Subgroups — #14537870

Order 2176.

364 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×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11762940	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762938	C₂	1	2	yes	maximal, complemented
4	#11762937	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762933	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762865	C₂	1	1	yes	maximal, characteristic
8	#11762864	C₂	1	2	yes	maximal
9	#11762861	C₂	1	2	yes	maximal
10	#11762860	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₇	C₄	1	1	yes	characteristic
15	C₄⋊C₄×D₁₇	C₄	1	2	yes	—
16	C₄⋊C₄×D₁₇	C₄	1	1	yes	characteristic
17	#10587573	C₄	1	2	yes	—
18	#10587573	C₄	1	1	yes	characteristic
19	C₄²×D₁₇	C₂²	1	1	yes	characteristic
20	C₄²×D₁₇	C₂²	1	2	yes	—
21	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₄	C₂²	1	2	yes	—
23	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₇×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	#10587491	C₂²	1	1	yes	characteristic
28	#10587491	C₂²	1	1	yes	characteristic
29	#10587491	C₂²	1	2	yes	—
30	#10587486	C₂²	1	2	yes	—
31	#10587485	C₂²	1	1	yes	characteristic
32	#10587485	C₂²	1	1	yes	characteristic
33	#10587485	C₂²	1	2	yes	—
34	#10587485	C₂²	1	2	yes	—
35	C₈×Dic₁₇	C₂²	1	2	yes	—
36	C₈×Dic₁₇	C₂²	1	2	yes	—
37	C₄⋊#3622	C₂²	1	2	yes	—
38	C₄⋊#3622	C₂²	1	1	yes	characteristic
39	#10587474	C₂²	1	2	yes	—
40	C₄×#3622	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₁₇	—	2	4	no	—
43	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
49	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂³	1	2	yes	—
52	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂³	1	1	yes	characteristic
56	C₄²×C₁₇	C₂³	1	2	yes	—
57	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
59	#63225	C₂×C₄	1	2	yes	—
60	#63225	C₂×C₄	1	2	yes	—
61	#63225	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂³	1	2	yes	—
67	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
68	C₂×#3622	C₂³	1	1	yes	characteristic
69	C₂×#3622	C₂³	1	1	yes	characteristic
70	C₂×#3622	C₂³	1	2	yes	—
71	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
83	C₄×D₁₇	—	2	2	no	—
84	C₄×D₁₇	—	2	4	no	—
85	C₄×D₁₇	—	2	2	no	—
86	C₄×D₁₇	—	2	1	no	—
87	C₄×D₁₇	—	2	1	no	—
88	C₄×D₁₇	—	2	4	no	—
89	Dic₃₄	—	2	4	no	—
90	Dic₃₄	M₁₆	1	4	yes	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	—	2	1	no	—
93	C₈×C₁₇	—	2	2	no	—
94	#3622	—	2	1	no	—
95	#3622	—	2	2	no	—
96	#3622	—	2	1	no	—
97	#2984	—	17	1	no	maximal, Sylow
98	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₁₇	—	2	1	no	—
100	C₂×D₁₇	—	2	1	no	—
101	C₂×D₁₇	#131	1	2	yes	—
102	C₄×C₁₇	—	2	2	no	—
103	C₄×C₁₇	—	2	1	no	—
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
106	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₁₇	—	2	1	no	—
109	Dic₁₇	—	2	1	no	—
110	Dic₁₇	—	2	2	no	—
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×Q₈	—	17	1	no	—
114	#446	—	17	1	no	—
115	#446	—	17	2	no	—
116	#446	—	17	2	no	—
117	#446	D₁₇	1	1	yes	characteristic, complemented
118	C₈×Q₈	—	17	1	no	—
119	C₈×Q₈	—	17	1	no	—
120	#432	—	17	1	no	—
121	#432	—	17	2	no	—
122	#431	—	17	2	no	—
123	#431	—	17	1	no	—
124	C₂×C₁₇	#519	1	1	yes	characteristic
125	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
126	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
127	D₁₇	—	2	2	no	—
128	C₂²×Q₈	—	17	1	no	—
129	C₄×Q₈	—	17	1	no	—
130	C₄×Q₈	—	17	1	no	—
131	C₄×Q₈	—	17	2	no	—
132	C₄×Q₈	—	17	1	no	—
133	C₄×Q₈	—	17	2	no	—
134	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	17	2	no	—
136	C₂×C₄⋊C₄	—	17	1	no	—
137	C₂×C₄²	—	17	2	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₄⋊C₈	—	17	1	no	—
140	C₄⋊C₈	—	17	1	no	—
141	C₄⋊C₈	—	17	2	no	—
142	C₄⋊C₈	—	17	2	no	—
143	C₄⋊C₈	—	17	2	no	—
144	C₄⋊C₈	—	17	1	no	—
145	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
146	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
147	C₂²⋊C₈	—	17	1	no	—
148	C₂²⋊C₈	—	17	2	no	—
149	C₂²⋊C₈	—	17	1	no	—
150	#97	—	17	2	no	—
151	#97	—	17	2	no	—
152	#97	C₂×D₁₇	1	2	yes	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	17	2	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₁₇	#2984	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	17	1	no	—
159	C₂×Q₈	—	17	2	no	—
160	C₂×Q₈	—	34	4	no	—
161	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
162	C₂²×C₄	—	17	2	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	2	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₇	1	2	yes	—
172	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
173	C₄⋊C₄	—	17	2	no	—
174	C₄⋊C₄	—	17	2	no	—
175	C₄⋊C₄	—	17	2	no	—
176	C₄⋊C₄	—	17	2	no	—
177	C₄⋊C₄	—	17	1	no	—
178	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
179	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
180	C₄²	—	17	2	no	—
181	C₄²	—	17	2	no	—
182	C₄²	—	17	1	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	2	no	—
185	C₄²	—	17	1	no	—
186	C₄²	C₂²×D₁₇	1	1	yes	characteristic
187	C₄²	C₂²×D₁₇	1	2	yes	—
188	C₂³	—	17	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	17	4	no	—
191	Q₈	—	34	4	no	—
192	C₂×C₄	—	17	2	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	34	4	no	—
198	C₂×C₄	—	34	2	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	34	1	no	—
212	C₈	—	34	2	no	—
213	C₈	—	34	1	no	—
214	C₂²	—	17	2	no	—
215	C₂²	—	34	1	no	—
216	C₂²	—	34	1	no	—
217	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	17	4	no	—
222	C₄	—	34	1	no	—
223	C₄	—	34	1	no	—
224	C₄	—	34	2	no	—
225	C₄	—	34	1	no	—
226	C₄	M₁₆×D₁₇	1	2	yes	—
227	C₄	#10587694	1	1	yes	characteristic
228	C₄	#10587697	1	1	yes	characteristic
229	C₂	—	34	2	no	—
230	C₂	#11763285	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
232	C₂	#131×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2984
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11762940	C₂	semidirect
3. #11762938	C₂	semidirect
4. #11762937	C₂	semidirect
5. C₈×Dic₃₄	C₂	semidirect
6. #11762933	C₂	semidirect
11. C₁₇×#446	C₂	semidirect
117. #446	D₁₇	semidirect
157. C₁₇	#2984	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₇	2
#11762940	2
#11762938	2
#11762937	2
C₈×Dic₃₄	2
#11762933	2
#11762865	2
#11762864	2
#11762861	2
#11762860	2
C₁₇×#446	2
#2984	17