Subgroups — #17962219

Order 4352.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545125	C₂	1	2	yes	maximal
2	C₂×#11763731	C₂	1	1	yes	maximal, characteristic
3	#14544990	C₂	1	1	yes	maximal, characteristic
4	#14543835	C₂	1	2	yes	maximal
5	#14528666	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂×#10587538	C₂²	1	1	yes	characteristic
8	C₂×#10587538	C₂²	1	1	yes	characteristic
9	#11763731	—	2	4	no	—
10	#11763731	—	2	1	no	—
11	#11763731	—	2	1	no	—
12	#11763698	—	2	1	no	—
13	#11763695	—	2	1	no	—
14	#11762847	C₄	1	1	yes	characteristic
15	#11762767	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762570	C₄	1	2	yes	—
18	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	#10587612	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	#10587538	—	2	1	no	—
25	#10587538	—	2	1	no	—
26	#10587538	—	2	2	no	—
27	#10587538	—	2	2	no	—
28	#10587538	D₄	1	2	yes	—
29	#10587538	Q₈	1	2	yes	—
30	C₂×#63216	—	2	1	no	—
31	C₂×#63216	—	2	1	no	—
32	C₂×#63216	—	2	2	no	—
33	C₁₇×#108	C₈	1	2	yes	—
34	C₁₇×#108	C₈	1	2	yes	—
35	#10587489	—	2	1	no	—
36	#10587488	—	2	1	no	—
37	#10587486	—	2	2	no	—
38	C₈×Dic₁₇	—	2	2	no	—
39	C₈×Dic₁₇	D₄	1	1	yes	characteristic
40	C₈×Dic₁₇	D₄	1	1	yes	characteristic
41	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
42	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	2	1	no	—
45	M₁₆×C₁₇	—	2	2	no	—
46	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
54	C₂²⋊Dic₁₇	—	2	2	no	—
55	C₄⋊Dic₁₇	—	2	2	no	—
56	C₄×Dic₁₇	—	2	1	no	—
57	C₄×Dic₁₇	—	2	2	no	—
58	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#3622	C₂×D₄	1	1	yes	characteristic
61	C₂×#3622	C₂×D₄	1	1	yes	characteristic
62	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
63	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
64	C₂×#3622	#41	1	1	yes	characteristic
65	C₂×#3622	#41	1	1	yes	characteristic
66	#63216	—	2	2	no	—
67	#63216	—	2	2	no	—
68	#63216	—	4	2	no	—
69	#13164	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	4	2	no	—
80	C₂×Dic₁₇	—	4	2	no	—
81	C₈×C₁₇	—	2	4	no	—
82	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
86	#3622	#121	1	1	yes	characteristic
87	#3622	C₂²⋊Q₈	1	1	yes	characteristic
88	#3622	#125	1	1	yes	characteristic
89	#3622	C₄⋊Q₈	1	1	yes	characteristic
90	#2153	—	17	2	no	—
91	C₂×#364	—	17	1	no	—
92	C₄×M₃₂	—	17	1	no	—
93	#1772	—	17	1	no	—
94	#1378	—	17	2	no	—
95	C₂²×C₁₇	—	4	1	no	—
96	C₂²×C₁₇	#424	1	1	yes	characteristic
97	C₂²×C₁₇	#436	1	1	yes	characteristic
98	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
99	C₄×C₁₇	#389	1	1	yes	characteristic
100	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
102	C₄×C₁₇	#446	1	1	yes	characteristic
103	Dic₁₇	—	2	2	no	—
104	Dic₁₇	—	4	1	no	—
105	Dic₁₇	—	8	2	no	—
106	C₂×M₃₂	—	17	1	no	—
107	C₂×M₃₂	—	17	1	no	—
108	C₂×M₃₂	—	17	2	no	—
109	C₂×#108	#595	1	1	yes	characteristic
110	#427	—	17	1	no	—
111	#405	—	17	2	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	#364	—	34	1	no	—
114	#364	—	34	1	no	—
115	#364	—	34	4	no	—
