Subgroups — C₂×#11762577 #14528519

Order 2176.

432 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#63223	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762577	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₇	C₄	1	2	yes	—
7	C₂×M₁₆×C₁₇	—	2	2	no	—
8	C₂×M₁₆×C₁₇	—	2	1	no	—
9	C₂×M₁₆×C₁₇	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₇	—	2	1	no	—
12	C₂×C₄⋊Dic₁₇	—	2	1	no	—
13	C₂×C₄⋊Dic₁₇	—	2	1	no	—
14	C₂×C₄⋊Dic₁₇	—	2	1	no	—
15	C₂×C₄⋊Dic₁₇	—	2	1	no	—
16	C₂×C₄⋊Dic₁₇	C₄	1	2	yes	—
17	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
18	C₂×#63223	—	2	1	no	—
19	C₂×#63223	—	2	2	no	—
20	C₂×#63223	C₄	1	4	yes	complemented
21	C₂×#63223	C₂²	1	2	yes	—
22	C₂²×#3622	—	2	1	no	—
23	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₇	—	2	2	no	—
35	C₂²×Dic₁₇	—	2	2	no	—
36	C₂²×Dic₁₇	—	4	4	no	—
37	C₂²×Dic₁₇	C₂×C₄	1	4	yes	—
38	M₁₆×C₁₇	—	2	4	no	—
39	M₁₆×C₁₇	—	2	2	no	—
40	M₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
41	C₂×C₈×C₁₇	—	2	4	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₄⋊Dic₁₇	—	2	2	no	—
44	C₄⋊Dic₁₇	—	2	2	no	—
45	C₄⋊Dic₁₇	—	2	2	no	—
46	C₄⋊Dic₁₇	—	2	2	no	—
47	#63223	—	2	4	no	—
48	#63223	—	2	2	no	—
49	#63223	C₂×C₄	1	4	yes	complemented
50	C₂×#3622	—	2	4	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂³×C₁₇	—	2	2	no	—
53	C₂³×C₁₇	—	2	2	no	—
54	C₂³×C₁₇	C₄²	1	2	yes	—
55	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
58	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄²	1	2	yes	—
60	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	4	no	—
78	C₂×Dic₁₇	—	4	4	no	—
79	C₂×Dic₁₇	—	4	4	no	—
80	C₂×Dic₁₇	—	4	4	no	—
81	C₈×C₁₇	—	2	4	no	—
82	#3622	—	2	4	no	—
83	C₂×#343	—	17	1	no	maximal, Sylow
84	C₂²×C₁₇	—	2	4	no	—
85	C₂²×C₁₇	—	2	4	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	—	2	1	no	—
88	C₂²×C₁₇	—	2	1	no	—
89	C₂²×C₁₇	—	2	1	no	—
90	C₂²×C₁₇	#95	1	2	yes	—
91	C₂²×C₁₇	#95	1	2	yes	—
92	C₂²×C₁₇	#100	1	2	yes	—
93	C₂²×C₁₇	#101	1	2	yes	—
94	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
95	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₁₇	#95	1	2	yes	—
98	C₄×C₁₇	#95	1	2	yes	—
99	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	Dic₁₇	—	4	4	no	—
104	C₂²×M₁₆	—	17	1	no	—
105	C₂²×M₁₆	D₁₇	1	1	yes	characteristic
106	C₂²×C₄⋊C₄	—	17	1	no	—
107	#343	—	17	4	no	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	—	2	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	#343	1	4	yes	complemented
112	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
113	C₂×C₁₇	C₂×#100	1	1	yes	characteristic
114	C₂×C₁₇	C₂×#101	1	1	yes	characteristic
115	C₂³×C₄	—	17	2	no	—
116	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	2	1	no	—
119	C₂×M₁₆	—	17	4	no	—
120	C₂×M₁₆	—	17	2	no	—
121	C₂×M₁₆	—	34	1	no	—
122	C₂×M₁₆	—	34	2	no	—
123	C₂×M₁₆	Dic₁₇	1	4	yes	complemented
124	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	34	1	no	—
127	C₂×C₄⋊C₄	—	17	2	no	—
128	C₂×C₄⋊C₄	—	17	2	no	—
129	C₂×C₄⋊C₄	—	34	1	no	—
130	C₂×C₄⋊C₄	—	34	1	no	—
131	C₂×C₄⋊C₄	—	34	1	no	—
132	C₂×C₄⋊C₄	—	34	1	no	—
133	C₁₇	C₂×#343	1	1	yes	characteristic, complemented, Sylow
134	C₂⁴	C₄×D₁₇	1	1	yes	characteristic
135	C₂²×C₄	—	17	4	no	—
136	C₂²×C₄	—	34	2	no	—
137	C₂²×C₄	—	34	2	no	—
