Subgroups — C₂×#14558102 #18007907

Order 4736.

1896 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	C₂²×#10596962	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558102	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11924533	C₂	1	2	yes	maximal, complemented
5	C₂×#11924524	C₂	1	2	yes	maximal, complemented
6	C₂×#11924511	C₂	1	4	yes	maximal, complemented
7	C₂×#11924506	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₃₇	C₂²	1	2	yes	—
10	C₂×#10596962	—	2	8	no	—
11	C₂×#10596962	C₂²	1	8	yes	complemented
12	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₁₄₈	—	2	8	no	—
14	C₂²×D₁₄₈	C₂²	1	2	yes	—
15	C₂²×D₁₄₈	C₂²	1	2	yes	—
16	C₂²×D₁₄₈	C₂²	1	8	yes	complemented
17	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₁₄₈	C₂²	1	8	yes	complemented
19	C₂²×C₂²⋊D₃₇	C₂²	1	2	yes	—
20	C₂²×C₄×D₃₇	C₂²	1	2	yes	complemented
21	#11924533	C₂²	1	16	yes	complemented
22	#11924524	C₂²	1	16	yes	complemented
23	#11924511	C₂²	1	32	yes	complemented
24	#11924506	C₂²	1	32	yes	complemented
25	C₂×#10596938	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
27	#117×C₃₇	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₃₇	C₂²	1	2	yes	complemented
31	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
32	C₂³×D₃₇	—	2	8	no	—
33	C₂³×D₃₇	—	4	8	no	—
34	C₂³×D₃₇	C₂³	1	4	yes	—
35	C₂³×D₃₇	C₂³	1	2	yes	—
36	C₂³×D₃₇	C₂³	1	8	yes	—
37	C₂²×C₄×C₃₇	D₄	1	8	yes	complemented
38	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
39	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₃₇	—	2	8	no	—
43	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
44	C₂²×Dic₃₇	C₂³	1	2	yes	—
45	#10596962	—	2	32	no	—
46	C₂×D₁₄₈	—	2	8	no	—
47	C₂×D₁₄₈	—	2	8	no	—
48	C₂×D₁₄₈	—	2	4	no	—
49	C₂×D₁₄₈	—	2	32	no	—
50	C₂×D₁₄₈	—	4	16	no	—
51	C₂×D₁₄₈	C₂³	1	8	yes	—
52	C₂×D₁₄₈	C₂³	1	8	yes	—
53	C₂×D₁₄₈	C₂³	1	4	yes	complemented
54	C₂×D₁₄₈	C₂³	1	32	yes	complemented
55	C₂×C₄×D₃₇	—	2	8	no	—
56	C₂×C₄×D₃₇	C₂³	1	8	yes	complemented
57	C₂×Dic₇₄	—	2	4	no	—
58	C₂×Dic₇₄	C₂³	1	4	yes	complemented
59	C₄⋊C₄×C₃₇	C₂³	1	8	yes	complemented
60	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
61	C₄²×C₃₇	C₂³	1	8	yes	complemented
62	#10596938	C₂³	1	32	yes	complemented
63	C₂³×C₃₇	—	2	4	no	—
64	C₂³×C₃₇	C₂×D₄	1	4	yes	—
65	C₂³×C₃₇	C₂⁴	1	2	yes	—
66	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
67	C₂²×D₃₇	—	2	4	no	—
68	C₂²×D₃₇	—	2	16	no	—
69	C₂²×D₃₇	—	2	8	no	—
70	C₂²×D₃₇	—	2	8	no	—
71	C₂²×D₃₇	—	2	16	no	—
72	C₂²×D₃₇	—	4	16	no	—
73	C₂²×D₃₇	—	4	32	no	—
74	C₂²×D₃₇	C₂⁴	1	8	yes	—
75	C₂²×D₃₇	C₂⁴	1	4	yes	—
76	C₂×C₄×C₃₇	—	2	8	no	—
77	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
78	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₃₇	C₂×D₄	1	16	yes	complemented
80	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
81	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
82	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
