Subgroups — C₂×#14530579 #17187380

Order 4352.

642 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	C₂²×#10587490	C₂	1	2	yes	maximal
2	#14530579	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11762639	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	—	2	2	no	—
6	C₂×#10587490	—	2	4	no	—
7	C₂×#10587490	—	2	2	no	—
8	C₂×#10587490	C₂²	1	4	yes	—
9	C₂×#10587490	C₂²	1	8	yes	—
10	#11762639	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×M₃₂	C₄	1	4	yes	—
13	C₂×C₁₇×M₃₂	C₄	1	4	yes	—
14	C₂×C₁₇×M₃₂	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
25	C₂×#63223	—	2	4	no	—
26	C₂×#63223	—	2	8	no	—
27	C₂×#63223	—	4	2	no	—
28	C₂×#63223	—	4	4	no	—
29	C₂²×#3622	—	4	2	no	—
30	C₁₇×M₃₂	C₂×C₄	1	16	yes	—
31	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
34	#10587490	—	2	4	no	—
35	#10587490	—	2	8	no	—
36	#10587490	C₂³	1	8	yes	—
37	C₂⁴×C₁₇	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₇	D₈	1	2	yes	—
41	C₂²×C₄×C₁₇	D₈	1	2	yes	—
42	C₂²×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂²×C₄	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	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
59	#63223	—	2	8	no	—
60	#63223	—	4	8	no	—
61	#63223	—	4	4	no	—
62	C₂×#3622	—	4	8	no	—
63	C₂×#3622	—	4	4	no	—
64	C₁₆×C₁₇	C₂²×C₄	1	8	yes	—
65	C₂×#2161	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	—	4	2	no	—
67	C₂³×C₁₇	#107	1	2	yes	—
68	C₂³×C₁₇	#107	1	2	yes	—
69	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂³×C₁₇	C₂×Q₁₆	1	2	yes	—
71	C₂×C₄×C₁₇	#107	1	4	yes	—
72	C₂×C₄×C₁₇	#107	1	2	yes	—
73	C₂×C₄×C₁₇	#107	1	4	yes	—
74	C₂×C₄×C₁₇	#107	1	2	yes	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
84	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₈×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
89	#3622	—	4	8	no	—
90	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
91	C₂²×#108	—	17	2	no	—
92	#2161	—	17	8	no	—
93	C₂×#368	—	17	4	no	—
94	C₂²×C₁₇	—	2	4	no	—
95	C₂²×C₁₇	—	2	2	no	—
96	C₂²×C₁₇	—	4	2	no	—
97	C₂²×C₁₇	—	4	2	no	—
98	C₂²×C₁₇	C₂×#107	1	4	yes	—
99	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
100	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
101	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
102	C₄×C₁₇	C₂×#107	1	4	yes	—
103	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
105	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
107	C₂²×M₁₆	—	34	2	no	—
108	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
109	C₂×M₃₂	Dic₁₇	1	4	yes	—
110	C₂×M₃₂	Dic₁₇	1	4	yes	—
111	C₂×M₃₂	C₂×D₁₇	1	4	yes	—
112	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
113	C₂×#108	—	17	4	no	—
114	C₂×#108	—	17	8	no	—
115	C₂×#108	—	34	4	no	—
116	C₂×#108	—	34	2	no	—
117	#368	—	17	16	no	—
118	C₂×C₁₇	—	2	1	no	—
119	C₂×C₁₇	—	2	4	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	#2161	1	2	yes	complemented
122	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
123	C₂³×C₄	D₆₈	1	1	yes	characteristic
124	C₂×M₁₆	—	34	8	no	—
125	C₂×M₁₆	—	34	4	no	—
126	C₂×M₁₆	—	68	4	no	—
127	C₂×M₁₆	—	68	2	no	—
128	C₂²×C₈	—	68	2	no	—
129	C₂²×C₈	Dic₃₄	1	2	yes	—
130	C₂²×C₈	Dic₃₄	1	2	yes	—
131	C₂²×C₈	D₆₈	1	2	yes	—
132	C₂²×C₈	D₆₈	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
135	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
137	M₃₂	C₂×Dic₁₇	1	16	yes	—
138	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
139	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
140	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
141	#108	—	17	8	no	—
142	#108	—	34	4	no	—
143	#108	—	34	8	no	—
144	C₁₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	Dic₆₈	1	1	yes	characteristic
146	C₂²×C₄	D₁₃₆	1	2	yes	—
147	C₂²×C₄	D₁₃₆	1	2	yes	—
148	C₂²×C₄	Dic₆₈	1	1	yes	characteristic
149	C₂²×C₄	Dic₆₈	1	2	yes	—
150	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
151	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
152	C₂²×C₄	C₂×D₆₈	1	2	yes	—
153	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
154	M₁₆	—	34	8	no	—
155	M₁₆	—	68	8	no	—
156	M₁₆	—	68	4	no	—
157	C₂×C₈	—	68	8	no	—
158	C₂×C₈	—	68	4	no	—
159	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
160	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
161	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
163	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
166	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
167	C₂×C₈	C₂×D₆₈	1	2	yes	—
168	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
169	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
171	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
172	C₁₆	C₂²×Dic₁₇	1	8	yes	—
173	C₂³	—	4	2	no	—
174	C₂³	#10587489	1	2	yes	—
175	C₂³	#10587489	1	2	yes	—
176	C₂³	C₂×Dic₆₈	1	1	yes	characteristic
177	C₂³	C₂×Dic₆₈	1	2	yes	—
178	C₂×C₄	#10587489	1	2	yes	—
179	C₂×C₄	#10587489	1	4	yes	—
180	C₂×C₄	#10587489	1	2	yes	—
181	C₂×C₄	#10587489	1	4	yes	—
182	C₂×C₄	C₂×D₁₃₆	1	4	yes	—
183	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
184	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
186	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄⋊Dic₁₇	1	4	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
191	C₈	—	68	8	no	—
192	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
193	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
194	C₈	C₂×C₄⋊Dic₁₇	1	4	yes	—
195	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
196	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×#10587489	1	4	yes	—
202	C₂²	C₂×#10587489	1	1	yes	characteristic
203	C₂²	C₂×#10587489	1	1	yes	characteristic
204	C₂²	C₂²×Dic₆₈	1	1	yes	characteristic
205	C₄	C₂×#10587489	1	4	yes	—
206	C₄	C₂×#10587489	1	1	yes	characteristic
207	C₄	C₂×#10587489	1	1	yes	characteristic
208	C₄	C₂²×C₄⋊Dic₁₇	1	1	yes	characteristic
209	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	4	no	—
212	C₂	—	2	1	no	—
213	C₂	#14530579	1	2	yes	direct product factor
214	C₂	C₂²×#10587489	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂⁴×C₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	C₂×#2161
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #14530579	C₂	direct (N × H)
3. C₂×#11762639	C₂	semidirect
10. #11762639	C₂²	semidirect
121. C₂×C₁₇	#2161	semidirect
144. C₁₇	C₂×#2161	semidirect
213. C₂	#14530579	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×#10587490	2
#14530579	2
C₂×#11762639	2
C₂²×C₁₇×M₃₂	2
C₂×#2161	17

Subgroups — C2×#14530579 #17187380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530579 #17187380