Subgroups — C₂×#14531070 #17189821

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763211	C₂	1	1	yes	maximal, characteristic
3	#14531070	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762637	C₂	1	2	yes	maximal
5	C₂×#11762635	C₂	1	2	yes	maximal
6	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#10587555	—	2	1	no	—
8	C₂×#10587556	—	2	1	no	—
9	#11763214	—	2	2	no	—
10	#11763214	C₂²	1	4	yes	—
11	#11763211	—	2	2	no	—
12	#11763211	C₂²	1	4	yes	—
13	C₂×#10587489	—	2	1	no	—
14	C₂×#10587489	C₂²	1	1	yes	characteristic
15	C₂×#10587489	C₂²	1	2	yes	—
16	C₂×#10587489	C₂²	1	1	yes	characteristic
17	C₂×#10587488	—	2	1	no	—
18	#11762637	C₂²	1	8	yes	—
19	#11762635	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#346	C₂²	1	4	yes	—
23	C₂²×Dic₃₄	—	4	1	no	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	1	no	—
28	C₂×C₄⋊Dic₁₇	—	2	1	no	—
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₂×C₄⋊Dic₁₇	—	4	1	no	—
31	C₂×#63225	—	4	1	no	—
32	C₂×#63225	—	4	1	no	—
33	#10587556	—	2	4	no	—
34	#10587556	—	4	2	no	—
35	#10587555	—	2	4	no	—
36	#10587555	—	4	2	no	—
37	C₂×C₁₆×C₁₇	Q₈	1	8	yes	complemented
38	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
39	C₄×C₈×C₁₇	D₄	1	2	yes	—
40	C₄×C₈×C₁₇	C₂³	1	2	yes	—
41	#10587489	—	2	1	no	—
42	#10587489	—	2	1	no	—
43	#10587489	—	2	2	no	—
44	#10587489	—	2	4	no	—
45	#10587489	C₂³	1	2	yes	—
46	#10587489	C₂³	1	2	yes	—
47	#10587489	C₂³	1	4	yes	—
48	#10587488	—	2	4	no	—
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	4	2	no	—
53	C₂²×Dic₁₇	—	4	1	no	—
54	C₂²×Dic₁₇	—	4	1	no	—
55	C₂×Dic₃₄	—	4	4	no	—
56	C₂×Dic₃₄	—	8	4	no	—
57	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
62	C₄²×C₁₇	D₈	1	2	yes	—
63	C₄²×C₁₇	C₂×D₄	1	2	yes	—
64	C₄⋊Dic₁₇	—	2	2	no	—
65	C₄⋊Dic₁₇	—	2	2	no	—
66	C₄⋊Dic₁₇	—	2	4	no	—
67	C₄⋊Dic₁₇	—	4	4	no	—
68	C₄⋊Dic₁₇	—	4	2	no	—
69	C₄⋊Dic₁₇	—	4	1	no	—
70	C₄⋊Dic₁₇	—	4	1	no	—
71	#63225	—	4	4	no	—
72	#63225	—	4	4	no	—
73	C₁₆×C₁₇	C₂×Q₈	1	8	yes	complemented
74	C₂×#2255	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₁₇	—	4	4	no	—
85	C₂×Dic₁₇	—	4	2	no	—
86	C₂×Dic₁₇	—	4	2	no	—
87	C₂×Dic₁₇	—	8	4	no	—
88	C₂×Dic₁₇	—	8	2	no	—
89	C₂×Dic₁₇	—	8	2	no	—
90	Dic₃₄	—	8	8	no	—
91	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
92	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
93	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
94	C₂×#500	—	17	1	no	—
95	C₂×#499	—	17	1	no	—
96	#2255	—	17	8	no	—
97	C₂×#367	—	17	2	no	—
98	C₂×#366	—	17	2	no	—
99	C₂×#346	D₁₇	1	1	yes	characteristic
100	C₂²×C₁₇	#500	1	2	yes	—
101	C₂²×C₁₇	#509	1	2	yes	—
102	C₂²×C₁₇	#510	1	2	yes	—
103	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	—	2	2	no	—
106	C₄×C₁₇	#500	1	2	yes	—
107	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
109	Dic₁₇	—	8	4	no	—
110	Dic₁₇	—	8	2	no	—
111	Dic₁₇	—	8	2	no	—
112	C₂×C₄⋊Q₈	—	34	1	no	—
113	C₂×#125	—	34	1	no	—
114	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
115	#500	—	17	4	no	—
116	#500	—	34	2	no	—
117	#499	—	17	4	no	—
118	#499	—	34	2	no	—
119	C₂×#107	—	17	1	no	—
120	C₂×#107	—	17	1	no	—
121	C₂×#107	—	17	2	no	—
122	C₂×#107	—	34	1	no	—
123	C₂×#106	—	34	1	no	—
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	#367	—	17	8	no	—
126	#366	—	17	8	no	—
127	#346	C₂×D₁₇	1	4	yes	—
128	C₂×C₁₇	#2255	1	4	yes	complemented
