Subgroups — C₂×#22578064 #31794257

Order 17536.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555511	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578064	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17984077	C₂	1	3	yes	maximal, complemented
4	C₂×#17984064	C₂	1	3	yes	maximal, complemented
5	C₂×#17984062	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₂²⋊D₁₃₇	C₂	1	3	yes	maximal, complemented
8	C₂³×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555511	—	2	8	no	—
10	C₂×#14555511	C₂²	1	8	yes	complemented
11	C₂²×D₅₄₈	C₂²	1	3	yes	—
12	C₂²×C₂²⋊D₁₃₇	C₂²	1	3	yes	—
13	C₂²×C₄×D₁₃₇	C₂²	1	3	yes	—
14	#17984077	C₂²	1	24	yes	complemented
15	#17984064	C₂²	1	24	yes	complemented
16	#17984062	C₂²	1	24	yes	complemented
17	Q₈×Dic₁₃₇	C₂²	1	8	yes	complemented
18	C₄×C₂²⋊D₁₃₇	C₂²	1	24	yes	complemented
19	C₂×C₄⋊Dic₁₃₇	C₂²	1	3	yes	—
20	C₂×#14555489	C₂²	1	3	yes	—
21	C₂×C₂²⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#14555487	C₂²	1	3	yes	—
23	C₂×#14555487	C₂²	1	6	yes	—
24	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₃₇	—	2	8	no	—
26	C₂²×Q₈×C₁₃₇	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₁₃₇	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₁₃₇	C₂²	1	3	yes	—
30	C₂³×D₁₃₇	C₂³	1	3	yes	—
31	#14555511	—	2	32	no	—
32	C₂×D₅₄₈	—	2	12	no	—
33	C₂×D₅₄₈	C₂³	1	12	yes	—
34	C₂×C₂²⋊D₁₃₇	—	2	12	no	—
35	C₂×C₂²⋊D₁₃₇	C₂³	1	12	yes	—
36	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₃₇	C₂³	1	3	yes	—
38	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₁₃₇	—	2	12	no	—
40	C₂×C₄×D₁₃₇	C₂³	1	12	yes	—
41	C₄⋊Dic₁₃₇	C₂³	1	12	yes	—
42	#14555489	C₂³	1	12	yes	—
43	C₂²⋊Dic₁₃₇	C₂³	1	4	yes	—
44	#14555487	C₂³	1	12	yes	—
45	#14555487	C₂³	1	24	yes	—
46	C₄×Dic₁₃₇	C₂³	1	12	yes	—
47	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₃₇	—	2	32	no	—
49	C₂×Q₈×C₁₃₇	—	2	12	no	—
50	C₂×Q₈×C₁₃₇	D₄	1	8	yes	complemented
51	C₂×Q₈×C₁₃₇	C₂³	1	12	yes	—
52	C₂×Q₈×C₁₃₇	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₃₇	—	2	12	no	—
54	C₂²×C₄×C₁₃₇	C₂³	1	3	yes	—
55	C₂²×C₄×C₁₃₇	C₂³	1	12	yes	—
56	C₂²×C₄×C₁₃₇	C₂³	1	3	yes	—
57	C₂²×D₁₃₇	—	2	6	no	—
58	C₂²×D₁₃₇	—	2	6	no	—
59	C₂²×D₁₃₇	—	2	6	no	—
60	C₂²×D₁₃₇	C₂⁴	1	6	yes	—
61	D₅₄₈	—	2	24	no	—
62	C₂²⋊D₁₃₇	—	2	24	no	—
63	C₂×Dic₁₃₇	—	2	2	no	—
64	C₂×Dic₁₃₇	—	2	6	no	—
65	C₂×Dic₁₃₇	—	2	2	no	—
66	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
67	C₂×Dic₁₃₇	C₂⁴	1	6	yes	—
68	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
69	C₄×D₁₃₇	—	2	24	no	—
70	C₂³×C₁₃₇	—	2	2	no	—
71	C₂³×C₁₃₇	—	2	2	no	—
72	C₂³×C₁₃₇	#41	1	4	yes	complemented
73	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
74	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₃₇	—	2	24	no	—
