Subgroups — C₂×#22578366 #31797347

Order 17536.

592 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984167	C₂	1	1	yes	maximal, characteristic
3	#22578366	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	C₂²×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14555520	C₄	1	2	yes	complemented
8	D₁₃₇⋊M₁₆	C₄	1	4	yes	complemented
9	D₁₃₇⋊M₁₆	C₂²	1	4	yes	—
10	C₂×#14555519	C₄	1	2	yes	complemented
11	C₂×#14555518	C₂²	1	2	yes	—
12	#17984167	C₄	1	4	yes	—
13	#17984167	C₂²	1	4	yes	—
14	C₂×C₄⋊#10587739	C₄	1	2	yes	—
15	C₂×C₂²⋊#10587739	C₄	1	2	yes	—
16	C₂×C₄×#10587739	C₂²	1	2	yes	—
17	#17984142	C₂²	1	8	yes	complemented
18	C₈×#10587739	C₂²	1	8	yes	complemented
19	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
20	C₂²×#11764040	C₄	1	1	yes	characteristic, complemented
21	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
22	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
23	C₂×C₈×D₁₃₇	C₄	1	4	yes	complemented
24	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₃₇	C₂²	1	4	yes	—
27	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
28	C₂²×#10587739	C₂×C₄	1	4	yes	—
29	#14555520	C₂×C₄	1	8	yes	complemented
30	#14555519	C₂×C₄	1	8	yes	complemented
31	#14555518	C₂×C₄	1	4	yes	—
32	#14555518	C₂³	1	4	yes	—
33	C₂×#11764047	C₂×C₄	1	4	yes	—
34	C₄⋊#10587739	C₂×C₄	1	8	yes	—
35	C₂²⋊#10587739	C₂×C₄	1	8	yes	—
36	C₄×#10587739	C₂×C₄	1	4	yes	—
37	C₄×#10587739	C₂³	1	4	yes	—
38	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
48	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
49	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
50	C₂×#11764040	C₂×C₄	1	4	yes	complemented
51	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
52	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
53	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
54	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
55	C₈×D₁₃₇	C₂³	1	2	yes	—
56	C₈×D₁₃₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
61	C₂×#10587739	—	2	8	no	—
62	C₂×#10587739	C₂²×C₄	1	8	yes	—
63	#11764047	C₂²×C₄	1	8	yes	—
64	C₂²×D₁₃₇	—	2	4	no	—
65	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
66	C₂²×D₁₃₇	C₄²	1	2	yes	—
67	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₁₃₇	C₄²	1	2	yes	—
70	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
71	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
73	C₄×D₁₃₇	C₄²	1	2	yes	—
74	C₄×D₁₃₇	C₄²	1	2	yes	—
75	C₄×D₁₃₇	C₄²	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	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
82	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
83	#11764040	C₂²×C₄	1	2	yes	—
84	#11764040	C₂²×C₄	1	2	yes	—
85	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
90	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
91	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₁₃₇	—	2	2	no	—
93	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
94	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×D₁₃₇	—	2	2	no	—
96	C₂×D₁₃₇	—	2	4	no	—
97	C₂×D₁₃₇	—	2	4	no	—
98	C₂×D₁₃₇	C₂×C₄²	1	2	yes	—
99	C₂×D₁₃₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×D₁₃₇	#131	1	4	yes	—
101	C₂×D₁₃₇	C₂³×C₄	1	1	yes	characteristic
102	#10587739	—	2	8	no	—
103	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
104	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
105	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
106	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
107	Dic₁₃₇	C₂×C₄²	1	2	yes	—
108	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₁₃₇	—	2	2	no	—
110	C₂×C₁₃₇	#405	1	2	yes	complemented
111	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
112	D₁₃₇	—	2	2	no	—
113	D₁₃₇	#405	1	2	yes	complemented
114	D₁₃₇	C₂×#131	1	2	yes	—
115	C₁₃₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
116	C₂×#405	—	137	1	no	maximal, Sylow
117	C₂²×M₁₆	—	137	1	no	—
118	C₂³×C₈	—	137	1	no	—
119	C₂×#117	—	137	1	no	—
120	#405	—	137	8	no	—
121	C₂×#97	—	137	2	no	—
122	C₂×C₄×C₈	—	137	2	no	—
123	C₂³×C₄	—	137	1	no	—
124	C₂×M₁₆	—	137	4	no	—
125	C₂×M₁₆	—	137	2	no	—
126	C₂×M₁₆	—	137	4	no	—
127	C₂×M₁₆	—	137	2	no	—
