Subgroups — C₂×#22578413 #31797440

Order 17536.

512 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984167	C₂	1	2	yes	maximal
2	#22578413	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984157	C₂	1	2	yes	maximal, complemented
4	C₂×#17984156	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984167	—	2	4	no	—
7	#17984167	C₂²	1	8	yes	—
8	C₂×C₄⋊#10587739	C₂²	1	4	yes	—
9	C₂×C₂²⋊#10587739	—	2	2	no	—
10	C₂×C₄×#10587739	—	2	2	no	—
11	#17984157	C₂²	1	8	yes	complemented
12	#17984156	C₂²	1	8	yes	complemented
13	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×#11764040	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₁₃₇	C₄	1	4	yes	complemented
18	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃₇	C₂²	1	4	yes	—
21	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
22	C₂²×#10587739	—	2	4	no	—
23	C₄⋊#10587739	—	2	4	no	—
24	C₄⋊#10587739	C₂³	1	8	yes	—
25	C₂²⋊#10587739	—	2	8	no	—
26	C₄×#10587739	—	2	8	no	—
27	C₂³×D₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×Dic₁₃₇	Q₈	1	1	yes	characteristic
29	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
33	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
34	C₂×C₄×D₁₃₇	Q₈	1	1	yes	characteristic
35	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
37	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
38	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
39	C₂×#11764040	C₂×C₄	1	4	yes	complemented
40	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
41	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
42	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
43	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
44	C₈×D₁₃₇	C₂³	1	2	yes	—
45	C₈×D₁₃₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
50	C₂×#10587739	—	2	8	no	—
51	C₂×#10587739	—	4	8	no	—
52	C₂²×D₁₃₇	—	2	4	no	—
53	C₂²×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×D₁₃₇	C₄⋊C₄	1	2	yes	—
55	C₂²×D₁₃₇	C₂×D₄	1	2	yes	—
56	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
58	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₁₃₇	C₂×Q₈	1	2	yes	—
60	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
61	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
62	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
63	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
64	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
66	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
67	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
69	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
70	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
71	#11764040	C₂²×C₄	1	2	yes	—
72	#11764040	C₂²×C₄	1	2	yes	—
73	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₁₃₇	—	2	2	no	—
81	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×D₁₃₇	—	2	2	no	—
84	C₂×D₁₃₇	—	2	4	no	—
85	C₂×D₁₃₇	—	2	4	no	—
86	C₂×D₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×D₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₁₃₇	#135	1	4	yes	—
89	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
90	#10587739	—	4	8	no	—
91	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
94	Dic₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	Dic₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
96	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₁₃₇	—	2	2	no	—
98	C₂×C₁₃₇	#427	1	2	yes	complemented
99	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	D₁₃₇	—	2	2	no	—
101	D₁₃₇	#427	1	2	yes	complemented
102	D₁₃₇	C₂×#135	1	2	yes	—
103	C₁₃₇	C₂×#427	1	1	yes	characteristic, complemented, Sylow
104	C₂×#427	—	137	1	no	maximal, Sylow
105	C₂³×C₈	—	137	1	no	—
106	C₂×#117	—	137	2	no	—
107	#427	—	137	8	no	—
108	C₂×#107	—	137	2	no	—
109	C₂×#106	—	137	2	no	—
