Subgroups — C₂×#11762568 #14528489

Order 2176.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#11762568	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₇×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₇	—	2	1	no	—
7	C₁₇×#117	—	2	1	no	—
8	C₁₇×#117	—	2	1	no	—
9	C₁₇×#117	C₄	1	2	yes	complemented
10	C₁₇×#117	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₇	C₄	1	2	yes	complemented
12	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
13	C₂×C₄²×C₁₇	—	2	1	no	—
14	C₂×C₄×Dic₁₇	—	2	2	no	—
15	C₂×C₄×Dic₁₇	—	2	1	no	—
16	C₂×C₄×Dic₁₇	C₄	1	4	yes	complemented
17	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
18	C₂×#63223	—	2	2	no	—
19	C₂×#63223	—	2	1	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	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	—	2	2	no	—
25	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₂×C₄	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	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₇	—	2	8	no	—
36	C₂²×Dic₁₇	—	2	4	no	—
37	C₂²×Dic₁₇	—	2	2	no	—
38	C₄⋊C₄×C₁₇	—	2	2	no	—
39	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	complemented
40	C₂²⋊C₄×C₁₇	—	2	2	no	—
41	C₂²⋊C₄×C₁₇	—	2	2	no	—
42	C₄²×C₁₇	—	2	2	no	—
43	C₄²×C₁₇	—	2	2	no	—
44	C₄×Dic₁₇	—	2	4	no	—
45	C₄×Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	C₂×C₄	1	4	yes	complemented
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	4	no	—
53	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	—	4	4	no	—
60	C₂×C₄×C₁₇	C₄²	1	2	yes	—
61	C₂×C₄×C₁₇	C₄²	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	C₂²⋊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	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
78	C₂×Dic₁₇	—	2	16	no	—
79	C₂×Dic₁₇	—	2	8	no	—
80	C₂×Dic₁₇	—	2	4	no	—
81	#3622	—	2	4	no	—
82	C₂×#339	—	17	1	no	maximal, Sylow
83	C₂²×C₁₇	—	2	4	no	—
84	C₂²×C₁₇	—	2	4	no	—
85	C₂²×C₁₇	—	2	2	no	—
86	C₂²×C₁₇	—	2	2	no	—
87	C₂²×C₁₇	#95	1	2	yes	—
88	C₂²×C₁₇	#95	1	2	yes	—
89	C₂²×C₁₇	#104	1	4	yes	—
90	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₁₇	—	4	4	no	—
94	C₄×C₁₇	#95	1	2	yes	—
95	C₄×C₁₇	#95	1	2	yes	—
96	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
97	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	Dic₁₇	—	2	8	no	—
101	C₂²×M₁₆	—	17	1	no	—
102	C₂×#117	D₁₇	1	1	yes	characteristic
103	C₂²×C₄²	—	17	1	no	—
104	#339	—	17	4	no	—
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	—	2	2	no	—
107	C₂×C₁₇	—	2	1	no	—
108	C₂×C₁₇	#339	1	4	yes	complemented
109	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
110	C₂×C₁₇	C₂×#104	1	2	yes	—
111	C₂³×C₄	—	34	1	no	—
112	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
113	C₂×M₁₆	—	17	4	no	—
114	C₂×M₁₆	—	17	2	no	—
115	C₂×M₁₆	—	34	2	no	—
116	C₂×M₁₆	—	34	1	no	—
117	C₂²×C₈	—	34	1	no	—
118	#117	—	2	1	no	—
119	#117	—	2	1	no	—
120	#117	Dic₁₇	1	2	yes	complemented
121	#117	C₂×D₁₇	1	2	yes	—
122	C₂×C₄⋊C₄	Dic₁₇	1	2	yes	complemented
123	C₂×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	—	17	4	no	—
126	C₂×C₄²	—	17	2	no	—
127	C₂×C₄²	—	34	2	no	—
