Subgroups — #14537912

Order 2176.

394 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#63254	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762933	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762887	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762885	C₂	1	2	yes	maximal, complemented
6	#11762884	C₂	1	2	yes	maximal, complemented
7	#11762860	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762859	C₂	1	2	yes	maximal, complemented
9	#11762832	C₂	1	2	yes	maximal, complemented
10	#11762831	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#63254	C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
14	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
15	D₄×Dic₁₇	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
18	C₂×#63223	C₂²	1	2	yes	—
19	C₂×#63218	C₂²	1	2	yes	—
20	#10587578	C₄	1	1	yes	characteristic
21	#10587573	C₄	1	1	yes	characteristic
22	#10587567	C₄	1	2	yes	—
23	#10587563	C₄	1	2	yes	—
24	C₄²×D₁₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
26	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
29	C₂²⋊#3622	C₂²	1	2	yes	—
30	#10587491	C₂²	1	1	yes	characteristic
31	#10587491	C₂²	1	1	yes	characteristic
32	#10587485	C₂²	1	2	yes	—
33	#10587485	C₂²	1	1	yes	characteristic
34	#10587485	C₂²	1	2	yes	—
35	#10587485	C₂²	1	2	yes	—
36	#10587485	C₂²	1	1	yes	characteristic
37	C₈×Dic₁₇	C₂²	1	2	yes	—
38	C₈×Dic₁₇	C₂²	1	2	yes	—
39	C₄⋊#3622	C₂²	1	1	yes	characteristic
40	C₄×#3622	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
46	#63254	—	2	4	no	—
47	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
50	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
51	M₁₆×C₁₇	—	4	2	no	—
52	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂³	1	2	yes	—
54	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
57	C₄²×C₁₇	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₇	C₂×C₄	1	2	yes	—
59	#63227	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
61	#63225	C₂×C₄	1	2	yes	—
62	#63225	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₇	C₂³	1	2	yes	—
68	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
69	#63223	—	4	2	no	—
70	C₂×#3622	C₂³	1	1	yes	characteristic
71	C₂×#3622	C₂³	1	1	yes	characteristic
72	C₂×#3622	C₂³	1	2	yes	—
73	#63218	—	4	2	no	—
74	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
76	D₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₇	—	2	4	no	—
84	C₂×Dic₁₇	—	2	4	no	—
85	C₂×Dic₁₇	—	2	4	no	—
86	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₄×D₁₇	—	2	2	no	—
92	C₄×D₁₇	—	2	1	no	—
93	C₄×D₁₇	—	2	1	no	—
94	C₄×D₁₇	—	2	2	no	—
95	Dic₃₄	M₁₆	1	4	yes	complemented
96	C₈×C₁₇	—	2	1	no	—
97	C₈×C₁₇	—	2	2	no	—
98	C₈×C₁₇	—	2	1	no	—
99	#3622	—	2	2	no	—
100	#3622	—	2	1	no	—
101	#3622	—	2	1	no	—
102	#2991	—	17	1	no	maximal, Sylow
103	C₂²×C₁₇	—	2	2	no	—
104	C₂²×C₁₇	—	2	2	no	—
105	C₂²×C₁₇	—	2	2	no	—
106	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₁₇	—	2	1	no	—
108	C₂×D₁₇	—	2	1	no	—
109	C₂×D₁₇	—	2	1	no	—
110	C₄×C₁₇	—	2	2	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
113	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₁₇	—	2	1	no	—
116	Dic₁₇	—	2	1	no	—
117	Dic₁₇	—	2	2	no	—
118	Dic₁₇	C₂×M₁₆	1	4	yes	—
119	Dic₁₇	#131	1	2	yes	—
120	C₄×#41	—	17	1	no	—
121	C₈×Q₈	—	17	1	no	—
122	C₈×Q₈	—	17	1	no	—
123	#436	—	17	2	no	—
124	#436	—	17	1	no	—
125	#436	—	17	2	no	—
126	#436	D₁₇	1	1	yes	characteristic, complemented
127	#431	—	17	2	no	—
128	#431	—	17	1	no	—
129	#423	—	17	2	no	—
130	#423	—	17	1	no	—
131	C₂×C₁₇	—	4	2	no	—
132	C₂×C₁₇	#520	1	1	yes	characteristic
