Subgroups — C₂×#22578369 #31797351

Order 17536.

592 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555518	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984167	C₂	1	1	yes	maximal, characteristic
3	#22578369	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984142	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#10587739	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14555518	C₄	1	2	yes	complemented
8	C₂×#14555518	C₄	1	4	yes	complemented
9	C₂×#14555518	C₂²	1	1	yes	characteristic
10	C₂×#14555518	C₂²	1	1	yes	characteristic
11	C₂×#14555518	C₂²	1	4	yes	—
12	C₂²×#11764047	C₄	1	2	yes	complemented
13	#17984167	C₄	1	4	yes	—
14	#17984167	C₂²	1	4	yes	—
15	C₂×C₄⋊#10587739	C₄	1	2	yes	—
16	C₂×C₂²⋊#10587739	C₄	1	2	yes	—
17	C₂×C₄×#10587739	C₂²	1	2	yes	—
18	#17984142	C₂²	1	8	yes	complemented
19	C₈×#10587739	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#14555494	C₄	1	1	yes	characteristic, complemented
22	C₂×#14555493	C₄	1	2	yes	complemented
23	M₁₆×D₁₃₇	C₄	1	4	yes	complemented
24	M₁₆×D₁₃₇	C₂²	1	4	yes	—
25	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
26	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
27	C₂²×#10587739	C₂×C₄	1	4	yes	—
28	#14555518	C₂×C₄	1	2	yes	—
29	#14555518	C₂×C₄	1	2	yes	—
30	#14555518	C₂×C₄	1	8	yes	complemented
31	#14555518	C₂³	1	2	yes	—
32	#14555518	C₂³	1	2	yes	—
33	C₂×#11764047	C₂×C₄	1	2	yes	—
34	C₂×#11764047	C₂×C₄	1	2	yes	—
35	C₂×#11764047	C₂×C₄	1	8	yes	complemented
36	C₄⋊#10587739	C₂×C₄	1	8	yes	—
37	C₂²⋊#10587739	C₂×C₄	1	8	yes	—
38	C₄×#10587739	C₂×C₄	1	4	yes	—
39	C₄×#10587739	C₂³	1	4	yes	—
40	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
50	#14555494	C₂×C₄	1	4	yes	complemented
51	#14555493	C₂×C₄	1	8	yes	complemented
52	C₂×#11764040	C₂×C₄	1	2	yes	—
53	C₈×D₁₃₇	C₂×C₄	1	4	yes	—
54	C₈×D₁₃₇	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
57	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
58	C₂×#10587739	—	2	8	no	—
59	C₂×#10587739	C₂²×C₄	1	8	yes	—
60	#11764047	C₂²×C₄	1	4	yes	—
61	#11764047	C₂²×C₄	1	4	yes	—
62	C₂²×D₁₃₇	—	2	4	no	—
63	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
64	C₂²×D₁₃₇	C₄²	1	2	yes	—
65	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
66	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×Dic₁₃₇	C₄²	1	2	yes	—
68	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
69	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
71	C₄×D₁₃₇	C₄²	1	2	yes	—
72	C₄×D₁₃₇	C₄²	1	2	yes	—
73	C₄×D₁₃₇	C₄²	1	2	yes	—
74	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
76	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
78	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
80	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
81	#11764040	C₂²×C₄	1	4	yes	—
82	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
84	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
88	C₂²×C₁₃₇	—	2	2	no	—
89	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
90	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×D₁₃₇	—	2	2	no	—
92	C₂×D₁₃₇	—	2	4	no	—
93	C₂×D₁₃₇	—	2	4	no	—
94	C₂×D₁₃₇	C₂×C₄²	1	2	yes	—
95	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×D₁₃₇	#131	1	4	yes	—
97	C₂×D₁₃₇	C₂³×C₄	1	1	yes	characteristic
98	#10587739	—	2	8	no	—
99	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
100	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
101	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
102	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
103	Dic₁₃₇	C₂×C₄²	1	2	yes	—
104	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₁₃₇	—	2	2	no	—
106	C₂×C₁₃₇	#405	1	2	yes	complemented
107	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
108	D₁₃₇	—	2	2	no	—
109	D₁₃₇	#405	1	2	yes	complemented
110	D₁₃₇	C₂×#131	1	2	yes	—
111	C₁₃₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
112	C₂×#405	—	137	1	no	maximal, Sylow
113	C₂²×M₁₆	—	137	1	no	—
114	C₂³×C₈	—	137	1	no	—
115	C₂×#117	—	137	1	no	—
