Subgroups — C₂×#14531064 #17189808

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#14531064	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762636	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	C₂×#10587555	—	2	1	no	—
7	C₂²×#63214	C₂²	1	2	yes	—
8	#11763214	—	2	2	no	—
9	#11763214	C₂²	1	4	yes	—
10	C₂×#10587489	—	2	1	no	—
11	C₂×#10587489	C₂²	1	2	yes	—
12	#11762636	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	—	4	1	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	—	2	2	no	—
30	C₂×C₄⋊Dic₁₇	—	4	1	no	—
31	#10587555	—	2	4	no	—
32	#10587555	—	4	2	no	—
33	C₁₇×#107	—	2	2	no	—
34	C₁₇×#107	—	2	4	no	—
35	C₁₇×#107	C₂³	1	4	yes	—
36	C₄×C₈×C₁₇	D₄	1	2	yes	—
37	C₄×C₈×C₁₇	C₂³	1	2	yes	—
38	#10587489	—	2	2	no	—
39	#10587489	—	2	4	no	—
40	#10587489	C₂³	1	4	yes	—
41	C₂×#63214	Q₈	1	8	yes	complemented
42	C₂×#63214	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₇	—	4	4	no	—
44	C₂×Q₈×C₁₇	—	8	4	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	—	4	2	no	—
50	C₂×Dic₃₄	—	4	4	no	—
51	C₂×Dic₃₄	—	8	4	no	—
52	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
53	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
56	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₁₇	—	2	4	no	—
58	C₄⋊C₄×C₁₇	—	4	4	no	—
59	C₄⋊C₄×C₁₇	—	4	2	no	—
60	C₄²×C₁₇	D₈	1	2	yes	—
61	C₄²×C₁₇	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₁₇	—	2	4	no	—
63	C₄⋊Dic₁₇	—	4	4	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	#63214	C₂×Q₈	1	8	yes	complemented
66	C₂×#2254	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₇	—	8	8	no	—
69	C₂×C₄×C₁₇	—	4	4	no	—
70	C₂×C₄×C₁₇	—	8	4	no	—
71	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
72	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
73	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₁₇	—	4	4	no	—
80	C₂×Dic₁₇	—	8	4	no	—
81	Dic₃₄	—	8	8	no	—
82	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
83	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
84	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
85	C₂×#500	—	17	1	no	—
86	C₂×#500	D₁₇	1	1	yes	characteristic
87	#2254	—	17	8	no	—
88	C₂×#367	—	17	4	no	—
89	C₂×C₄×C₁₆	—	17	1	no	—
90	C₂²×C₁₇	#500	1	2	yes	—
91	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
92	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
93	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₁₇	—	8	4	no	—
95	C₄×C₁₇	#500	1	2	yes	—
96	C₄×C₁₇	C₂×SD₃₂	1	4	yes	—
97	C₄×C₁₇	C₂×SD₃₂	1	4	yes	—
98	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
100	Dic₁₇	—	8	4	no	—
101	C₂×C₄⋊Q₈	—	2	1	no	—
102	C₂×C₄⋊Q₈	—	34	1	no	—
103	C₂²×C₁₆	—	17	2	no	—
104	#500	—	2	2	no	—
105	#500	—	17	4	no	—
106	#500	—	34	2	no	—
107	#500	C₂×D₁₇	1	4	yes	—
108	C₂×#107	—	2	1	no	—
109	C₂×#107	—	17	2	no	—
110	C₂×#107	—	34	1	no	—
111	C₂×#107	C₂×D₁₇	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	#367	—	17	16	no	—
114	C₄×C₁₆	—	17	4	no	—
115	C₂×C₁₇	#2254	1	4	yes	complemented
116	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
117	C₂×C₁₇	C₂²×SD₃₂	1	1	yes	characteristic
118	C₂×C₁₇	C₂²×SD₃₂	1	1	yes	characteristic
119	C₂²×Q₈	—	4	1	no	—
120	C₂²×Q₈	—	68	1	no	—
121	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	2	no	—
125	C₄⋊Q₈	—	34	4	no	—
126	C₄⋊Q₈	—	68	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	34	2	no	—
130	C₂×C₄⋊C₄	—	68	1	no	—
131	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
132	C₂×C₁₆	—	17	8	no	—
133	C₂×C₁₆	—	17	4	no	—
134	#107	—	2	4	no	—
135	#107	—	2	2	no	—
136	#107	—	17	4	no	—
137	#107	—	34	2	no	—
138	#107	—	34	4	no	—
139	#107	C₂²×D₁₇	1	4	yes	—
140	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
141	C₄×C₈	C₂²×D₁₇	1	2	yes	—
142	C₁₇	C₂×#2254	1	1	yes	characteristic, complemented, Sylow
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	68	4	no	—
146	C₂×Q₈	—	136	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	68	2	no	—
149	C₂²×C₄	#63228	1	1	yes	characteristic
150	C₂²×C₄	#63228	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
152	C₂×C₈	#63231	1	4	yes	—
153	C₂×C₈	Q₈×D₁₇	1	4	yes	—
154	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
156	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	34	4	no	—
161	C₄⋊C₄	—	68	2	no	—
162	C₄⋊C₄	—	68	4	no	—
163	C₄²	#63228	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
165	C₁₆	—	17	8	no	—
166	C₂³	C₂×#63228	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	136	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	68	4	no	—
172	C₂×C₄	—	136	4	no	—
173	C₂×C₄	#10587498	1	4	yes	—
174	C₂×C₄	#10587499	1	4	yes	—
175	C₂×C₄	C₂×#63228	1	1	yes	characteristic
176	C₂×C₄	C₂×#63228	1	2	yes	—
177	C₂×C₄	C₂×#63228	1	2	yes	—
178	C₂×C₄	C₂×#63231	1	2	yes	—
179	C₂×C₄	#10587630	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
181	C₈	C₂×#63231	1	4	yes	—
182	C₈	#10587630	1	2	yes	—
183	C₈	C₂×Q₈×D₁₇	1	2	yes	—
184	C₂²	#11763212	1	2	yes	—
185	C₂²	C₂×#10587499	1	2	yes	—
186	C₂²	C₂×#10587498	1	2	yes	—
187	C₂²	C₂²×#63228	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	136	4	no	—
190	C₄	#11763212	1	2	yes	—
191	C₄	C₂×#10587499	1	4	yes	—
192	C₄	C₂×#10587498	1	4	yes	—
193	C₄	C₂×#10587630	1	1	yes	characteristic
194	C₄	C₂²×#63231	1	1	yes	characteristic
195	C₂	#14531064	1	4	yes	direct product factor
196	C₂	C₂×#11763212	1	1	yes	characteristic
197	C₂	C₂²×#10587499	1	1	yes	characteristic
198	C₂	C₂²×#10587498	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×#2254
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. #14531064	C₂	direct (N × H)
41. C₂×#63214	Q₈	semidirect
65. #63214	C₂×Q₈	semidirect
115. C₂×C₁₇	#2254	semidirect
142. C₁₇	C₂×#2254	semidirect
195. C₂	#14531064	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
#14531064	2
C₂×#11762636	2
C₂×C₄×#63214	2
C₂×C₁₇×#500	2
C₂×#2254	17

Subgroups — C2×#14531064 #17189808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14531064 #17189808