Subgroups — C₂×#14531013 #17189614

Order 4352.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763189	C₂	1	1	yes	maximal, characteristic
2	#14531013	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762612	C₂	1	4	yes	maximal
4	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	—	2	1	no	—
7	C₂²×Dic₆₈	C₂²	1	2	yes	—
8	C₂×#10587555	—	2	1	no	—
9	C₂²×#63214	C₂²	1	2	yes	—
10	#11763189	—	2	2	no	—
11	#11763189	C₂²	1	4	yes	—
12	#11762612	C₂²	1	16	yes	—
13	C₄×#63214	C₂²	1	4	yes	—
14	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
15	C₁₇×#500	—	2	2	no	—
16	C₁₇×#500	C₂²	1	4	yes	—
17	C₂×C₁₇×#107	—	2	1	no	—
18	C₂×C₁₇×#107	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	—	4	1	no	—
21	C₂²×Dic₃₄	—	2	2	no	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₁₇×C₄⋊Q₈	—	2	4	no	—
25	C₁₇×C₄⋊Q₈	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	4	1	no	—
30	C₂×C₄⋊Dic₁₇	—	4	1	no	—
31	C₂×Dic₆₈	—	2	4	no	—
32	C₂×Dic₆₈	—	2	2	no	—
33	C₂×Dic₆₈	—	2	8	no	—
34	C₂×Dic₆₈	—	4	4	no	—
35	C₂×Dic₆₈	C₂³	1	4	yes	—
36	#10587555	—	2	4	no	—
37	#10587555	—	4	2	no	—
38	C₁₇×#107	—	2	2	no	—
39	C₁₇×#107	—	2	4	no	—
40	C₁₇×#107	C₂³	1	4	yes	—
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	C₄×C₈×C₁₇	C₂³	1	2	yes	—
43	C₂×#63214	—	2	4	no	—
44	C₂×#63214	C₂³	1	4	yes	—
45	C₂×Q₈×C₁₇	—	4	4	no	—
46	C₂×Q₈×C₁₇	—	8	4	no	—
47	C₂²×C₄×C₁₇	—	4	2	no	—
48	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	—	4	2	no	—
52	C₂×Dic₃₄	—	2	4	no	—
53	C₂×Dic₃₄	—	4	4	no	—
54	C₂×Dic₃₄	—	4	4	no	—
55	C₂×Dic₃₄	—	4	2	no	—
56	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₇	#41	1	4	yes	—
60	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₁₇	—	2	4	no	—
62	C₄⋊C₄×C₁₇	—	4	4	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄²×C₁₇	D₈	1	2	yes	—
65	C₄²×C₁₇	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₁₇	—	4	4	no	—
67	C₄⋊Dic₁₇	—	4	4	no	—
68	Dic₆₈	—	2	8	no	—
69	Dic₆₈	—	4	4	no	—
70	Dic₆₈	—	4	8	no	—
71	#63214	—	2	4	no	—
72	C₂×#2241	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
74	Q₈×C₁₇	—	8	8	no	—
75	C₂×C₄×C₁₇	—	4	4	no	—
76	C₂×C₄×C₁₇	—	8	4	no	—
77	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
78	C₂×C₄×C₁₇	Q₃₂	1	4	yes	—
79	C₂×C₄×C₁₇	#124	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₁₇	—	4	4	no	—
86	C₂×Dic₁₇	—	8	4	no	—
87	Dic₃₄	—	4	4	no	—
88	Dic₃₄	—	4	4	no	—
89	Dic₃₄	—	8	4	no	—
90	C₈×C₁₇	#124	1	2	yes	—
91	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
92	C₈×C₁₇	C₂×#41	1	2	yes	—
93	C₂×#500	D₁₇	1	1	yes	characteristic
94	C₂×#494	—	17	1	no	—
95	#2241	—	17	8	no	—
96	C₂×#358	—	17	4	no	—
97	C₂×C₄×C₁₆	—	17	1	no	—
98	C₂²×C₁₇	#486	1	2	yes	—
99	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
100	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
101	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₁₇	—	8	4	no	—
103	C₄×C₁₇	#486	1	2	yes	—
104	C₄×C₁₇	C₂×SD₃₂	1	4	yes	—
105	C₄×C₁₇	C₂×Q₃₂	1	4	yes	—
106	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
107	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
108	Dic₁₇	—	8	4	no	—
109	C₂²×Q₁₆	—	17	2	no	—
110	C₂²×Q₁₆	—	34	1	no	—
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂×C₄⋊Q₈	—	34	1	no	—
113	C₂²×C₁₆	—	17	2	no	—
114	#500	—	2	2	no	—
