Subgroups — #14537736

Order 2176.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 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	#11762831	C₂	1	1	yes	maximal, characteristic
8	#11762831	C₂	1	2	yes	maximal
9	#11762761	C₂	1	2	yes	maximal
10	C₄×#63218	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	C₂×#63218	C₂²	1	1	yes	characteristic
16	C₂×#63218	C₂²	1	2	yes	—
17	C₂×#63218	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₁₇	C₄	1	2	yes	—
19	C₄⋊C₄×D₁₇	C₄	1	1	yes	characteristic
20	#10587573	C₄	1	2	yes	—
21	#10587573	C₄	1	1	yes	characteristic
22	C₄²×D₁₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₇	C₂²	1	2	yes	—
24	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₄	C₂²	1	2	yes	—
26	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
28	C₁₇×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
30	#10587486	C₂²	1	1	yes	characteristic
31	#10587486	C₂²	1	1	yes	characteristic
32	#10587485	C₂²	1	2	yes	—
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	1	yes	characteristic
38	C₄⋊#3622	C₂²	1	2	yes	—
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	2	yes	—
51	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	2	yes	—
70	C₂×#3622	C₂³	1	1	yes	characteristic
71	#63218	—	2	2	no	—
72	#63218	—	2	4	no	—
73	#63218	Q₈	1	4	yes	complemented
74	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
86	C₄×D₁₇	—	2	2	no	—
87	C₄×D₁₇	—	2	4	no	—
88	C₄×D₁₇	—	4	1	no	—
89	C₄×D₁₇	—	4	2	no	—
90	C₄×D₁₇	C₂×Q₈	1	2	yes	—
91	C₄×D₁₇	#41	1	2	yes	—
92	Dic₃₄	—	2	4	no	—
93	Dic₃₄	M₁₆	1	4	yes	complemented
94	C₈×C₁₇	—	2	2	no	—
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	C₂×Q₈	1	2	yes	complemented
97	#3622	—	2	1	no	—
98	#3622	—	2	2	no	—
99	#3622	C₂×Q₈	1	2	yes	complemented
100	Q₈×M₁₆	—	17	1	no	maximal, Sylow
101	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₇	—	2	1	no	—
103	C₂×D₁₇	—	2	1	no	—
104	C₂×D₁₇	C₄×Q₈	1	2	yes	—
105	C₄×C₁₇	—	2	2	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
109	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	—	2	2	no	—
113	Dic₁₇	—	2	1	no	—
114	Dic₁₇	C₄×Q₈	1	2	yes	—
115	Dic₁₇	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×Q₈	—	17	1	no	—
117	#446	—	17	1	no	—
118	#446	—	17	2	no	—
119	#446	—	17	2	no	—
120	#446	D₁₇	1	1	yes	characteristic, complemented
121	C₈×Q₈	—	17	1	no	—
122	C₈×Q₈	—	17	1	no	—
123	#423	—	17	2	no	—
124	#423	—	17	1	no	—
125	C₄×M₁₆	—	17	2	no	—
126	C₄×M₁₆	—	17	1	no	—
127	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×C₁₇	#568	1	1	yes	characteristic
130	D₁₇	—	2	2	no	—
131	C₂²×Q₈	—	17	1	no	—
132	C₂×M₁₆	—	17	1	no	—
133	C₂×M₁₆	—	17	2	no	—
134	C₂×M₁₆	—	17	1	no	—
135	C₄×Q₈	—	17	1	no	—
136	C₄×Q₈	—	17	1	no	—
137	C₄×Q₈	—	17	2	no	—
138	C₄×Q₈	—	17	1	no	—
139	C₄×Q₈	—	17	2	no	—
140	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	17	2	no	—
142	C₂×C₄⋊C₄	—	17	1	no	—
143	C₂×C₄²	—	17	2	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₄⋊C₈	—	17	2	no	—
146	C₄⋊C₈	—	17	2	no	—
147	C₄⋊C₈	—	17	2	no	—
148	C₄⋊C₈	—	17	1	no	—
149	C₄⋊C₈	—	17	2	no	—
150	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
151	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
152	#97	—	17	1	no	—
153	#97	—	17	1	no	—
154	#97	—	17	2	no	—
155	#97	C₂×D₁₇	1	2	yes	—
156	C₄×C₈	—	17	2	no	—
157	C₄×C₈	—	17	2	no	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
160	C₁₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₈	—	17	1	no	—
162	C₂×Q₈	—	17	2	no	—
163	C₂×Q₈	—	34	4	no	—
164	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	2	no	—
167	C₂²×C₄	—	17	1	no	—
168	C₂²×C₄	—	17	2	no	—
169	C₂²×C₄	—	17	1	no	—
170	M₁₆	—	17	4	no	—
171	M₁₆	—	34	4	no	—
172	M₁₆	—	34	2	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	17	2	no	—
175	C₂×C₈	—	17	1	no	—
176	C₂×C₈	C₂²×D₁₇	1	2	yes	—
177	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
179	C₄⋊C₄	—	17	2	no	—
180	C₄⋊C₄	—	17	2	no	—
181	C₄⋊C₄	—	17	2	no	—
182	C₄⋊C₄	—	17	2	no	—
183	C₄⋊C₄	—	17	1	no	—
184	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
185	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
186	C₄²	—	17	2	no	—
187	C₄²	—	17	2	no	—
188	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	17	2	no	—
191	C₄²	—	17	1	no	—
192	C₄²	C₂²×D₁₇	1	1	yes	characteristic
193	C₄²	C₂²×D₁₇	1	2	yes	—
194	C₂³	—	17	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	17	4	no	—
197	Q₈	—	34	4	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	2	no	—
200	C₂×C₄	—	17	2	no	—
201	C₂×C₄	—	17	2	no	—
202	C₂×C₄	—	17	2	no	—
203	C₂×C₄	—	17	1	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	34	4	no	—
206	C₂×C₄	—	34	2	no	—
207	C₂×C₄	—	68	1	no	—
208	C₂×C₄	—	68	2	no	—
209	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	17	2	no	—
217	C₈	—	34	2	no	—
218	C₈	—	34	1	no	—
219	C₈	Q₈×D₁₇	1	2	yes	complemented
220	C₂²	—	17	2	no	—
221	C₂²	—	34	1	no	—
222	C₂²	—	34	1	no	—
223	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	17	2	no	—
228	C₄	—	17	4	no	—
229	C₄	—	34	1	no	—
230	C₄	—	34	2	no	—
231	C₄	—	34	1	no	—
232	C₄	M₁₆×D₁₇	1	2	yes	—
233	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
234	C₄	#41×D₁₇	1	1	yes	characteristic
235	C₂	—	34	2	no	—
236	C₂	C₄×Q₈×D₁₇	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
238	C₂	#11763574	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	Q₈×M₁₆
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
73. #63218	Q₈	semidirect
93. Dic₃₄	M₁₆	semidirect
96. C₈×C₁₇	C₂×Q₈	semidirect
99. #3622	C₂×Q₈	semidirect
120. #446	D₁₇	semidirect
160. C₁₇	Q₈×M₁₆	semidirect
219. C₈	Q₈×D₁₇	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
#11762831	2
#11762761	2
C₄×#63218	2
C₁₇×#446	2
Q₈×M₁₆	17