83	C₂²⋊D₃₇	—	2	16	no	—
84	C₂×Dic₃₇	—	2	4	no	—
85	C₂×Dic₃₇	C₂⁴	1	4	yes	—
86	D₁₄₈	—	2	16	no	—
87	D₁₄₈	—	2	16	no	—
88	D₁₄₈	—	2	8	no	—
89	D₁₄₈	—	4	32	no	—
90	C₄×D₃₇	—	2	16	no	—
91	Dic₇₄	—	2	8	no	—
92	C₂²×C₃₇	—	2	8	no	—
93	C₂²×C₃₇	—	2	4	no	—
94	C₂²×C₃₇	C₂²×D₄	1	4	yes	—
95	C₂²×C₃₇	C₂²×D₄	1	2	yes	—
96	C₂²×C₃₇	H(5,𝔽₂)	1	4	yes	—
97	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₃₇	—	2	4	no	—
99	C₂×D₃₇	—	2	8	no	—
100	C₂×D₃₇	—	2	8	no	—
101	C₂×D₃₇	—	2	16	no	—
102	C₂×D₃₇	—	4	32	no	—
103	C₂×D₃₇	—	4	8	no	—
104	C₂×D₃₇	—	4	16	no	—
105	C₄×C₃₇	—	2	8	no	—
106	C₄×C₃₇	C₂²×D₄	1	4	yes	—
107	C₄×C₃₇	C₂²×D₄	1	4	yes	—
108	Dic₃₇	—	2	4	no	—
109	C₂×#535	—	37	1	no	maximal, Sylow
110	C₂×C₃₇	—	2	4	no	—
111	C₂×C₃₇	#535	1	4	yes	complemented
112	C₂×C₃₇	C₂³×D₄	1	1	yes	characteristic
113	C₂×C₃₇	C₂×H(5,𝔽₂)	1	2	yes	—
114	D₃₇	—	4	4	no	—
115	D₃₇	—	4	8	no	—
116	C₂²×#41	—	37	1	no	—
117	C₂³×D₄	—	37	1	no	—
118	#535	—	37	16	no	—
119	C₂×#127	—	37	2	no	—
120	C₂×#124	—	37	2	no	—
121	C₂×#121	—	37	4	no	—
122	C₂×#120	—	37	4	no	—
123	C₂×#117	D₃₇	1	1	yes	characteristic, complemented
124	C₃₇	C₂×#535	1	1	yes	characteristic, complemented, Sylow
125	C₂⁵	—	37	2	no	—
126	C₂×#41	—	37	8	no	—
127	C₂×#41	—	74	8	no	—
128	C₂²×Q₈	—	37	1	no	—
129	C₂²×D₄	—	37	2	no	—
130	C₂²×D₄	—	37	2	no	—
131	C₂²×D₄	—	37	2	no	—
132	C₂²×D₄	—	37	1	no	—
133	C₂²×D₄	—	37	8	no	—
134	C₂²×D₄	—	37	8	no	—
135	C₂²×D₄	—	74	8	no	—
136	C₂³×C₄	—	37	2	no	—
137	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
138	#127	—	37	16	no	—
139	#124	—	37	16	no	—
140	#121	—	37	32	no	—
141	#120	—	37	32	no	—
142	#117	C₂×D₃₇	1	8	yes	complemented
143	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	complemented
144	C₂×C₂²⋊C₄	—	37	8	no	—
145	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
146	C₂×C₄²	C₂×D₃₇	1	2	yes	complemented
147	C₂⁴	—	37	4	no	—
148	C₂⁴	—	37	2	no	—
149	C₂⁴	—	37	8	no	—
150	C₂⁴	—	74	8	no	—
151	C₂⁴	—	148	8	no	—
152	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
153	#41	—	74	32	no	—
154	C₂×Q₈	—	37	4	no	—
155	C₂×Q₈	—	74	4	no	—
156	C₂×D₄	—	37	8	no	—
157	C₂×D₄	—	37	8	no	—
158	C₂×D₄	—	37	8	no	—
159	C₂×D₄	—	37	4	no	—
160	C₂×D₄	—	37	32	no	—
161	C₂×D₄	—	74	8	no	—
162	C₂×D₄	—	74	8	no	—
163	C₂×D₄	—	74	8	no	—
164	C₂×D₄	—	74	4	no	—
165	C₂×D₄	—	74	32	no	—
166	C₂×D₄	—	148	16	no	—
167	C₂²×C₄	—	37	2	no	—
168	C₂²×C₄	—	37	8	no	—
169	C₂²×C₄	—	74	8	no	—
170	C₂²×C₄	D₁₄₈	1	8	yes	complemented
171	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
172	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
175	C₄⋊C₄	C₂²×D₃₇	1	8	yes	complemented
176	C₂²⋊C₄	—	37	32	no	—
177	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	—
178	C₄²	C₂²×D₃₇	1	8	yes	complemented
179	C₂³	—	2	4	no	—
180	C₂³	—	37	8	no	—
181	C₂³	—	37	4	no	—
182	C₂³	—	74	4	no	—
183	C₂³	—	74	16	no	—
184	C₂³	—	74	8	no	—
185	C₂³	—	74	8	no	—
186	C₂³	—	74	16	no	—
187	C₂³	—	148	16	no	—
188	C₂³	—	148	32	no	—