138	C₂²×C₄	—	68	4	no	—
139	C₂²×C₄	Dic₃₄	1	2	yes	—
140	C₂²×C₄	C₄×D₁₇	1	2	yes	—
141	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
142	C₂²×C₄	D₆₈	1	1	yes	characteristic
143	C₂²×C₄	D₆₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₁₇	1	2	yes	—
145	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
148	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
149	M₁₆	—	2	4	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	17	4	no	—
152	M₁₆	—	34	2	no	—
153	M₁₆	—	34	4	no	—
154	M₁₆	C₂×Dic₁₇	1	4	yes	complemented
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	4	no	—
159	C₄⋊C₄	—	34	2	no	—
160	C₄⋊C₄	—	34	2	no	—
161	C₄⋊C₄	—	34	2	no	—
162	C₄⋊C₄	—	34	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₄×Dic₁₇	1	2	yes	—
166	C₂³	#63225	1	2	yes	—
167	C₂³	#63227	1	1	yes	characteristic
168	C₂³	#63227	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
170	C₂×C₄	—	34	4	no	—
171	C₂×C₄	—	68	4	no	—
172	C₂×C₄	—	68	4	no	—
173	C₂×C₄	—	68	4	no	—
174	C₂×C₄	C₄×Dic₁₇	1	2	yes	—
175	C₂×C₄	#63225	1	2	yes	—
176	C₂×C₄	#63225	1	2	yes	—
177	C₂×C₄	#63225	1	2	yes	—
178	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
179	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
180	C₂×C₄	#63227	1	1	yes	characteristic
181	C₂×C₄	#63227	1	1	yes	characteristic
182	C₂×C₄	#63227	1	2	yes	—
183	C₂×C₄	#63227	1	2	yes	—
184	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
185	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
186	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
188	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	34	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	4	no	—
200	C₂²	#10587494	1	2	yes	—
201	C₂²	#10587495	1	2	yes	—
202	C₂²	#10587502	1	2	yes	—
203	C₂²	#10587502	1	2	yes	—
204	C₂²	C₂×C₄×Dic₁₇	1	1	yes	characteristic
205	C₂²	C₂×#63225	1	1	yes	characteristic
206	C₂²	C₂×#63227	1	1	yes	characteristic
207	C₄	—	68	4	no	—
208	C₄	#10587502	1	2	yes	—
209	C₄	#10587502	1	2	yes	—
210	C₄	C₂×#63225	1	1	yes	characteristic
211	C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
212	C₄	C₂×#63227	1	1	yes	characteristic
213	C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#11762577	1	4	yes	direct product factor
218	C₂	C₂×#10587502	1	1	yes	characteristic
219	C₂	C₂×#10587494	1	1	yes	characteristic
220	C₂	C₂×#10587495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#63227
commutator	C₂²×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂²×M₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂²⋊C₄
prime radical	C₂⁴×C₁₇	C₂×C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂²×M₁₆	D₁₇
hypocenter	C₁₇	C₂×#343
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11762577	C₂	direct (N × H)
9. C₂×M₁₆×C₁₇	C₄	semidirect
20. C₂×#63223	C₄	semidirect
40. M₁₆×C₁₇	C₂×C₄	semidirect
49. #63223	C₂×C₄	semidirect
111. C₂×C₁₇	#343	semidirect
123. C₂×M₁₆	Dic₁₇	semidirect
133. C₁₇	C₂×#343	semidirect
154. M₁₆	C₂×Dic₁₇	semidirect
217. C₂	#11762577	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#63223	2
C₂²×C₄⋊Dic₁₇	2
#11762577	2
C₂²×M₁₆×C₁₇	2
C₂×#343	17

Subgroups — C2×#11762577 #14528519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762577 #14528519