76	Q₈×C₁₃₇	C₂×D₄	1	16	yes	complemented
77	C₂×C₄×C₁₃₇	—	2	6	no	—
78	C₂×C₄×C₁₃₇	—	2	24	no	—
79	C₂×C₄×C₁₃₇	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₁₃₇	C₂⁴	1	6	yes	—
81	C₂×C₄×C₁₃₇	C₂⁴	1	6	yes	—
82	C₂²×C₁₃₇	—	2	8	no	—
83	C₂²×C₁₃₇	—	2	2	no	—
84	C₂²×C₁₃₇	—	2	2	no	—
85	C₂²×C₁₃₇	C₂²×D₄	1	2	yes	—
86	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
87	C₂²×C₁₃₇	C₂×#41	1	4	yes	complemented
88	C₂²×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
89	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
90	C₂×D₁₃₇	—	2	6	no	—
91	C₂×D₁₃₇	—	2	6	no	—
92	C₂×D₁₃₇	—	2	6	no	—
93	C₂×D₁₃₇	—	4	12	no	—
94	C₄×C₁₃₇	—	2	6	no	—
95	C₄×C₁₃₇	C₂²×D₄	1	12	yes	—
96	Dic₁₃₇	—	2	2	no	—
97	Dic₁₃₇	—	2	6	no	—
98	Dic₁₃₇	—	2	2	no	—
99	C₂×C₁₃₇	—	2	4	no	—
100	C₂×C₁₃₇	#548	1	4	yes	complemented
101	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
102	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
103	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
104	D₁₃₇	—	4	6	no	—
105	C₁₃₇	C₂×#548	1	1	yes	characteristic, complemented, Sylow
106	C₂×#548	—	137	1	no	maximal, Sylow
107	C₂²×#41	—	137	1	no	—
108	C₂³×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
109	#548	—	137	16	no	—
110	C₂×#124	—	137	3	no	—
111	C₂×C₂²⋊Q₈	—	137	3	no	—
112	C₂×#121	—	137	3	no	—
113	C₂×C₄×Q₈	—	137	1	no	—
114	C₂×C₄×D₄	—	137	3	no	—
115	C₂×#41	—	274	8	no	—
116	C₂×#41	—	137	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂×D₁₃₇	1	3	yes	—
119	C₂²×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
120	C₂²×Q₈	C₂×D₁₃₇	1	8	yes	complemented
121	C₂²×D₄	—	137	3	no	—
122	C₂²×D₄	—	137	3	no	—
123	C₂³×C₄	—	137	3	no	—
124	C₂³×C₄	C₂×D₁₃₇	1	3	yes	—
125	#124	—	137	24	no	—
126	C₂²⋊Q₈	—	137	24	no	—
127	#121	—	137	24	no	—
128	C₄×Q₈	—	137	8	no	—
129	C₄×D₄	—	137	24	no	—
130	C₂×C₄⋊C₄	—	137	3	no	—
131	C₂×C₄⋊C₄	—	137	3	no	—
132	C₂×C₂²⋊C₄	—	137	1	no	—
133	C₂×C₂²⋊C₄	—	137	3	no	—
134	C₂×C₂²⋊C₄	—	137	6	no	—
135	C₂×C₄²	—	137	3	no	—
136	C₂⁴	—	137	3	no	—
137	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
138	#41	—	274	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	C₂²⋊D₁₃₇	1	8	yes	complemented
142	C₂×Q₈	C₂²×D₁₃₇	1	12	yes	—
143	C₂×Q₈	C₂²×D₁₃₇	1	4	yes	—
144	C₂×D₄	—	274	12	no	—
145	C₂×D₄	—	274	12	no	—
146	C₂×D₄	—	137	12	no	—
147	C₂×D₄	—	137	12	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	274	12	no	—
150	C₂²×C₄	—	137	1	no	—
151	C₂²×C₄	—	137	3	no	—
152	C₂²×C₄	—	137	1	no	—
153	C₂²×C₄	—	137	12	no	—
154	C₂²×C₄	C₂²×D₁₃₇	1	12	yes	—
155	C₂²×C₄	C₂²×D₁₃₇	1	3	yes	—
156	C₂²×C₄	C₂²×D₁₃₇	1	3	yes	—
157	C₄⋊C₄	—	137	12	no	—
158	C₄⋊C₄	—	137	12	no	—
159	C₂²⋊C₄	—	137	4	no	—
160	C₂²⋊C₄	—	137	12	no	—
161	C₂²⋊C₄	—	137	24	no	—
162	C₄²	—	137	12	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	274	6	no	—
