Subgroups — C₂×#22578126 #31794338

Order 17536.

1112 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	#22578126	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984083	C₂	1	4	yes	maximal, complemented
4	C₂×#17984076	C₂	1	2	yes	maximal
5	C₂×#17984064	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₂₇₄	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#17984046	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
11	C₂²×D₅₄₈	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
13	#17984088	C₂²	1	8	yes	complemented
14	#17984083	C₂²	1	32	yes	complemented
15	#17984076	C₂²	1	16	yes	—
16	#17984064	C₂²	1	16	yes	complemented
17	C₄×Dic₂₇₄	C₂²	1	16	yes	complemented
18	C₄×D₅₄₈	C₂²	1	8	yes	—
19	#17984046	C₂²	1	16	yes	complemented
20	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×#14555489	C₂²	1	4	yes	—
24	C₂×#14555487	C₂²	1	4	yes	—
25	C₂×#14555487	C₂²	1	4	yes	—
26	C₂×#14555487	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
28	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₁₃₇	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₂³×D₁₃₇	C₂³	1	2	yes	—
35	C₂×Dic₂₇₄	—	2	8	no	—
36	C₂×Dic₂₇₄	C₂³	1	8	yes	—
37	C₂×D₅₄₈	—	2	4	no	—
38	C₂×D₅₄₈	C₂³	1	4	yes	—
39	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄×D₁₃₇	—	2	8	no	—
42	C₂×C₄×D₁₃₇	C₂³	1	8	yes	—
43	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
44	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
45	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
46	#14555489	C₂³	1	16	yes	—
47	#14555487	C₂³	1	16	yes	—
48	#14555487	C₂³	1	8	yes	—
49	#14555487	C₂³	1	16	yes	—
50	C₄×Dic₁₃₇	C₂³	1	16	yes	—
51	C₂×Q₈×C₁₃₇	—	2	4	no	—
52	C₂×Q₈×C₁₃₇	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
59	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
60	C₄²×C₁₃₇	C₂³	1	4	yes	—
61	C₄²×C₁₃₇	C₂³	1	8	yes	—
62	C₂²×D₁₃₇	—	2	4	no	—
63	C₂²×D₁₃₇	—	2	4	no	—
64	C₂²×D₁₃₇	—	2	4	no	—
65	C₂²×D₁₃₇	C₂⁴	1	4	yes	—
66	Dic₂₇₄	—	2	16	no	—
67	D₅₄₈	—	2	8	no	—
68	C₂×Dic₁₃₇	—	2	4	no	—
69	C₂×Dic₁₃₇	—	2	8	no	—
70	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
71	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
72	C₄×D₁₃₇	—	2	16	no	—
73	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	Q₈×C₁₃₇	—	2	8	no	—
75	C₂×C₄×C₁₃₇	—	2	4	no	—
76	C₂×C₄×C₁₃₇	—	2	4	no	—
77	C₂×C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄×C₁₃₇	#41	1	4	yes	—
79	C₂×C₄×C₁₃₇	#41	1	4	yes	—
80	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
81	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
82	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
83	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
85	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
86	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
87	C₂²×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
88	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
89	C₂×D₁₃₇	—	2	4	no	—
90	C₂×D₁₃₇	—	2	4	no	—
91	C₂×D₁₃₇	—	2	4	no	—
92	C₂×D₁₃₇	—	4	8	no	—
93	C₄×C₁₃₇	—	2	4	no	—
94	C₄×C₁₃₇	—	2	4	no	—
95	C₄×C₁₃₇	—	2	2	no	—
96	C₄×C₁₃₇	C₂×#41	1	4	yes	—
97	C₄×C₁₃₇	C₂×#41	1	4	yes	—
98	Dic₁₃₇	—	2	4	no	—
99	Dic₁₃₇	—	2	8	no	—
100	C₂×C₁₃₇	#556	1	4	yes	complemented
101	C₂×C₁₃₇	C₂²×#41	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	4	no	—
105	C₁₃₇	C₂×#556	1	1	yes	characteristic, complemented, Sylow
106	C₂×#556	—	137	1	no	maximal, Sylow
107	#556	—	137	16	no	—
108	C₂×C₄⋊Q₈	—	137	1	no	—
109	C₂×#126	—	137	4	no	—
110	C₂×#124	—	137	2	no	—
111	C₂×C₂²⋊Q₈	—	137	2	no	—
112	C₂×C₄×Q₈	—	137	2	no	—
113	C₂×C₄×Q₈	D₁₃₇	1	1	yes	characteristic, complemented
114	C₂×C₄×D₄	—	137	1	no	—
115	C₂×#117	—	137	2	no	—
116	C₂²×Q₈	—	137	2	no	—
117	C₂²×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
