂×C₄	Q₈×#595	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	17	1	no	—
218	C₈	—	17	1	no	—
219	C₈	—	17	1	no	—
220	C₈	—	17	1	no	—
221	C₈	—	17	2	no	—
222	C₈	—	17	2	no	—
223	C₈	—	34	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#11763773	1	1	yes	characteristic
226	C₂²	#11763836	1	1	yes	characteristic
227	C₂²	#11763858	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	68	2	no	—
231	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
232	C₄	C₄×#63242	1	1	yes	characteristic
233	C₄	#11763836	1	1	yes	characteristic
234	C₄	#11763858	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	#14544576	1	1	yes	characteristic
238	C₂	#14545172	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#339	C₄
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	#339	#595
hypocenter	C₁₇	#12664
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
12. #11763732	C₄	semidirect
17. C₄×#63216	C₄	semidirect
173. C₁₇	#12664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763732	2
#14544995	2
#14544993	2
C₂×C₄×#63216	2
#14543839	2
#14543838	2
#14528662	2
#12664	17