116	#346	—	34	1	no	—
117	C₄×C₁₆	—	34	1	no	—
118	#339	—	17	2	no	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	#1939	1	1	yes	characteristic
123	C₂×M₁₆	C₂×#595	1	2	yes	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	C₂×#595	1	1	yes	characteristic
126	#117	—	17	2	no	—
127	C₂×C₄²	—	17	1	no	—
128	M₃₂	—	17	2	no	—
129	M₃₂	—	17	2	no	—
130	M₃₂	—	34	1	no	—
131	M₃₂	—	34	1	no	—
132	M₃₂	—	34	2	no	—
133	M₃₂	—	34	2	no	—
134	C₂×C₁₆	—	34	1	no	—
135	C₂×C₁₆	—	34	1	no	—
136	C₂×C₁₆	—	34	2	no	—
137	#108	#3624	1	2	yes	—
138	#108	#3624	1	2	yes	—
139	#107	—	34	1	no	—
140	#106	—	34	1	no	—
141	#97	—	34	2	no	—
142	C₄×C₈	—	17	1	no	—
143	C₄×C₈	—	17	1	no	—
144	C₄×C₈	—	17	1	no	—
145	C₄×C₈	—	17	1	no	—
146	C₄×C₈	—	34	2	no	—
147	C₁₇	#13164	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₄	—	34	1	no	—
149	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×#3624	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	17	1	no	—
157	C₂×C₈	—	17	1	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	#63243	1	1	yes	characteristic
160	C₂×C₈	#63243	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
162	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
163	C₂×C₈	C₂×#3624	1	1	yes	characteristic
164	C₂×C₈	C₂×#3624	1	1	yes	characteristic
165	C₄⋊C₄	—	34	2	no	—
166	C₂²⋊C₄	—	34	2	no	—
167	C₄²	—	17	1	no	—
168	C₄²	—	17	1	no	—
169	C₄²	—	34	1	no	—
170	C₄²	—	34	2	no	—
171	C₁₆	—	34	2	no	—
172	C₁₆	—	34	2	no	—
173	C₁₆	—	68	2	no	—
174	C₂³	#10587680	1	1	yes	characteristic
175	C₂×C₄	—	34	1	no	—
176	C₂×C₄	—	34	1	no	—
177	C₂×C₄	—	68	2	no	—
178	C₂×C₄	—	68	2	no	—
179	C₂×C₄	C₂×#63243	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
181	C₂×C₄	#10587678	1	1	yes	characteristic
182	C₂×C₄	D₄×#595	1	1	yes	characteristic
183	C₂×C₄	Q₈×#595	1	1	yes	characteristic
184	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	17	1	no	—
187	C₈	—	17	1	no	—
188	C₈	—	17	1	no	—
189	C₈	—	17	1	no	—
190	C₈	C₄⋊#3624	1	1	yes	characteristic
191	C₈	C₄⋊#3624	1	1	yes	characteristic
192	C₈	C₄⋊#3624	1	1	yes	characteristic
193	C₈	C₄⋊#3624	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11763816	1	1	yes	characteristic
196	C₂²	#11763843	1	1	yes	characteristic
197	C₂²	Q₈×#3624	1	1	yes	characteristic
198	C₄	—	34	2	no	—
199	C₄	—	68	1	no	—
200	C₄	—	136	2	no	—
201	C₄	#11763760	1	1	yes	characteristic
202	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
203	C₄	D₄×#3624	1	1	yes	characteristic
204	C₄	#11763859	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#14544881	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763760
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#108	C₄
socle	C₂×C₁₇	#1939
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×#108	#595
hypocenter	C₁₇	#13164
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
147. C₁₇	#13164	semidirect

Maximal subgroups

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

subgroup	index
#14545125	2
C₂×#11763731	2
#14544990	2
#14543835	2
#14528666	2
#13164	17