Subgroups — #16944151

Order 4352.

322 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530540	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
3	#14526766	C₂	1	4	yes	maximal, complemented
4	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	—	2	1	no	—
6	C₂²×#63214	—	2	1	no	—
7	C₄⋊#63214	—	2	1	no	—
8	C₄⋊#63214	C₂²	1	2	yes	—
9	#11762585	—	2	1	no	—
10	#11762585	C₂²	1	2	yes	—
11	C₁₇×#533	C₄	1	1	yes	characteristic
12	C₁₇×#446	—	2	1	no	—
13	C₁₇×#446	C₂²	1	2	yes	—
14	C₈×Q₈×C₁₇	—	2	1	no	—
15	C₈×Q₈×C₁₇	C₄	1	2	yes	—
16	C₁₇×#424	—	2	1	no	—
17	C₁₇×#423	C₄	1	1	yes	characteristic
18	C₁₇×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₁₇×#125	—	2	1	no	—
26	C₁₇×C₂²⋊Q₈	—	2	1	no	—
27	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
28	C₄×Q₈×C₁₇	—	2	1	no	—
29	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₁₇×#117	—	2	1	no	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₁₇×C₄⋊C₈	—	2	2	no	—
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₁₇×#97	—	2	2	no	—
37	C₄×C₈×C₁₇	—	2	2	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	#10587483	—	4	2	no	—
43	C₂×#63214	—	2	4	no	—
44	C₂×#63214	—	2	2	no	—
45	C₂×#63214	—	2	2	no	—
46	C₂×Q₈×C₁₇	—	2	1	no	—
47	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
51	M₁₆×C₁₇	—	4	2	no	—
52	C₂×C₈×C₁₇	—	2	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₇	—	2	1	no	—
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	—	2	1	no	—
63	C₄⋊C₄×C₁₇	—	2	1	no	—
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
66	C₂²⋊C₄×C₁₇	—	2	2	no	—
67	C₄²×C₁₇	—	2	2	no	—
68	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
71	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	#63214	—	2	4	no	—
73	#63214	—	4	2	no	—
74	#11612	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₁₇	—	4	2	no	—
77	Q₈×C₁₇	—	4	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	4	1	no	—
81	C₂×C₄×C₁₇	—	4	1	no	—
82	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#103	1	2	yes	—
86	C₂×C₄×C₁₇	#104	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₁₇	—	2	1	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	—	4	2	no	—
93	C₈×C₁₇	#103	1	2	yes	—
94	C₈×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
96	#2960	D₁₇	1	1	yes	characteristic
97	#2150	—	17	1	no	—
98	C₂×#346	—	17	1	no	—
99	#1338	—	17	4	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	—	2	1	no	—
102	C₂²×C₁₇	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	1	no	—
108	C₄×C₁₇	—	4	2	no	—
109	C₄×C₁₇	—	4	2	no	—
110	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
111	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
112	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
113	#533	Dic₁₇	1	1	yes	characteristic
114	C₂×M₃₂	—	34	1	no	—
115	C₂²×C₁₆	—	34	1	no	—
116	#446	—	2	1	no	—
117	#446	C₂×D₁₇	1	2	yes	—
118	C₈×Q₈	—	2	1	no	—
119	C₈×Q₈	Dic₁₇	1	2	yes	—
120	#424	—	2	1	no	—
121	#423	Dic₁₇	1	1	yes	characteristic
122	#405	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
124	C₄⋊C₁₆	—	17	2	no	—
125	C₄⋊C₁₆	—	34	1	no	—
126	#346	—	17	2	no	—
127	#346	—	34	1	no	—
128	C₂×C₁₇	—	2	2	no	—
129	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
130	C₂×C₁₇	#2115	1	1	yes	characteristic
131	C₂×C₁₇	#2117	1	1	yes	characteristic
132	C₂×M₁₆	—	2	1	no	—
133	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₄⋊Q₈	#3622	1	1	yes	characteristic
136	C₄⋊Q₈	#3622	1	1	yes	characteristic
137	#125	—	2	1	no	—
138	C₂²⋊Q₈	—	2	1	no	—
139	C₂²⋊Q₈	#3622	1	2	yes	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
142	#117	—	2	1	no	—
143	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
144	M₃₂	—	68	2	no	—
145	C₂×C₁₆	—	34	4	no	—
146	C₂×C₁₆	—	34	2	no	—
147	C₂×C₁₆	—	34	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
158	C₁₇	#11612	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	C₂×#3622	1	2	yes	—
161	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
162	C₂²×C₄	#63223	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	#63230	1	2	yes	—
168	C₂×C₈	#63231	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂×#3622	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	C₂×#3622	1	1	yes	characteristic
182	C₄²	#63223	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
184	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
185	C₁₆	—	34	4	no	—
186	C₁₆	—	68	2	no	—
187	C₂³	C₂²⋊#3622	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
197	C₂×C₄	#10587506	1	2	yes	—
198	C₂×C₄	#10587508	1	2	yes	—
199	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
200	C₂×C₄	C₂×#63223	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#10587506	1	2	yes	—
206	C₈	C₂×#63230	1	1	yes	characteristic
207	C₈	C₂×#63231	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	Q₈⋊#3622	1	2	yes	—
211	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	Q₈⋊#3622	1	2	yes	—
219	C₄	C₂×#10587506	1	1	yes	characteristic
220	C₄	C₂×#10587508	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×Q₈⋊#3622	1	1	yes	characteristic
223	C₂	#14530405	1	1	yes	characteristic
224	C₂	#14530412	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#11612
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #14526766	C₂	semidirect
158. C₁₇	#11612	semidirect

Maximal subgroups

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

subgroup	index
#14530540	2
C₂×#11762585	2
#14526766	2
C₁₇×#2960	2
#11612	17