128	C₂×C₄²	—	34	1	no	—
129	C₁₇	C₂×#339	1	1	yes	characteristic, complemented, Sylow
130	C₂⁴	C₂²⋊D₁₇	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	34	8	no	—
133	C₂²×C₄	—	34	2	no	—
134	C₂²×C₄	—	34	4	no	—
135	C₂²×C₄	Dic₃₄	1	2	yes	—
136	C₂²×C₄	C₄×D₁₇	1	2	yes	—
137	C₂²×C₄	C₄×D₁₇	1	2	yes	—
138	C₂²×C₄	D₆₈	1	2	yes	—
139	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
145	M₁₆	—	17	4	no	—
146	M₁₆	—	34	2	no	—
147	M₁₆	—	34	4	no	—
148	C₂×C₈	—	34	2	no	—
149	C₂×C₈	—	34	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	C₂×Dic₁₇	1	4	yes	complemented
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	17	4	no	—
157	C₄²	—	34	2	no	—
158	C₄²	—	34	4	no	—
159	C₂³	—	2	4	no	—
160	C₂³	#63225	1	2	yes	—
161	C₂³	#63227	1	2	yes	—
162	C₂³	C₂²⋊Dic₁₇	1	1	yes	characteristic
163	C₂³	C₂²⋊Dic₁₇	1	1	yes	characteristic
164	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	34	8	no	—
168	C₂×C₄	—	34	16	no	—
169	C₂×C₄	—	34	4	no	—
170	C₂×C₄	C₄×Dic₁₇	1	2	yes	—
171	C₂×C₄	C₄×Dic₁₇	1	2	yes	—
172	C₂×C₄	#63225	1	2	yes	—
173	C₂×C₄	#63225	1	2	yes	—
174	C₂×C₄	#63225	1	2	yes	—
175	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
176	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
177	C₂×C₄	#63227	1	2	yes	—
178	C₂×C₄	#63227	1	2	yes	—
179	C₂×C₄	#63227	1	2	yes	—
180	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
181	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
185	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
188	C₈	—	34	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	#10587502	1	2	yes	—
194	C₂²	#10587502	1	2	yes	—
195	C₂²	#10587508	1	4	yes	—
196	C₂²	C₂×#63225	1	1	yes	characteristic
197	C₂²	C₂×#63227	1	1	yes	characteristic
198	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
199	C₄	—	4	4	no	—
200	C₄	—	34	8	no	—
201	C₄	#10587502	1	2	yes	—
202	C₄	#10587502	1	2	yes	—
203	C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
204	C₄	C₂×#63225	1	1	yes	characteristic
205	C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
206	C₄	C₂×#63227	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#11762568	1	4	yes	direct product factor
211	C₂	C₂×#10587502	1	1	yes	characteristic
212	C₂	C₂×#10587508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#117	C₂
socle	C₂³×C₁₇	C₂²⋊C₄
prime radical	C₂⁴×C₁₇	D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#117	D₁₇
hypocenter	C₁₇	C₂×#339
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #11762568	C₂	direct (N × H)
9. C₁₇×#117	C₄	semidirect
11. C₂×C₄⋊C₄×C₁₇	C₄	semidirect
16. C₂×C₄×Dic₁₇	C₄	semidirect
20. C₂×#63223	C₄	semidirect
39. C₄⋊C₄×C₁₇	C₂×C₄	semidirect
46. C₄×Dic₁₇	C₂×C₄	semidirect
49. #63223	C₂×C₄	semidirect
108. C₂×C₁₇	#339	semidirect
120. #117	Dic₁₇	semidirect
122. C₂×C₄⋊C₄	Dic₁₇	semidirect
129. C₁₇	C₂×#339	semidirect
151. C₄⋊C₄	C₂×Dic₁₇	semidirect
210. C₂	#11762568	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
#11762568	2
C₂×C₁₇×#117	2
C₂×#339	17

Subgroups — C2×#11762568 #14528489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762568 #14528489