Subgroups — C₂×#22577953 #31794072

Order 17536.

1728 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555510	C₂	1	2	yes	maximal, complemented
2	#22577953	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17984085	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17984076	C₂	1	2	yes	maximal, complemented
6	C₂×#17984062	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×D₅₄₈	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555510	—	2	16	no	—
10	C₂×#14555510	C₂²	1	16	yes	complemented
11	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
12	C₂²×D₅₄₈	C₂²	1	2	yes	—
13	C₂²×D₅₄₈	C₂²	1	4	yes	—
14	C₂²×C₂²⋊D₁₃₇	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₃₇	C₂²	1	4	yes	—
16	#17984088	C₂²	1	8	yes	complemented
17	#17984085	C₂²	1	8	yes	complemented
18	#17984076	C₂²	1	16	yes	complemented
19	#17984062	C₂²	1	32	yes	complemented
20	C₄×D₅₄₈	C₂²	1	32	yes	complemented
21	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
22	C₂×#14555487	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₃₇	C₂²	1	8	yes	complemented
29	C₂³×D₁₃₇	C₂³	1	4	yes	—
30	#14555510	—	2	64	no	—
31	C₂×Dic₂₇₄	—	2	8	no	—
32	C₂×Dic₂₇₄	C₂³	1	8	yes	—
33	C₂×D₅₄₈	—	2	8	no	—
34	C₂×D₅₄₈	—	2	16	no	—
35	C₂×D₅₄₈	C₂³	1	16	yes	—
36	C₂×D₅₄₈	C₂³	1	8	yes	—
37	C₂×C₂²⋊D₁₃₇	—	2	16	no	—
38	C₂×C₂²⋊D₁₃₇	C₂³	1	16	yes	—
39	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
40	C₂×C₄×D₁₃₇	—	2	16	no	—
41	C₂×C₄×D₁₃₇	C₂³	1	16	yes	—
42	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
43	#14555487	C₂³	1	32	yes	—
44	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	—	2	8	no	—
46	C₂²×C₄×C₁₃₇	D₄	1	8	yes	complemented
47	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃₇	C₂³	1	8	yes	—
49	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₄²×C₁₃₇	C₂³	1	8	yes	—
54	C₄²×C₁₃₇	C₂³	1	4	yes	—
55	C₄²×C₁₃₇	C₂³	1	4	yes	—
56	C₂²×D₁₃₇	—	2	8	no	—
57	C₂²×D₁₃₇	—	2	16	no	—
58	C₂²×D₁₃₇	C₂⁴	1	8	yes	—
59	Dic₂₇₄	—	2	16	no	—
60	D₅₄₈	—	2	16	no	—
61	D₅₄₈	—	2	32	no	—
62	C₂²⋊D₁₃₇	—	2	32	no	—
63	C₂×Dic₁₃₇	—	2	8	no	—
64	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
65	C₄×D₁₃₇	—	2	32	no	—
66	C₂³×C₁₃₇	—	2	4	no	—
67	C₂³×C₁₃₇	C₂×D₄	1	4	yes	—
68	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
69	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	—	2	4	no	—
71	C₂×C₄×C₁₃₇	—	2	16	no	—
72	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₁₃₇	C₂×D₄	1	16	yes	complemented
75	C₂×C₄×C₁₃₇	#41	1	8	yes	complemented
76	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
77	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
80	C₂²×C₁₃₇	—	2	8	no	—
81	C₂²×C₁₃₇	—	2	4	no	—
82	C₂²×C₁₃₇	C₂²×D₄	1	4	yes	—
83	C₂²×C₁₃₇	C₂²×D₄	1	2	yes	—
84	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
85	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
86	C₂×D₁₃₇	—	2	8	no	—
87	C₂×D₁₃₇	—	2	16	no	—
88	C₂×D₁₃₇	—	4	16	no	—
89	C₄×C₁₃₇	—	2	4	no	—
90	C₄×C₁₃₇	C₂²×D₄	1	4	yes	—
91	C₄×C₁₃₇	C₂²×D₄	1	4	yes	—
92	C₄×C₁₃₇	C₂×#41	1	8	yes	complemented
93	Dic₁₃₇	—	2	8	no	—
94	C₂×C₁₃₇	—	2	4	no	—
95	C₂×C₁₃₇	#532	1	4	yes	complemented
96	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
97	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
98	D₁₃₇	—	4	8	no	—
99	C₁₃₇	C₂×#532	1	1	yes	characteristic, complemented, Sylow
100	C₂×#532	—	137	1	no	maximal, Sylow
101	C₂²×#41	—	137	2	no	—
102	#532	—	137	16	no	—
103	C₂×C₄⋊Q₈	—	137	1	no	—
104	C₂×#127	—	137	1	no	—
105	C₂×#124	—	137	2	no	—
106	C₂×#121	—	137	4	no	—
107	C₂×C₄×D₄	—	137	4	no	—
108	C₂²×C₄²	D₁₃₇	1	1	yes	characteristic, complemented
109	C₂×#41	—	274	16	no	—
110	C₂×#41	—	137	16	no	—
111	C₂²×Q₈	—	137	2	no	—
112	C₂²×D₄	—	137	4	no	—
113	C₂²×D₄	—	137	4	no	—
114	C₂²×D₄	—	137	2	no	—
115	C₂³×C₄	—	137	4	no	—
116	C₂³×C₄	C₂×D₁₃₇	1	2	yes	—
117	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