116	#405	—	137	8	no	—
117	C₂×#97	—	137	2	no	—
118	C₂×C₄×C₈	—	137	2	no	—
119	C₂³×C₄	—	137	1	no	—
120	C₂×M₁₆	—	137	1	no	—
121	C₂×M₁₆	—	137	4	no	—
122	C₂×M₁₆	—	137	4	no	—
123	C₂×M₁₆	—	137	2	no	—
124	C₂×M₁₆	#10587739	1	1	yes	characteristic, complemented
125	C₂²×C₈	—	137	1	no	—
126	C₂²×C₈	—	137	1	no	—
127	C₂²×C₈	—	137	4	no	—
128	C₂²×C₈	—	137	2	no	—
129	C₂²×C₈	—	137	2	no	—
130	C₂²×C₈	—	137	4	no	—
131	C₂²×C₈	—	137	2	no	—
132	#117	—	137	4	no	—
133	#117	—	137	4	no	—
134	C₂×C₄⋊C₄	—	137	2	no	—
135	C₂×C₂²⋊C₄	—	137	2	no	—
136	C₂×C₄²	—	137	2	no	—
137	#97	—	137	8	no	—
138	C₄×C₈	—	137	8	no	—
139	C₂⁴	—	137	1	no	—
140	C₂²×C₄	—	137	4	no	—
141	C₂²×C₄	—	137	1	no	—
142	C₂²×C₄	—	137	1	no	—
143	C₂²×C₄	—	137	2	no	—
144	C₂²×C₄	—	137	2	no	—
145	C₂²×C₄	—	137	1	no	—
146	C₂²×C₄	—	137	2	no	—
147	C₂²×C₄	—	137	2	no	—
148	C₂²×C₄	—	137	1	no	—
149	C₂²×C₄	—	137	1	no	—
150	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
151	M₁₆	—	137	4	no	—
152	M₁₆	—	137	8	no	—
153	M₁₆	C₂×#10587739	1	4	yes	complemented
154	C₂×C₈	—	137	2	no	—
155	C₂×C₈	—	137	2	no	—
156	C₂×C₈	—	137	2	no	—
157	C₂×C₈	—	137	2	no	—
158	C₂×C₈	—	137	8	no	—
159	C₂×C₈	—	137	2	no	—
160	C₂×C₈	—	137	2	no	—
161	C₂×C₈	—	137	8	no	—
162	C₂×C₈	—	137	2	no	—
163	C₂×C₈	—	137	4	no	—
164	C₂×C₈	—	137	4	no	—
165	C₂×C₈	C₂×#10587739	1	2	yes	—
166	C₄⋊C₄	—	137	8	no	—
167	C₂²⋊C₄	—	137	8	no	—
168	C₄²	—	137	4	no	—
169	C₄²	—	137	4	no	—
170	C₂³	—	274	4	no	—
171	C₂³	—	137	2	no	—
172	C₂³	—	137	2	no	—
173	C₂³	—	137	1	no	—
174	C₂³	—	137	1	no	—
175	C₂³	C₄×#10587739	1	1	yes	characteristic
176	C₂×C₄	—	274	8	no	—
177	C₂×C₄	—	137	8	no	—
178	C₂×C₄	—	137	2	no	—
179	C₂×C₄	—	137	1	no	—
180	C₂×C₄	—	137	2	no	—
181	C₂×C₄	—	137	1	no	—
182	C₂×C₄	—	137	1	no	—
183	C₂×C₄	—	137	2	no	—
184	C₂×C₄	—	137	2	no	—
185	C₂×C₄	—	137	1	no	—
186	C₂×C₄	—	137	2	no	—
187	C₂×C₄	—	137	2	no	—
188	C₂×C₄	—	137	2	no	—
189	C₂×C₄	—	137	1	no	—
190	C₂×C₄	—	137	1	no	—
191	C₂×C₄	—	137	2	no	—
192	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
193	C₂×C₄	C₄×#10587739	1	2	yes	—
194	C₂×C₄	C₂²×#10587739	1	2	yes	—
195	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
196	C₈	—	137	4	no	—
197	C₈	—	137	4	no	—
198	C₈	—	137	4	no	—
199	C₈	C₂²×#10587739	1	4	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	274	4	no	—
202	C₂²	—	274	4	no	—
203	C₂²	—	274	2	no	—
204	C₂²	—	137	2	no	—
205	C₂²	—	137	1	no	—
206	C₂²	—	137	1	no	—
207	C₂²	—	137	4	no	—
208	C₂²	C₂×C₄×#10587739	1	2	yes	—
209	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
210	C₄	—	274	8	no	—
211	C₄	—	137	1	no	—
212	C₄	—	137	1	no	—
213	C₄	—	137	2	no	—
214	C₄	C₂×C₄×#10587739	1	2	yes	—
215	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
216	C₄	C₂³×#10587739	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	274	2	no	—
219	C₂	—	137	2	no	—
220	C₂	—	137	2	no	—
221	C₂	#22578369	1	2	yes	direct product factor
222	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	C₂×#405
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22578369	C₂	direct (N × H)
4. C₂×#17984142	C₂	semidirect
5. C₂×C₈×#10587739	C₂	semidirect
7. C₂×#14555518	C₄	semidirect
8. C₂×#14555518	C₄	semidirect
12. C₂²×#11764047	C₄	semidirect
18. #17984142	C₂²	semidirect
19. C₈×#10587739	C₂²	semidirect
21. C₂×#14555494	C₄	semidirect
22. C₂×#14555493	C₄	semidirect
23. M₁₆×D₁₃₇	C₄	semidirect
26. C₂×M₁₆×C₁₃₇	C₄	semidirect
30. #14555518	C₂×C₄	semidirect
35. C₂×#11764047	C₂×C₄	semidirect
50. #14555494	C₂×C₄	semidirect
51. #14555493	C₂×C₄	semidirect
56. M₁₆×C₁₃₇	C₂×C₄	semidirect
106. C₂×C₁₃₇	#405	semidirect
109. D₁₃₇	#405	semidirect
111. C₁₃₇	C₂×#405	semidirect
124. C₂×M₁₆	#10587739	semidirect
153. M₁₆	C₂×#10587739	semidirect
221. C₂	#22578369	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×#14555518	2
C₂×#17984167	2
#22578369	2
C₂×#17984142	2
C₂×C₈×#10587739	2
C₂×M₁₆×D₁₃₇	2
C₂×#405	137

Subgroups — C2×#22578369 #31797351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578369 #31797351