115	#500	C₂×D₁₇	1	4	yes	—
116	#494	—	17	4	no	—
117	#494	—	34	2	no	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	C₂×D₁₇	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
121	#358	—	17	16	no	—
122	C₄×C₁₆	—	17	4	no	—
123	C₂×C₁₇	#2241	1	4	yes	complemented
124	C₂×C₁₇	C₂×#486	1	1	yes	characteristic
125	C₂×C₁₇	C₂²×SD₃₂	1	1	yes	characteristic
126	C₂×C₁₇	C₂²×Q₃₂	1	1	yes	characteristic
127	C₂²×Q₈	—	4	1	no	—
128	C₂²×Q₈	—	34	2	no	—
129	C₂×Q₁₆	—	17	4	no	—
130	C₂×Q₁₆	—	34	8	no	—
131	C₂×Q₁₆	—	34	2	no	—
132	C₂×Q₁₆	—	34	4	no	—
133	C₂×Q₁₆	—	68	4	no	—
134	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	C₄⋊Q₈	—	34	4	no	—
139	C₄⋊Q₈	—	68	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	68	1	no	—
143	C₂×C₄⋊C₄	—	68	1	no	—
144	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
145	C₂×C₁₆	—	17	4	no	—
146	C₂×C₁₆	—	34	4	no	—
147	#107	—	2	4	no	—
148	#107	—	2	2	no	—
149	#107	C₂²×D₁₇	1	4	yes	—
150	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
151	C₄×C₈	C₂²×D₁₇	1	2	yes	—
152	C₁₇	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×Q₈	—	34	4	no	—
156	C₂×Q₈	—	68	2	no	—
157	C₂×Q₈	—	68	4	no	—
158	C₂×Q₈	—	68	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	68	2	no	—
161	C₂²×C₄	#63228	1	1	yes	characteristic
162	C₂²×C₄	#63228	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
164	Q₁₆	—	34	8	no	—
165	Q₁₆	—	68	8	no	—
166	Q₁₆	—	68	4	no	—
167	C₂×C₈	#63230	1	4	yes	—
168	C₂×C₈	#63256	1	4	yes	—
169	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
170	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	68	4	no	—
176	C₄⋊C₄	—	68	4	no	—
177	C₄²	#63228	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
179	C₁₆	—	34	4	no	—
180	C₂³	C₂×#63228	1	1	yes	characteristic
181	Q₈	—	8	8	no	—
182	Q₈	—	68	4	no	—
183	Q₈	—	68	4	no	—
184	Q₈	—	136	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	68	4	no	—
188	C₂×C₄	—	136	4	no	—
189	C₂×C₄	#10587498	1	4	yes	—
190	C₂×C₄	#10587500	1	4	yes	—
191	C₂×C₄	C₂×#63228	1	2	yes	—
192	C₂×C₄	C₂×#63228	1	2	yes	—
193	C₂×C₄	C₂×#63228	1	1	yes	characteristic
194	C₂×C₄	C₂×#63230	1	2	yes	—
195	C₂×C₄	#10587633	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
197	C₈	C₂×#63230	1	4	yes	—
198	C₈	#10587633	1	2	yes	—
199	C₈	C₂×#63256	1	2	yes	—
200	C₂²	#11763157	1	2	yes	—
201	C₂²	C₂×#10587498	1	2	yes	—
202	C₂²	C₂×#10587500	1	2	yes	—
203	C₂²	C₂²×#63228	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	136	4	no	—
206	C₄	#11763157	1	2	yes	—
207	C₄	C₂×#10587498	1	4	yes	—
208	C₄	C₂×#10587500	1	4	yes	—
209	C₄	C₂×#10587633	1	1	yes	characteristic
210	C₄	C₂²×#63230	1	1	yes	characteristic
211	C₂	#14531013	1	4	yes	direct product factor
212	C₂	C₂×#11763157	1	1	yes	characteristic
213	C₂	C₂²×#10587498	1	1	yes	characteristic
214	C₂	C₂²×#10587500	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	C₂×#2241
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #14531013	C₂	direct (N × H)
123. C₂×C₁₇	#2241	semidirect
152. C₁₇	C₂×#2241	semidirect
211. C₂	#14531013	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×#11763189	2
#14531013	2
C₂×#11762612	2
C₂×C₄×#63214	2
C₂×C₁₇×#500	2
C₂×#2241	17

Subgroups — C2×#14531013 #17189614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531013 #17189614