128	C₂²×C₈	—	137	2	no	—
129	C₂²×C₈	—	137	1	no	—
130	C₂²×C₈	—	137	1	no	—
131	C₂²×C₈	—	137	1	no	—
132	C₂²×C₈	—	137	1	no	—
133	C₂²×C₈	—	137	1	no	—
134	C₂²×C₈	—	137	4	no	—
135	C₂²×C₈	—	137	4	no	—
136	C₂²×C₈	#10587739	1	1	yes	characteristic, complemented
137	#117	—	137	4	no	—
138	#117	—	137	4	no	—
139	C₂×C₄⋊C₄	—	137	2	no	—
140	C₂×C₂²⋊C₄	—	137	2	no	—
141	C₂×C₄²	—	137	2	no	—
142	#97	—	137	8	no	—
143	C₄×C₈	—	137	8	no	—
144	C₂⁴	—	137	1	no	—
145	C₂²×C₄	—	137	4	no	—
146	C₂²×C₄	—	137	1	no	—
147	C₂²×C₄	—	137	1	no	—
148	C₂²×C₄	—	137	2	no	—
149	C₂²×C₄	—	137	1	no	—
150	C₂²×C₄	—	137	2	no	—
151	C₂²×C₄	—	137	1	no	—
152	C₂²×C₄	—	137	2	no	—
153	C₂²×C₄	—	137	1	no	—
154	C₂²×C₄	—	137	2	no	—
155	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
156	M₁₆	—	137	8	no	—
157	M₁₆	—	137	8	no	—
158	C₂×C₈	—	137	4	no	—
159	C₂×C₈	—	137	4	no	—
160	C₂×C₈	—	137	4	no	—
161	C₂×C₈	—	137	1	no	—
162	C₂×C₈	—	137	1	no	—
163	C₂×C₈	—	137	4	no	—
164	C₂×C₈	—	137	2	no	—
165	C₂×C₈	—	137	2	no	—
166	C₂×C₈	—	137	2	no	—
167	C₂×C₈	—	137	2	no	—
168	C₂×C₈	—	137	4	no	—
169	C₂×C₈	—	137	4	no	—
170	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
171	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
172	C₂×C₈	C₂×#10587739	1	4	yes	complemented
173	C₄⋊C₄	—	137	8	no	—
174	C₂²⋊C₄	—	137	8	no	—
175	C₄²	—	137	4	no	—
176	C₄²	—	137	4	no	—
177	C₂³	—	274	4	no	—
178	C₂³	—	137	2	no	—
179	C₂³	—	137	2	no	—
180	C₂³	—	137	1	no	—
181	C₂³	—	137	1	no	—
182	C₂³	C₄×#10587739	1	1	yes	characteristic
183	C₂×C₄	—	274	8	no	—
184	C₂×C₄	—	137	8	no	—
185	C₂×C₄	—	137	1	no	—
186	C₂×C₄	—	137	2	no	—
187	C₂×C₄	—	137	2	no	—
188	C₂×C₄	—	137	1	no	—
189	C₂×C₄	—	137	1	no	—
190	C₂×C₄	—	137	2	no	—
191	C₂×C₄	—	137	1	no	—
192	C₂×C₄	—	137	2	no	—
193	C₂×C₄	—	137	2	no	—
194	C₂×C₄	—	137	2	no	—
195	C₂×C₄	—	137	2	no	—
196	C₂×C₄	—	137	1	no	—
197	C₂×C₄	—	137	2	no	—
198	C₂×C₄	—	137	1	no	—
199	C₂×C₄	C₄×#10587739	1	2	yes	—
200	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
201	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
202	C₂×C₄	C₂²×#10587739	1	2	yes	—
203	C₈	—	137	8	no	—
204	C₈	—	137	2	no	—
205	C₈	—	137	2	no	—
206	C₈	C₂²×#10587739	1	2	yes	—
207	C₈	C₂²×#10587739	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	274	4	no	—
210	C₂²	—	274	4	no	—
211	C₂²	—	274	2	no	—
212	C₂²	—	137	2	no	—
213	C₂²	—	137	1	no	—
214	C₂²	—	137	1	no	—
215	C₂²	—	137	4	no	—
216	C₂²	C₂×C₄×#10587739	1	2	yes	—
217	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
218	C₄	—	274	8	no	—
219	C₄	—	137	1	no	—
220	C₄	—	137	1	no	—
221	C₄	—	137	2	no	—
222	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
223	C₄	C₂×C₄×#10587739	1	2	yes	—
224	C₄	C₂³×#10587739	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	274	2	no	—
227	C₂	—	137	2	no	—
228	C₂	—	137	2	no	—
229	C₂	#22578366	1	2	yes	direct product factor
230	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#10587739
commutator	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄²
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	C₂×#405
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #22578366	C₂	direct (N × H)
4. C₂×#17984142	C₂	semidirect
5. C₂×C₈×#10587739	C₂	semidirect
7. C₂×#14555520	C₄	semidirect
8. D₁₃₇⋊M₁₆	C₄	semidirect
10. C₂×#14555519	C₄	semidirect
17. #17984142	C₂²	semidirect
18. C₈×#10587739	C₂²	semidirect
20. C₂²×#11764040	C₄	semidirect
21. C₂×C₈×D₁₃₇	C₄	semidirect
22. C₂×C₈×D₁₃₇	C₄	semidirect
23. C₂×C₈×D₁₃₇	C₄	semidirect
27. C₂²×C₈×C₁₃₇	C₄	semidirect
29. #14555520	C₂×C₄	semidirect
30. #14555519	C₂×C₄	semidirect
50. C₂×#11764040	C₂×C₄	semidirect
53. C₈×D₁₃₇	C₂×C₄	semidirect
54. C₈×D₁₃₇	C₂×C₄	semidirect
60. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
110. C₂×C₁₃₇	#405	semidirect
113. D₁₃₇	#405	semidirect
115. C₁₃₇	C₂×#405	semidirect
136. C₂²×C₈	#10587739	semidirect
172. C₂×C₈	C₂×#10587739	semidirect
229. C₂	#22578366	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×D₁₃₇⋊M₁₆	2
C₂×#17984167	2
#22578366	2
C₂×#17984142	2
C₂×C₈×#10587739	2
C₂²×C₈×D₁₃₇	2
C₂×#405	137

Subgroups — C2×#22578366 #31797347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578366 #31797347