118	C₄⋊Q₈	—	137	8	no	—
119	#127	—	137	8	no	—
120	#124	—	137	16	no	—
121	#121	—	137	32	no	—
122	C₄×D₄	—	137	32	no	—
123	C₂×C₄⋊C₄	—	137	4	no	—
124	C₂×C₂²⋊C₄	—	137	8	no	—
125	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
127	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
128	C₂×C₄²	C₂×D₁₃₇	1	8	yes	complemented
129	C₂⁴	—	137	4	no	—
130	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
131	#41	—	274	64	no	—
132	C₂×Q₈	—	274	8	no	—
133	C₂×Q₈	—	137	8	no	—
134	C₂×D₄	—	274	16	no	—
135	C₂×D₄	—	274	8	no	—
136	C₂×D₄	—	274	16	no	—
137	C₂×D₄	—	137	16	no	—
138	C₂×D₄	—	137	16	no	—
139	C₂×D₄	—	137	8	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	274	16	no	—
142	C₂²×C₄	—	137	4	no	—
143	C₂²×C₄	—	137	16	no	—
144	C₂²×C₄	D₅₄₈	1	8	yes	complemented
145	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
146	C₂²×C₄	C₂²×D₁₃₇	1	8	yes	—
147	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
149	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
150	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
151	C₄⋊C₄	—	137	16	no	—
152	C₂²⋊C₄	—	137	32	no	—
153	C₄²	C₂²×D₁₃₇	1	8	yes	—
154	C₄²	C₂²×D₁₃₇	1	4	yes	—
155	C₄²	C₂²×D₁₃₇	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	—	274	8	no	—
158	C₂³	—	274	16	no	—
159	C₂³	—	137	8	no	—
160	C₂³	C₂×D₅₄₈	1	4	yes	—
161	C₂³	C₂³×D₁₃₇	1	2	yes	—
162	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
163	Q₈	—	274	16	no	—
164	D₄	—	274	32	no	—
165	D₄	—	274	16	no	—
166	D₄	—	274	32	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	274	8	no	—
170	C₂×C₄	—	274	32	no	—
171	C₂×C₄	—	137	8	no	—
172	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
173	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
174	C₂×C₄	C₂×D₅₄₈	1	16	yes	complemented
175	C₂×C₄	#14555510	1	8	yes	complemented
176	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
177	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
178	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
179	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	274	8	no	—
183	C₂²	—	274	16	no	—
184	C₂²	—	548	16	no	—
185	C₂²	C₂²×D₅₄₈	1	4	yes	—
186	C₂²	C₂²×D₅₄₈	1	2	yes	—
187	C₂²	C₂×#14555510	1	4	yes	—
188	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	274	8	no	—
191	C₄	C₂²×D₅₄₈	1	4	yes	—
192	C₄	C₂²×D₅₄₈	1	4	yes	—
193	C₄	C₂×#14555510	1	8	yes	complemented
194	C₂	—	2	4	no	—
195	C₂	—	548	8	no	—
196	C₂	#22577953	1	4	yes	direct product factor
197	C₂	C₂³×D₅₄₈	1	1	yes	characteristic
198	C₂	C₂²×#14555510	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂²×C₄²×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₃₇
hypocenter	C₁₃₇	C₂×#532
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555510	C₂	semidirect
2. #22577953	C₂	direct (N × H)
3. C₂×#17984088	C₂	semidirect
4. C₂×#17984085	C₂	semidirect
5. C₂×#17984076	C₂	semidirect
6. C₂×#17984062	C₂	semidirect
7. C₂×C₄×D₅₄₈	C₂	semidirect
8. C₂²×C₄²×C₁₃₇	C₂	semidirect
10. C₂×#14555510	C₂²	semidirect
16. #17984088	C₂²	semidirect
17. #17984085	C₂²	semidirect
18. #17984076	C₂²	semidirect
19. #17984062	C₂²	semidirect
20. C₄×D₅₄₈	C₂²	semidirect
28. C₂×C₄²×C₁₃₇	C₂²	semidirect
46. C₂²×C₄×C₁₃₇	D₄	semidirect
74. C₂×C₄×C₁₃₇	C₂×D₄	semidirect
75. C₂×C₄×C₁₃₇	#41	semidirect
92. C₄×C₁₃₇	C₂×#41	semidirect
95. C₂×C₁₃₇	#532	semidirect
99. C₁₃₇	C₂×#532	semidirect
108. C₂²×C₄²	D₁₃₇	semidirect
128. C₂×C₄²	C₂×D₁₃₇	semidirect
144. C₂²×C₄	D₅₄₈	semidirect
174. C₂×C₄	C₂×D₅₄₈	semidirect
175. C₂×C₄	#14555510	semidirect
193. C₄	C₂×#14555510	semidirect
196. C₂	#22577953	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×#14555510	2
#22577953	2
C₂×#17984088	2
C₂×#17984085	2
C₂×#17984076	2
C₂×#17984062	2
C₂×C₄×D₅₄₈	2
C₂²×C₄²×C₁₃₇	2
C₂×#532	137

Subgroups — C2×#22577953 #31794072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577953 #31794072