129	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#509	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#510	1	1	yes	characteristic
132	C₂²×Q₈	—	68	1	no	—
133	C₂²×C₈	D₆₈	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	34	4	no	—
136	C₄⋊Q₈	—	68	2	no	—
137	#125	—	34	4	no	—
138	#125	—	68	2	no	—
139	C₂×C₄⋊C₄	—	34	1	no	—
140	C₂×C₄⋊C₄	—	34	1	no	—
141	C₂×C₄⋊C₄	—	34	2	no	—
142	C₂×C₄⋊C₄	—	68	1	no	—
143	C₂×C₄⋊C₄	—	68	1	no	—
144	C₂×C₄⋊C₄	—	68	1	no	—
145	C₂×C₄²	D₆₈	1	1	yes	characteristic
146	C₂×C₁₆	Dic₃₄	1	8	yes	complemented
147	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
148	#107	—	17	2	no	—
149	#107	—	17	4	no	—
150	#107	—	17	2	no	—
151	#107	—	34	1	no	—
152	#107	—	34	1	no	—
153	#107	—	34	2	no	—
154	#107	—	34	4	no	—
155	#106	—	34	4	no	—
156	C₄×C₈	D₆₈	1	2	yes	—
157	C₄×C₈	C₂²×D₁₇	1	2	yes	—
158	C₁₇	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₈	—	68	4	no	—
160	C₂×Q₈	—	136	4	no	—
161	C₂²×C₄	—	68	1	no	—
162	C₂²×C₄	—	68	1	no	—
163	C₂²×C₄	—	68	2	no	—
164	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
165	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
167	C₂×C₈	Dic₆₈	1	4	yes	—
168	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
169	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
170	C₂×C₈	C₂×D₆₈	1	2	yes	—
171	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
172	C₄⋊C₄	—	34	2	no	—
173	C₄⋊C₄	—	34	2	no	—
174	C₄⋊C₄	—	34	4	no	—
175	C₄⋊C₄	—	68	1	no	—
176	C₄⋊C₄	—	68	1	no	—
177	C₄⋊C₄	—	68	2	no	—
178	C₄⋊C₄	—	68	4	no	—
179	C₄⋊C₄	—	68	4	no	—
180	C₄⋊C₄	—	68	4	no	—
181	C₄²	D₁₃₆	1	2	yes	—
182	C₄²	C₂×D₆₈	1	2	yes	—
183	C₁₆	C₂×Dic₃₄	1	8	yes	complemented
184	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
185	Q₈	—	136	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	68	2	no	—
189	C₂×C₄	—	68	2	no	—
190	C₂×C₄	—	68	4	no	—
191	C₂×C₄	—	136	4	no	—
192	C₂×C₄	—	136	2	no	—
193	C₂×C₄	—	136	2	no	—
194	C₂×C₄	#10587555	1	2	yes	—
195	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
196	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
197	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
199	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
200	C₈	#10587555	1	2	yes	—
201	C₈	C₂×Dic₆₈	1	4	yes	—
202	C₈	C₂²×Dic₃₄	1	2	yes	—
203	C₂²	#11763214	1	2	yes	—
204	C₂²	#11763247	1	2	yes	—
205	C₂²	#11763252	1	2	yes	—
206	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	136	4	no	—
210	C₄	—	136	2	no	—
211	C₄	—	136	2	no	—
212	C₄	#11763214	1	2	yes	—
213	C₄	C₂×#10587555	1	1	yes	characteristic
214	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
215	C₂	#14531070	1	4	yes	direct product factor
216	C₂	C₂×#11763214	1	1	yes	characteristic
217	C₂	C₂×#11763247	1	1	yes	characteristic
218	C₂	C₂×#11763252	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	C₂×#2255
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #14531070	C₂	direct (N × H)
37. C₂×C₁₆×C₁₇	Q₈	semidirect
73. C₁₆×C₁₇	C₂×Q₈	semidirect
128. C₂×C₁₇	#2255	semidirect
146. C₂×C₁₆	Dic₃₄	semidirect
158. C₁₇	C₂×#2255	semidirect
183. C₁₆	C₂×Dic₃₄	semidirect
215. C₂	#14531070	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
C₂×#11763211	2
#14531070	2
C₂×#11762637	2
C₂×#11762635	2
C₂×C₁₇×#346	2
C₂×#2255	17

Subgroups — C2×#14531070 #17189821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531070 #17189821