Subgroups — C₂×#14543901 #17962581

Order 4352.

294 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×#11763741	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10587467	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543901	C₂	1	4	yes	maximal, direct product factor
4	C₂×#11763226	C₂	1	1	yes	maximal, characteristic
5	#11763741	—	2	1	no	—
6	#11763741	C₂²	1	2	yes	complemented
7	C₂×#10587467	—	2	1	no	—
8	C₂×#10587467	—	2	1	no	—
9	C₂×#10587467	—	2	1	no	—
10	C₂×#10587467	C₂²	1	2	yes	complemented
11	#11763226	—	2	1	no	—
12	#11763226	—	2	1	no	—
13	#11763226	C₄	1	2	yes	—
14	#11763226	C₂²	1	2	yes	—
15	C₂×#10587483	—	2	1	no	—
16	C₂×#10587483	C₄	1	1	yes	characteristic
17	C₂²×#63214	—	2	1	no	—
18	C₂²×#63214	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×#131	C₄	1	1	yes	characteristic
20	C₂×#41×C₁₇	C₈	1	1	yes	characteristic
21	C₁₇×#131	—	2	1	no	—
22	C₁₇×#131	—	2	1	no	—
23	C₁₇×#131	C₈	1	2	yes	—
24	C₁₇×#131	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₇	—	2	1	no	—
26	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	—	2	1	no	—
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	#10587483	—	2	2	no	—
30	#10587483	—	2	2	no	—
31	#10587483	—	2	1	no	—
32	#10587483	C₂×C₄	1	2	yes	—
33	C₂×#63214	—	2	2	no	—
34	C₂×#63214	—	2	2	no	—
35	C₂×#63214	—	2	1	no	—
36	C₂×#63214	C₂×C₄	1	1	yes	characteristic
37	C₂×#63214	D₄	1	1	yes	characteristic
38	C₂×#63214	D₄	1	1	yes	characteristic
39	C₂×#63214	D₄	1	1	yes	characteristic
40	C₂×#63214	D₄	1	1	yes	characteristic
41	C₂×#63214	C₂³	1	1	yes	characteristic
42	#10587467	—	2	2	no	—
43	#10587467	—	2	2	no	—
44	#41×C₁₇	—	2	1	no	—
45	#41×C₁₇	—	2	1	no	—
46	#41×C₁₇	C₁₆	1	2	yes	—
47	#41×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×Q₈×C₁₇	C₁₆	1	1	yes	characteristic
49	C₂×D₄×C₁₇	—	2	1	no	—
50	C₂×D₄×C₁₇	C₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	—	2	1	no	—
52	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₁₇	—	2	1	no	—
54	M₁₆×C₁₇	—	2	2	no	—
55	M₁₆×C₁₇	—	2	2	no	—
56	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	2	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	#63214	—	2	2	no	—
67	#63214	C₂²⋊C₄	1	1	yes	characteristic
68	#63214	C₂²⋊C₄	1	1	yes	characteristic
69	#63214	C₂×D₄	1	1	yes	characteristic
70	#63214	C₂×D₄	1	1	yes	characteristic
71	C₂×#1401	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	—	2	1	no	—
73	C₂³×C₁₇	M₃₂	1	1	yes	characteristic
74	Q₈×C₁₇	—	2	1	no	—
75	Q₈×C₁₇	C₂×C₁₆	1	2	yes	—
76	D₄×C₁₇	—	2	2	no	—
77	D₄×C₁₇	—	2	2	no	—
78	D₄×C₁₇	—	2	1	no	—
79	D₄×C₁₇	C₂×C₁₆	1	2	yes	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	2	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₈×C₁₇	—	2	2	no	—
90	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
92	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×#504	—	17	1	no	—
95	C₂×M₆₄	—	17	1	no	—
96	C₂²×C₃₂	—	17	1	no	—
97	#1401	—	17	4	no	—
98	C₂²×C₁₇	—	2	2	no	—
99	C₂²×C₁₇	—	2	2	no	—
100	C₂²×C₁₇	—	4	2	no	—
101	C₂²×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₂²×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
106	C₄×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
107	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
109	C₂×#131	#595	1	1	yes	characteristic
110	#504	—	17	2	no	—
111	#504	—	17	2	no	—
112	#504	—	34	1	no	—
113	#504	—	34	1	no	—
114	C₂×M₃₂	—	17	1	no	—
115	C₂×M₃₂	—	34	1	no	—