110	C₂³×C₄	—	137	1	no	—
111	C₂²×C₈	—	137	1	no	—
112	C₂²×C₈	—	137	1	no	—
113	C₂²×C₈	—	137	1	no	—
114	C₂²×C₈	—	137	1	no	—
115	C₂²×C₈	—	137	1	no	—
116	C₂²×C₈	—	137	4	no	—
117	C₂²×C₈	—	137	4	no	—
118	C₂²×C₈	#10587739	1	1	yes	characteristic, complemented
119	#117	—	274	4	no	—
120	#117	—	137	8	no	—
121	C₂×C₄⋊C₄	—	137	4	no	—
122	C₂×C₂²⋊C₄	—	274	2	no	—
123	C₂×C₄²	—	274	2	no	—
124	#107	—	137	8	no	—
125	#106	—	137	8	no	—
126	C₂⁴	—	137	1	no	—
127	C₂²×C₄	—	274	4	no	—
128	C₂²×C₄	—	137	1	no	—
129	C₂²×C₄	—	137	2	no	—
130	C₂²×C₄	—	137	2	no	—
131	C₂²×C₄	—	137	1	no	—
132	C₂²×C₄	—	137	1	no	—
133	C₂²×C₄	—	137	1	no	—
134	C₂²×C₄	—	137	2	no	—
135	C₂²×C₄	—	137	1	no	—
136	C₂²×C₄	—	137	2	no	—
137	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
138	C₂×C₈	—	137	1	no	—
139	C₂×C₈	—	137	1	no	—
140	C₂×C₈	—	137	4	no	—
141	C₂×C₈	—	137	2	no	—
142	C₂×C₈	—	137	2	no	—
143	C₂×C₈	—	137	4	no	—
144	C₂×C₈	—	137	4	no	—
145	C₂×C₈	—	137	2	no	—
146	C₂×C₈	—	137	2	no	—
147	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
148	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
149	C₂×C₈	C₂×#10587739	1	4	yes	complemented
150	C₄⋊C₄	—	274	4	no	—
151	C₄⋊C₄	—	137	8	no	—
152	C₂²⋊C₄	—	274	8	no	—
153	C₄²	—	274	8	no	—
154	C₂³	—	274	4	no	—
155	C₂³	—	137	2	no	—
156	C₂³	—	137	2	no	—
157	C₂³	—	137	1	no	—
158	C₂³	—	137	1	no	—
159	C₂³	C₄⋊#10587739	1	1	yes	characteristic
160	C₂×C₄	—	274	8	no	—
161	C₂×C₄	—	548	8	no	—
162	C₂×C₄	—	137	2	no	—
163	C₂×C₄	—	137	1	no	—
164	C₂×C₄	—	137	2	no	—
165	C₂×C₄	—	137	1	no	—
166	C₂×C₄	—	137	2	no	—
167	C₂×C₄	—	137	1	no	—
168	C₂×C₄	—	137	2	no	—
169	C₂×C₄	—	137	2	no	—
170	C₂×C₄	—	137	2	no	—
171	C₂×C₄	—	137	1	no	—
172	C₂×C₄	—	137	2	no	—
173	C₂×C₄	—	137	1	no	—
174	C₂×C₄	—	137	2	no	—
175	C₂×C₄	—	137	1	no	—
176	C₂×C₄	C₄⋊#10587739	1	2	yes	—
177	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
178	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
179	C₂×C₄	C₂²×#10587739	1	2	yes	—
180	C₈	—	137	2	no	—
181	C₈	—	137	2	no	—
182	C₈	C₂²×#10587739	1	2	yes	—
183	C₈	C₂²×#10587739	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	274	4	no	—
186	C₂²	—	274	4	no	—
187	C₂²	—	274	2	no	—
188	C₂²	—	137	2	no	—
189	C₂²	—	137	1	no	—
190	C₂²	—	137	1	no	—
191	C₂²	—	137	4	no	—
192	C₂²	C₂×C₄⋊#10587739	1	2	yes	—
193	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
194	C₄	—	548	8	no	—
195	C₄	—	137	2	no	—
196	C₄	—	137	1	no	—
197	C₄	—	137	1	no	—
198	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
199	C₄	C₂×C₄⋊#10587739	1	2	yes	—
200	C₄	C₂³×#10587739	1	1	yes	characteristic
201	C₂	—	2	2	no	—
202	C₂	—	274	2	no	—
203	C₂	—	137	2	no	—
204	C₂	—	137	2	no	—
205	C₂	#22578413	1	2	yes	direct product factor
206	C₂	C₂²×C₄⋊#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃₇	D₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	C₂×#427
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #22578413	C₂	direct (N × H)
3. C₂×#17984157	C₂	semidirect
4. C₂×#17984156	C₂	semidirect
11. #17984157	C₂²	semidirect
12. #17984156	C₂²	semidirect
14. C₂²×#11764040	C₄	semidirect
15. C₂×C₈×D₁₃₇	C₄	semidirect
16. C₂×C₈×D₁₃₇	C₄	semidirect
17. C₂×C₈×D₁₃₇	C₄	semidirect
21. C₂²×C₈×C₁₃₇	C₄	semidirect
39. C₂×#11764040	C₂×C₄	semidirect
42. C₈×D₁₃₇	C₂×C₄	semidirect
43. C₈×D₁₃₇	C₂×C₄	semidirect
49. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
98. C₂×C₁₃₇	#427	semidirect
101. D₁₃₇	#427	semidirect
103. C₁₃₇	C₂×#427	semidirect
118. C₂²×C₈	#10587739	semidirect
149. C₂×C₈	C₂×#10587739	semidirect
205. C₂	#22578413	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×#17984167	2
#22578413	2
C₂×#17984157	2
C₂×#17984156	2
C₂²×C₈×D₁₃₇	2
C₂×#427	137

Subgroups — C2×#22578413 #31797440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578413 #31797440