133	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
135	D₁₇	—	4	1	no	—
136	C₂×#41	—	17	1	no	—
137	C₂×M₁₆	—	17	2	no	—
138	C₂×M₁₆	—	17	2	no	—
139	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
140	C₄×Q₈	—	17	1	no	—
141	C₄×Q₈	—	17	1	no	—
142	C₄×D₄	—	17	1	no	—
143	C₄×D₄	—	17	2	no	—
144	C₄×D₄	—	17	2	no	—
145	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
146	#117	—	17	1	no	—
147	#117	—	17	2	no	—
148	C₂×C₄²	—	17	2	no	—
149	C₂×C₄²	—	17	1	no	—
150	C₄⋊C₈	—	17	2	no	—
151	C₄⋊C₈	—	17	1	no	—
152	C₄⋊C₈	—	17	2	no	—
153	C₄⋊C₈	—	17	2	no	—
154	C₄⋊C₈	—	17	1	no	—
155	C₄⋊C₈	—	17	1	no	—
156	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₂²⋊C₈	—	17	2	no	—
158	C₂²⋊C₈	—	17	1	no	—
159	C₂²⋊C₈	—	17	1	no	—
160	C₂²⋊C₈	C₂×D₁₇	1	2	yes	—
161	C₄×C₈	—	17	2	no	—
162	C₄×C₈	—	17	2	no	—
163	C₄×C₈	—	17	1	no	—
164	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
165	C₁₇	#2991	1	1	yes	characteristic, complemented, Sylow
166	#41	—	34	4	no	—
167	C₂×Q₈	—	17	1	no	—
168	C₂×D₄	—	17	2	no	—
169	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
170	C₂²×C₄	—	17	2	no	—
171	C₂²×C₄	—	17	2	no	—
172	C₂²×C₄	—	17	1	no	—
173	C₂²×C₄	—	17	1	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	68	2	no	—
178	M₁₆	—	68	2	no	—
179	C₂×C₈	—	17	1	no	—
180	C₂×C₈	—	17	2	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₇	1	2	yes	—
184	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
185	C₄⋊C₄	—	17	2	no	—
186	C₄⋊C₄	—	17	2	no	—
187	C₄⋊C₄	—	17	1	no	—
188	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	17	2	no	—
190	C₂²⋊C₄	—	17	2	no	—
191	C₂²⋊C₄	C₄×D₁₇	1	2	yes	—
192	C₄²	—	17	2	no	—
193	C₄²	—	17	1	no	—
194	C₄²	—	17	2	no	—
195	C₄²	—	17	1	no	—
196	C₄²	—	17	2	no	—
197	C₄²	—	17	1	no	—
198	C₄²	C₂²×D₁₇	1	1	yes	characteristic
199	C₂³	—	17	1	no	—
200	C₂³	C₂×C₄×D₁₇	1	2	yes	—
201	Q₈	—	17	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	34	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	17	1	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	1	no	—
209	C₂×C₄	—	17	1	no	—
210	C₂×C₄	—	17	2	no	—
211	C₂×C₄	—	34	2	no	—
212	C₂×C₄	—	34	4	no	—
213	C₂×C₄	—	34	4	no	—
214	C₂×C₄	—	34	1	no	—
215	C₂×C₄	—	34	1	no	—
216	C₂×C₄	—	34	2	no	—
217	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	34	2	no	—
225	C₈	—	34	1	no	—
226	C₈	—	34	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	34	1	no	—
231	C₂²	—	34	1	no	—
232	C₂²	—	34	1	no	—
233	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	17	4	no	—
237	C₄	—	17	2	no	—
238	C₄	—	34	1	no	—
239	C₄	—	34	1	no	—
240	C₄	—	34	2	no	—
241	C₄	M₁₆×D₁₇	1	2	yes	—
242	C₄	#10587691	1	1	yes	characteristic
243	C₄	#10587697	1	1	yes	characteristic
244	C₂	—	4	2	no	—
245	C₂	—	68	1	no	—
246	C₂	#11763288	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
248	C₂	#131×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#436	C₂
prime radical	C₂×#63254	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#436	D₁₇
hypocenter	C₁₇	#2991
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₃₄	C₂	semidirect
3. #11762933	C₂	semidirect
4. #11762887	C₂	semidirect
5. #11762885	C₂	semidirect
6. #11762884	C₂	semidirect
7. #11762860	C₂	semidirect
8. #11762859	C₂	semidirect
9. #11762832	C₂	semidirect
10. #11762831	C₂	semidirect
11. C₁₇×#436	C₂	semidirect
95. Dic₃₄	M₁₆	semidirect
126. #436	D₁₇	semidirect
165. C₁₇	#2991	semidirect

Maximal subgroups

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

subgroup	index
C₄×#63254	2
C₈×Dic₃₄	2
#11762933	2
#11762887	2
#11762885	2
#11762884	2
#11762860	2
#11762859	2
#11762832	2
#11762831	2
C₁₇×#436	2
#2991	17