166	C₂³	—	274	6	no	—
167	C₂³	—	274	6	no	—
168	C₂³	—	137	6	no	—
169	C₂³	#14555511	1	4	yes	complemented
170	C₂³	C₂³×D₁₃₇	1	2	yes	—
171	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×C₂²⋊D₁₃₇	1	16	yes	complemented
174	D₄	—	274	24	no	—
175	D₄	—	274	24	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	274	2	no	—
179	C₂×C₄	—	274	6	no	—
180	C₂×C₄	—	274	2	no	—
181	C₂×C₄	—	274	24	no	—
182	C₂×C₄	—	137	2	no	—
183	C₂×C₄	—	137	6	no	—
184	C₂×C₄	—	137	2	no	—
185	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	12	yes	—
186	C₂×C₄	C₂³×D₁₃₇	1	6	yes	—
187	C₂×C₄	C₂³×D₁₃₇	1	6	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	274	6	no	—
192	C₂²	—	274	6	no	—
193	C₂²	—	274	6	no	—
194	C₂²	—	548	12	no	—
195	C₂²	C₂²×C₂²⋊D₁₃₇	1	2	yes	—
196	C₂²	C₂×#14555511	1	2	yes	—
197	C₂²	C₂×#14555511	1	4	yes	complemented
198	C₂²	#17984136	1	2	yes	—
199	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
200	C₄	—	2	6	no	—
201	C₄	—	274	2	no	—
202	C₄	—	274	6	no	—
203	C₄	—	274	2	no	—
204	C₄	C₂²×C₂²⋊D₁₃₇	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	548	6	no	—
207	C₂	#22578064	1	4	yes	direct product factor
208	C₂	C₂³×C₂²⋊D₁₃₇	1	1	yes	characteristic
209	C₂	C₂²×#14555511	1	1	yes	characteristic
210	C₂	C₂×#17984136	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂³×Q₈×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
hypercenter	C₂³×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#548
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555511	C₂	semidirect
2. #22578064	C₂	direct (N × H)
3. C₂×#17984077	C₂	semidirect
4. C₂×#17984064	C₂	semidirect
5. C₂×#17984062	C₂	semidirect
6. C₂×Q₈×Dic₁₃₇	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₃₇	C₂	semidirect
8. C₂³×Q₈×C₁₃₇	C₂	semidirect
10. C₂×#14555511	C₂²	semidirect
14. #17984077	C₂²	semidirect
15. #17984064	C₂²	semidirect
16. #17984062	C₂²	semidirect
17. Q₈×Dic₁₃₇	C₂²	semidirect
18. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
28. C₂²×Q₈×C₁₃₇	C₂²	semidirect
50. C₂×Q₈×C₁₃₇	D₄	semidirect
72. C₂³×C₁₃₇	#41	semidirect
76. Q₈×C₁₃₇	C₂×D₄	semidirect
87. C₂²×C₁₃₇	C₂×#41	semidirect
100. C₂×C₁₃₇	#548	semidirect
105. C₁₃₇	C₂×#548	semidirect
108. C₂³×Q₈	D₁₃₇	semidirect
120. C₂²×Q₈	C₂×D₁₃₇	semidirect
141. C₂×Q₈	C₂²⋊D₁₃₇	semidirect
169. C₂³	#14555511	semidirect
173. Q₈	C₂×C₂²⋊D₁₃₇	semidirect
197. C₂²	C₂×#14555511	semidirect
207. C₂	#22578064	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×#14555511	2
#22578064	2
C₂×#17984077	2
C₂×#17984064	2
C₂×#17984062	2
C₂×Q₈×Dic₁₃₇	2
C₂×C₄×C₂²⋊D₁₃₇	2
C₂³×Q₈×C₁₃₇	2
C₂×#548	137

Subgroups — C2×#22578064 #31794257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578064 #31794257