116	C₂²×C₁₆	—	17	1	no	—
117	C₂²×C₁₆	—	34	1	no	—
118	M₆₄	—	17	2	no	—
119	M₆₄	—	34	1	no	—
120	C₂×C₃₂	—	17	2	no	—
121	C₂×C₃₂	—	34	1	no	—
122	C₂×C₃₂	—	34	1	no	—
123	C₂×C₃₂	—	34	1	no	—
124	C₂×C₁₇	—	2	1	no	—
125	C₂×C₁₇	—	2	1	no	—
126	C₂×C₁₇	—	4	2	no	—
127	C₂×C₁₇	#1401	1	2	yes	complemented
128	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
129	C₂×#41	#3624	1	1	yes	characteristic
130	#131	—	2	1	no	—
131	#131	—	2	1	no	—
132	#131	#3624	1	2	yes	—
133	#131	C₂×#595	1	2	yes	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	#3624	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₂×#595	1	1	yes	characteristic
138	M₃₂	—	17	2	no	—
139	M₃₂	—	34	2	no	—
140	M₃₂	—	34	2	no	—
141	M₃₂	—	34	1	no	—
142	C₂×C₁₆	—	17	1	no	—
143	C₂×C₁₆	—	17	1	no	—
144	C₂×C₁₆	—	17	1	no	—
145	C₂×C₁₆	—	17	1	no	—
146	C₂×C₁₆	—	17	1	no	—
147	C₂×C₁₆	—	17	1	no	—
148	C₂×C₁₆	—	34	2	no	—
149	C₂×C₁₆	—	34	2	no	—
150	C₂×C₁₆	—	34	1	no	—
151	C₃₂	—	34	2	no	—
152	C₃₂	—	34	2	no	—
153	C₁₇	C₂×#1401	1	1	yes	characteristic, complemented, Sylow
154	#41	—	2	1	no	—
155	#41	—	2	1	no	—
156	#41	#63216	1	2	yes	—
157	#41	C₂×#3624	1	2	yes	—
158	C₂×Q₈	#63216	1	1	yes	characteristic
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	#63216	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	2	2	no	—
166	M₁₆	C₂×#3624	1	2	yes	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂×#3624	1	1	yes	characteristic
171	C₂×C₈	#63247	1	1	yes	characteristic
172	C₂×C₈	#63247	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
175	C₂×C₈	C₂²×#595	1	1	yes	characteristic
176	C₁₆	—	17	1	no	—
177	C₁₆	—	17	1	no	—
178	C₁₆	—	17	1	no	—
179	C₁₆	—	17	1	no	—
180	C₁₆	—	34	2	no	—
181	C₂³	—	2	1	no	—
182	C₂³	#10587538	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×#63216	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×#63216	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂×#63216	1	1	yes	characteristic
193	C₂×C₄	C₂×#63216	1	1	yes	characteristic
194	C₂×C₄	#10587538	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
197	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊#3624	1	1	yes	characteristic
200	C₈	C₂²⋊#3624	1	1	yes	characteristic
201	C₈	C₂×#63247	1	1	yes	characteristic
202	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊#63216	1	1	yes	characteristic
207	C₂²	C₂²⋊#63216	1	1	yes	characteristic
208	C₂²	C₂×#10587538	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊#63216	1	1	yes	characteristic
211	C₄	C₂²⋊#63216	1	1	yes	characteristic
212	C₄	C₂×C₂²⋊#3624	1	1	yes	characteristic
213	C₄	C₂²×#63216	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	4	2	no	—
217	C₂	#14543901	1	2	yes	direct product factor
218	C₂	C₂×C₂²⋊#63216	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂×C₁₇×#131	C₄
socle	C₂²×C₁₇	C₂²⋊C₁₆
prime radical	C₂×#41×C₁₇	C₈
cosocle	C₂×#63214	C₂³
hypercenter	C₂×#131	#595
hypocenter	C₁₇	C₂×#1401
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#11763741	C₂	semidirect
2. C₂²×#10587467	C₂	semidirect
3. #14543901	C₂	direct (N × H)
6. #11763741	C₂²	semidirect
10. C₂×#10587467	C₂²	semidirect
127. C₂×C₁₇	#1401	semidirect
153. C₁₇	C₂×#1401	semidirect
217. C₂	#14543901	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×#11763741	2
C₂²×#10587467	2
#14543901	2
C₂×#11763226	2
C₂×#1401	17

Subgroups — C2×#14543901 #17962581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543901 #17962581