Subgroups — #16945716

Order 4352.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530583	C₂	1	1	yes	maximal, characteristic
2	#14527543	C₂	1	1	yes	maximal, characteristic
3	#14526833	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic
5	#11762833	—	2	1	no	—
6	C₄⋊#63214	C₂²	1	2	yes	—
7	C₂²⋊#63214	—	2	1	no	—
8	C₄×#63214	—	2	1	no	—
9	#11762562	—	2	1	no	—
10	#11762562	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₇	—	2	1	no	—
12	C₂²×Q₁₆×C₁₇	C₄	1	2	yes	—
13	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
14	C₁₇×#494	—	2	1	no	—
15	C₁₇×#494	—	2	1	no	—
16	C₁₇×#494	C₄	1	2	yes	—
17	C₁₇×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	—	2	2	no	—
20	C₂×Q₁₆×C₁₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₇	—	4	4	no	—
24	C₂×Q₁₆×C₁₇	—	4	4	no	—
25	C₂×Q₁₆×C₁₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₁₇×C₄⋊Q₈	—	2	2	no	—
29	C₁₇×C₄⋊Q₈	—	2	2	no	—
30	C₁₇×C₄⋊Q₈	—	4	1	no	—
31	C₁₇×C₄⋊Q₈	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
34	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
35	C₂×#63223	—	4	1	no	—
36	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×#63214	—	2	2	no	—
41	C₄⋊#3622	—	2	2	no	—
42	C₄⋊#3622	—	4	1	no	—
43	C₂×Q₈×C₁₇	—	2	4	no	—
44	C₂×Q₈×C₁₇	—	4	4	no	—
45	C₂×Q₈×C₁₇	—	4	4	no	—
46	C₂×Q₈×C₁₇	—	4	2	no	—
47	C₂²×C₄×C₁₇	—	4	2	no	—
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₁₇	—	4	4	no	—
52	Q₁₆×C₁₇	—	4	4	no	—
53	Q₁₆×C₁₇	—	4	4	no	—
54	Q₁₆×C₁₇	—	4	4	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₇	—	4	2	no	—
62	C₄⋊C₄×C₁₇	—	4	2	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄⋊C₄×C₁₇	—	4	2	no	—
65	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₇	D₈	1	1	yes	characteristic
67	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
68	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
69	#63223	—	8	2	no	—
70	C₂×#3622	—	4	2	no	—
71	#63214	—	4	2	no	—
72	#11962	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	#102	1	1	yes	characteristic
74	Q₈×C₁₇	—	4	4	no	—
75	Q₈×C₁₇	—	4	4	no	—
76	Q₈×C₁₇	—	8	4	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	1	no	—
79	C₂×C₄×C₁₇	—	2	1	no	—
80	C₂×C₄×C₁₇	—	4	4	no	—
81	C₂×C₄×C₁₇	—	8	4	no	—
82	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
86	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₁₇	—	2	1	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	#100	1	2	yes	—
93	C₈×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
94	#3622	—	8	2	no	—
95	C₂×#494	D₁₇	1	1	yes	characteristic
96	#2162	—	17	1	no	—
97	#1567	—	17	1	no	—
98	#1362	—	17	4	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	#358	1	2	yes	—
102	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
103	C₄×C₁₇	—	2	1	no	—
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	8	4	no	—
106	C₄×C₁₇	#358	1	2	yes	—
107	C₄×C₁₇	#362	1	2	yes	—
108	C₄×C₁₇	C₂×#100	1	1	yes	characteristic
109	C₄×C₁₇	#418	1	1	yes	characteristic
110	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
111	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
112	C₂²×Q₁₆	—	2	1	no	—
113	C₂²×Q₁₆	Dic₁₇	1	2	yes	—
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	#494	—	2	1	no	—
116	#494	—	2	1	no	—
117	#494	Dic₁₇	1	2	yes	—
118	#494	C₂×D₁₇	1	2	yes	—
119	#423	—	34	1	no	—
120	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
121	C₄⋊C₁₆	—	17	2	no	—
122	C₂²⋊C₁₆	—	34	1	no	—
123	C₄×C₁₆	—	34	1	no	—
124	#335	—	17	2	no	—
125	#335	—	34	1	no	—
126	C₂×C₁₇	—	2	2	no	—
127	C₂×C₁₇	#1545	1	1	yes	characteristic
128	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	4	4	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	C₂×Dic₁₇	1	4	yes	—
137	C₂×M₁₆	—	68	1	no	—
138	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	4	1	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂×C₁₆	—	34	2	no	—
148	C₄⋊C₈	—	34	2	no	—
149	C₄⋊C₈	—	68	1	no	—
150	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
153	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
154	C₁₇	#11962	1	1	yes	characteristic, complemented, Sylow
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#63228	1	1	yes	characteristic
161	C₂²×C₄	#63229	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	136	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	68	2	no	—
171	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	#63228	1	1	yes	characteristic
180	C₄²	#63229	1	1	yes	characteristic
181	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
183	C₁₆	—	68	2	no	—
184	C₂³	#10587503	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#10587499	1	2	yes	—
194	C₂×C₄	#10587500	1	2	yes	—
195	C₂×C₄	#10587503	1	1	yes	characteristic
196	C₂×C₄	#10587503	1	1	yes	characteristic
197	C₂×C₄	#10587503	1	1	yes	characteristic
198	C₂×C₄	C₂×#63228	1	1	yes	characteristic
199	C₂×C₄	C₂×#63229	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	136	2	no	—
204	C₈	#10587504	1	2	yes	—
205	C₈	#10587618	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11762613	1	2	yes	—
209	C₂²	C₂×#10587503	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#11762613	1	2	yes	—
214	C₄	#11762625	1	2	yes	—
215	C₄	C₂×#10587504	1	1	yes	characteristic
216	C₄	#11762814	1	1	yes	characteristic
217	C₄	C₂×#10587499	1	1	yes	characteristic
218	C₄	C₂×#10587500	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#14527452	1	1	yes	characteristic
221	C₂	C₂×#11762613	1	1	yes	characteristic
222	C₂	C₂×#11762625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587503
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#11962
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #14526833	C₂	semidirect
154. C₁₇	#11962	semidirect

Maximal subgroups

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

subgroup	index
#14530583	2
#14527543	2
#14526833	2
C₂×C₁₇×#494	2
#11962	17