Subgroups — C₂×#22577451 #31789395

Order 17536.

460 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555494	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555494	C₂	1	1	yes	maximal, characteristic
3	#22577451	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984023	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	—	2	1	no	—
7	C₂×#14555494	—	2	2	no	—
8	C₂×#14555494	C₄	1	2	yes	—
9	C₂×#14555494	C₄	1	2	yes	—
10	C₂×#14555494	C₄	1	4	yes	—
11	C₂×#14555494	C₂²	1	2	yes	—
12	C₂×#14555494	C₂²	1	4	yes	—
13	C₂×#14555494	C₂²	1	4	yes	—
14	C₂²×#11764040	—	2	1	no	—
15	C₂²×#11764040	C₂²	1	2	yes	—
16	#17984023	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₃₇	—	2	1	no	—
19	C₂×M₁₆×C₁₃₇	—	2	2	no	—
20	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
21	C₂×M₁₆×C₁₃₇	C₂²	1	4	yes	—
22	C₂²×C₈×C₁₃₇	—	2	1	no	—
23	#14555494	—	2	4	no	—
24	#14555494	—	2	2	no	—
25	#14555494	C₂×C₄	1	8	yes	—
26	#14555494	C₂×C₄	1	8	yes	—
27	#14555494	C₂³	1	4	yes	—
28	C₂×#11764040	—	2	4	no	—
29	C₂×#11764040	—	2	2	no	—
30	C₂×#11764040	C₂×C₄	1	4	yes	—
31	C₂×#11764040	C₂×C₄	1	4	yes	—
32	C₂×#11764040	C₂³	1	4	yes	—
33	C₂⁴×C₁₃₇	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
40	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
42	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
44	M₁₆×C₁₃₇	—	2	4	no	—
45	M₁₆×C₁₃₇	—	2	2	no	—
46	M₁₆×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₈×C₁₃₇	—	2	4	no	—
48	C₂×C₈×C₁₃₇	—	2	2	no	—
49	#11764040	—	2	4	no	—
50	#11764040	C₂²×C₄	1	8	yes	—
51	C₂³×C₁₃₇	—	4	2	no	—
52	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
55	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₁₃₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
75	C₈×C₁₃₇	—	2	4	no	—
76	C₂²×C₁₃₇	—	2	2	no	—
77	C₂²×C₁₃₇	—	2	2	no	—
78	C₂²×C₁₃₇	—	2	2	no	—
79	C₂²×C₁₃₇	—	4	2	no	—
80	C₂²×C₁₃₇	—	4	2	no	—
81	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₁₃₇	—	2	2	no	—
93	C₂×C₁₃₇	—	2	2	no	—
94	C₂×C₁₃₇	—	2	1	no	—
95	C₂×C₁₃₇	—	2	1	no	—
96	C₂×C₁₃₇	#430	1	2	yes	complemented
97	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₁₃₇	C₂×#430	1	1	yes	characteristic, complemented, Sylow
99	C₂×#430	—	137	1	no	maximal, Sylow
100	C₂²×M₁₆	—	137	1	no	—
101	C₂²×M₁₆	—	137	1	no	—
102	C₂²×M₁₆	D₁₃₇	1	1	yes	characteristic
103	#430	—	137	8	no	—
104	C₂×#108	—	137	4	no	—
105	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂×M₁₆	—	274	2	no	—
109	C₂×M₁₆	—	274	1	no	—
110	C₂×M₁₆	—	137	2	no	—
111	C₂×M₁₆	—	137	4	no	—
112	C₂×M₁₆	—	137	2	no	—
113	C₂×M₁₆	—	137	2	no	—
114	C₂×M₁₆	—	137	4	no	—
115	C₂×M₁₆	—	137	4	no	—
116	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
117	C₂×M₁₆	C₂×D₁₃₇	1	4	yes	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	274	1	no	—
120	C₂²×C₈	—	137	2	no	—
121	#108	—	137	16	no	—
122	C₂⁴	Dic₂₇₄	1	1	yes	characteristic
123	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
125	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
127	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	Dic₂₇₄	1	1	yes	characteristic
130	C₂²×C₄	Dic₂₇₄	1	2	yes	—
131	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
132	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
133	M₁₆	—	2	4	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	—	274	4	no	—
136	M₁₆	—	274	2	no	—
137	M₁₆	—	137	4	no	—
138	M₁₆	—	137	8	no	—
139	M₁₆	—	137	8	no	—
140	M₁₆	C₂²×D₁₃₇	1	4	yes	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	274	4	no	—
144	C₂×C₈	—	274	2	no	—
145	C₂×C₈	—	137	4	no	—
146	C₂×C₈	—	137	4	no	—
147	C₂×C₈	—	137	4	no	—
148	C₂³	—	4	2	no	—
149	C₂³	#14555489	1	1	yes	characteristic
150	C₂³	#14555489	1	1	yes	characteristic
151	C₂³	#14555489	1	2	yes	—
152	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
153	C₂³	C₂×Dic₂₇₄	1	2	yes	—
154	C₂×C₄	#14555489	1	2	yes	—
155	C₂×C₄	#14555489	1	2	yes	—
156	C₂×C₄	#14555489	1	2	yes	—
157	C₂×C₄	#14555489	1	1	yes	characteristic
158	C₂×C₄	#14555489	1	1	yes	characteristic
159	C₂×C₄	#14555489	1	2	yes	—
160	C₂×C₄	#14555489	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
164	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
165	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
168	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
170	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
171	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	—	274	4	no	—
174	C₈	—	137	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	C₂×#14555489	1	1	yes	characteristic
181	C₂²	C₂×#14555489	1	2	yes	—
182	C₂²	C₂×#14555489	1	2	yes	—
183	C₂²	C₂×#14555489	1	1	yes	characteristic
184	C₂²	C₂²×Dic₂₇₄	1	1	yes	characteristic
185	C₄	C₂×#14555489	1	1	yes	characteristic
186	C₄	C₂×#14555489	1	2	yes	—
187	C₄	C₂×#14555489	1	2	yes	—
188	C₄	C₂×#14555489	1	1	yes	characteristic
189	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
190	C₄	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	2	no	—
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#22577451	1	2	yes	direct product factor
196	C₂	C₂²×#14555489	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂²×M₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃₇	Q₈
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₃₇
hypocenter	C₁₃₇	C₂×#430
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22577451	C₂	direct (N × H)
4. C₂×#17984023	C₂	semidirect
16. #17984023	C₂²	semidirect
96. C₂×C₁₃₇	#430	semidirect
98. C₁₃₇	C₂×#430	semidirect
195. C₂	#22577451	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×#14555494	2
#22577451	2
C₂×#17984023	2
C₂²×M₁₆×C₁₃₇	2
C₂×#430	137

Subgroups — C2×#22577451 #31789395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577451 #31789395