189	C₂³	C₂×D₁₄₈	1	4	yes	—
190	C₂³	C₂³×D₃₇	1	2	yes	—
191	C₂³	C₂³×D₃₇	1	1	yes	characteristic
192	Q₈	—	74	8	no	—
193	D₄	—	74	16	no	—
194	D₄	—	74	16	no	—
195	D₄	—	74	16	no	—
196	D₄	—	74	8	no	—
197	D₄	—	148	32	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	37	4	no	—
200	C₂×C₄	—	74	4	no	—
201	C₂×C₄	—	74	16	no	—
202	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
203	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
204	C₂×C₄	C₂×D₁₄₈	1	16	yes	complemented
205	C₂×C₄	C₂³×D₃₇	1	8	yes	—
206	C₂×C₄	C₂³×D₃₇	1	2	yes	—
207	C₂×C₄	C₂³×D₃₇	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	74	4	no	—
211	C₂²	—	74	8	no	—
212	C₂²	—	74	8	no	—
213	C₂²	—	74	16	no	—
214	C₂²	—	148	32	no	—
215	C₂²	—	148	16	no	—
216	C₂²	—	148	8	no	—
217	C₂²	C₂²×D₁₄₈	1	4	yes	—
218	C₂²	C₂²×D₁₄₈	1	2	yes	—
219	C₂²	#11924582	1	4	yes	—
220	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
221	C₄	—	2	8	no	—
222	C₄	—	74	4	no	—
223	C₄	C₂²×D₁₄₈	1	4	yes	—
224	C₄	C₂²×D₁₄₈	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	148	8	no	—
227	C₂	—	148	4	no	—
228	C₂	#14558102	1	4	yes	direct product factor
229	C₂	C₂³×D₁₄₈	1	1	yes	characteristic
230	C₂	C₂×#11924582	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#117×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	C₂×#535
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#10596962	C₂	semidirect
2. C₂³×D₁₄₈	C₂	semidirect
3. #14558102	C₂	direct (N × H)
4. C₂×#11924533	C₂	semidirect
5. C₂×#11924524	C₂	semidirect
6. C₂×#11924511	C₂	semidirect
7. C₂×#11924506	C₂	semidirect
8. C₂×#117×C₃₇	C₂	semidirect
11. C₂×#10596962	C₂²	semidirect
12. C₂²×Dic₇₄	C₂²	semidirect
16. C₂²×D₁₄₈	C₂²	semidirect
17. C₂²×D₁₄₈	C₂²	semidirect
18. C₂²×D₁₄₈	C₂²	semidirect
20. C₂²×C₄×D₃₇	C₂²	semidirect
21. #11924533	C₂²	semidirect
22. #11924524	C₂²	semidirect
23. #11924511	C₂²	semidirect
24. #11924506	C₂²	semidirect
25. C₂×#10596938	C₂²	semidirect
27. #117×C₃₇	C₂²	semidirect
28. C₂×C₄⋊C₄×C₃₇	C₂²	semidirect
30. C₂×C₄²×C₃₇	C₂²	semidirect
37. C₂²×C₄×C₃₇	D₄	semidirect
53. C₂×D₁₄₈	C₂³	semidirect
54. C₂×D₁₄₈	C₂³	semidirect
56. C₂×C₄×D₃₇	C₂³	semidirect
58. C₂×Dic₇₄	C₂³	semidirect
59. C₄⋊C₄×C₃₇	C₂³	semidirect
61. C₄²×C₃₇	C₂³	semidirect
62. #10596938	C₂³	semidirect
79. C₂×C₄×C₃₇	C₂×D₄	semidirect
111. C₂×C₃₇	#535	semidirect
123. C₂×#117	D₃₇	semidirect
124. C₃₇	C₂×#535	semidirect
142. #117	C₂×D₃₇	semidirect
143. C₂×C₄⋊C₄	C₂×D₃₇	semidirect
146. C₂×C₄²	C₂×D₃₇	semidirect
170. C₂²×C₄	D₁₄₈	semidirect
175. C₄⋊C₄	C₂²×D₃₇	semidirect
178. C₄²	C₂²×D₃₇	semidirect
204. C₂×C₄	C₂×D₁₄₈	semidirect
228. C₂	#14558102	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×#10596962	2
C₂³×D₁₄₈	2
#14558102	2
C₂×#11924533	2
C₂×#11924524	2
C₂×#11924511	2
C₂×#11924506	2
C₂×#117×C₃₇	2
C₂×#535	37

Subgroups — C2×#14558102 #18007907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558102 #18007907