118	C₂²×D₄	—	137	1	no	—
119	C₂³×C₄	—	137	2	no	—
120	C₄⋊Q₈	—	137	8	no	—
121	#126	—	137	32	no	—
122	#124	—	137	16	no	—
123	C₂²⋊Q₈	—	137	16	no	—
124	C₄×Q₈	—	137	16	no	—
125	C₄×Q₈	C₂×D₁₃₇	1	8	yes	complemented
126	C₄×D₄	—	137	8	no	—
127	#117	—	137	16	no	—
128	C₂×C₄⋊C₄	—	137	2	no	—
129	C₂×C₄⋊C₄	—	137	1	no	—
130	C₂×C₄⋊C₄	—	137	4	no	—
131	C₂×C₄⋊C₄	—	137	2	no	—
132	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
134	C₂×C₂²⋊C₄	—	137	4	no	—
135	C₂×C₂²⋊C₄	—	137	2	no	—
136	C₂×C₂²⋊C₄	—	137	4	no	—
137	C₂×C₄²	—	137	4	no	—
138	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
140	C₂⁴	—	137	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	274	8	no	—
143	C₂×Q₈	—	137	8	no	—
144	C₂×Q₈	C₂²×D₁₃₇	1	4	yes	—
145	C₂×D₄	—	274	4	no	—
146	C₂×D₄	—	137	4	no	—
147	C₂²×C₄	—	274	8	no	—
148	C₂²×C₄	—	137	2	no	—
149	C₂²×C₄	—	137	4	no	—
150	C₂²×C₄	—	137	8	no	—
151	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
154	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
156	C₄⋊C₄	—	137	16	no	—
157	C₄⋊C₄	—	137	8	no	—
158	C₄⋊C₄	—	137	4	no	—
159	C₄⋊C₄	—	137	8	no	—
160	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
161	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	—
162	C₂²⋊C₄	—	137	16	no	—
163	C₂²⋊C₄	—	137	8	no	—
164	C₂²⋊C₄	—	137	16	no	—
165	C₄²	—	137	16	no	—
166	C₄²	C₂²×D₁₃₇	1	4	yes	—
167	C₄²	C₂²×D₁₃₇	1	8	yes	—
168	C₂³	—	274	4	no	—
169	C₂³	—	274	4	no	—
170	C₂³	—	274	4	no	—
171	C₂³	—	137	4	no	—
172	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	274	16	no	—
175	D₄	—	274	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	274	4	no	—
180	C₂×C₄	—	274	8	no	—
181	C₂×C₄	—	274	16	no	—
182	C₂×C₄	—	137	4	no	—
183	C₂×C₄	—	137	8	no	—
184	C₂×C₄	#14555510	1	4	yes	—
185	C₂×C₄	#14555511	1	4	yes	—
186	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
187	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
188	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
189	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
190	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
191	C₂²	—	274	4	no	—
192	C₂²	—	274	4	no	—
193	C₂²	—	274	4	no	—
194	C₂²	—	548	8	no	—
195	C₂²	C₂×#14555510	1	2	yes	—
196	C₂²	C₂×#14555511	1	2	yes	—
197	C₂²	#17984135	1	2	yes	—
198	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	2	no	—
202	C₄	—	274	4	no	—
203	C₄	—	274	8	no	—
204	C₄	C₂×#14555510	1	4	yes	—
205	C₄	C₂×#14555511	1	4	yes	—
206	C₂	—	548	4	no	—
207	C₂	#22578126	1	4	yes	direct product factor
208	C₂	C₂²×#14555510	1	1	yes	characteristic
209	C₂	C₂²×#14555511	1	1	yes	characteristic
210	C₂	C₂×#17984135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×C₄×Q₈×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂²×D₅₄₈	C₂²
hypercenter	C₂×C₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#556
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578126	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
3. C₂×#17984083	C₂	semidirect
5. C₂×#17984064	C₂	semidirect
6. C₂×C₄×Dic₂₇₄	C₂	semidirect
8. C₂×#17984046	C₂	semidirect
9. C₂×C₄×Q₈×C₁₃₇	C₂	semidirect
13. #17984088	C₂²	semidirect
14. #17984083	C₂²	semidirect
16. #17984064	C₂²	semidirect
17. C₄×Dic₂₇₄	C₂²	semidirect
19. #17984046	C₂²	semidirect
29. C₄×Q₈×C₁₃₇	C₂²	semidirect
100. C₂×C₁₃₇	#556	semidirect
105. C₁₃₇	C₂×#556	semidirect
113. C₂×C₄×Q₈	D₁₃₇	semidirect
125. C₄×Q₈	C₂×D₁₃₇	semidirect
207. C₂	#22578126	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22578126	2
C₂×#17984088	2
C₂×#17984083	2
C₂×#17984076	2
C₂×#17984064	2
C₂×C₄×Dic₂₇₄	2
C₂×C₄×D₅₄₈	2
C₂×#17984046	2
C₂×C₄×Q₈×C₁₃₇	2
C₂×#556	137

Subgroups — C2×#22578126 #31794338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578126 #31794338