Subgroups — C₂×#22578110 #31794320

Order 17536.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578110	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984079	C₂	1	2	yes	maximal, complemented
4	C₂×#17984067	C₂	1	2	yes	maximal, complemented
5	C₂×#17984065	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×Dic₁₃₇	C₂	1	2	yes	maximal
8	C₂²×C₄⋊Dic₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
12	C₂³×Dic₁₃₇	C₂²	1	4	yes	—
13	#17984088	C₂²	1	8	yes	complemented
14	#17984079	C₂²	1	16	yes	complemented
15	#17984067	C₂²	1	16	yes	complemented
16	#17984065	C₂²	1	32	yes	complemented
17	C₄×Dic₂₇₄	C₂²	1	8	yes	complemented
18	D₄×Dic₁₃₇	C₂²	1	16	yes	—
19	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
21	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃₇	C₂²	1	16	yes	complemented
24	C₂×#14555489	C₂²	1	4	yes	—
25	C₂×#14555489	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₁₃₇	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
30	C₄×D₄×C₁₃₇	C₂²	1	8	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₂×Dic₂₇₄	—	2	4	no	—
35	C₂×Dic₂₇₄	—	4	4	no	—
36	C₂×Dic₂₇₄	C₂³	1	4	yes	—
37	C₂×Dic₂₇₄	C₂³	1	8	yes	—
38	C₂²×Dic₁₃₇	—	4	8	no	—
39	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
42	C₂²×Dic₁₃₇	C₂³	1	16	yes	—
43	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
44	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
45	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
46	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
47	#14555489	C₂³	1	8	yes	—
48	#14555489	C₂³	1	16	yes	—
49	C₂²⋊Dic₁₃₇	C₂³	1	16	yes	—
50	C₄×Dic₁₃₇	C₂³	1	8	yes	—
51	C₂⁴×C₁₃₇	C₂³	1	2	yes	—
52	C₂×D₄×C₁₃₇	Q₈	1	8	yes	complemented
53	C₂×D₄×C₁₃₇	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₃₇	—	2	8	no	—
55	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂³	1	8	yes	—
58	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
61	C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
62	C₄²×C₁₃₇	C₂³	1	4	yes	—
63	Dic₂₇₄	—	2	8	no	—
64	Dic₂₇₄	—	4	8	no	—
65	C₂×Dic₁₃₇	—	2	4	no	—
66	C₂×Dic₁₃₇	—	2	4	no	—
67	C₂×Dic₁₃₇	—	2	8	no	—
68	C₂×Dic₁₃₇	—	4	16	no	—
69	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
70	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
71	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
72	C₂³×C₁₃₇	—	2	4	no	—
73	C₂³×C₁₃₇	—	2	4	no	—
74	C₂³×C₁₃₇	C₂×Q₈	1	8	yes	—
75	C₂³×C₁₃₇	C₂⁴	1	4	yes	—
76	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	D₄×C₁₃₇	C₂×Q₈	1	16	yes	complemented
78	C₂×C₄×C₁₃₇	—	2	4	no	—
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	2	16	no	—
81	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
82	C₂×C₄×C₁₃₇	#41	1	4	yes	—
83	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
85	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
87	C₂²×C₁₃₇	—	2	16	no	—
88	C₂²×C₁₃₇	—	2	4	no	—
89	C₂²×C₁₃₇	—	2	4	no	—
90	C₂²×C₁₃₇	C₂²×Q₈	1	8	yes	—
91	C₂²×C₁₃₇	C₂²×Q₈	1	2	yes	—
92	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
93	C₂²×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
94	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
95	C₄×C₁₃₇	—	2	4	no	—
96	C₄×C₁₃₇	—	2	2	no	—
97	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
98	C₄×C₁₃₇	C₂×#41	1	4	yes	—
99	Dic₁₃₇	—	2	4	no	—
100	Dic₁₃₇	—	2	4	no	—
101	Dic₁₃₇	—	2	8	no	—
102	C₂×C₁₃₇	—	2	8	no	—
103	C₂×C₁₃₇	#554	1	4	yes	complemented
104	C₂×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
105	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
106	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₁₃₇	C₂×#554	1	1	yes	characteristic, complemented, Sylow
108	C₂×#554	—	137	1	no	maximal, Sylow
109	#554	—	137	16	no	—
110	C₂×C₄⋊Q₈	—	137	1	no	—
111	C₂×#125	—	137	2	no	—
112	C₂×C₂²⋊Q₈	—	137	2	no	—
113	C₂×C₂²⋊Q₈	—	137	4	no	—
114	C₂×C₄×Q₈	—	137	1	no	—
115	C₂×C₄×D₄	—	137	2	no	—
116	C₂×C₄×D₄	D₁₃₇	1	1	yes	characteristic
117	C₂²×C₄⋊C₄	—	137	2	no	—
118	C₂²×Q₈	—	137	1	no	—
119	C₂²×Q₈	—	137	2	no	—
120	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
121	C₂³×C₄	—	137	4	no	—
122	C₂³×C₄	C₂×D₁₃₇	1	2	yes	—
123	C₄⋊Q₈	—	137	8	no	—
124	#125	—	137	16	no	—
125	C₂²⋊Q₈	—	137	16	no	—
126	C₂²⋊Q₈	—	137	32	no	—
127	C₄×Q₈	—	137	8	no	—
128	C₄×D₄	—	137	16	no	—
129	C₄×D₄	C₂×D₁₃₇	1	8	yes	—
130	C₂×C₄⋊C₄	—	137	1	no	—
131	C₂×C₄⋊C₄	—	137	4	no	—
132	C₂×C₄⋊C₄	—	137	2	no	—
133	C₂×C₄⋊C₄	—	137	16	no	—
134	C₂×C₄⋊C₄	—	137	2	no	—
135	C₂×C₄⋊C₄	—	137	2	no	—
136	C₂×C₄⋊C₄	—	137	4	no	—
137	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
138	C₂×C₂²⋊C₄	—	137	4	no	—
139	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
140	C₂×C₄²	—	137	2	no	—
141	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
142	C₂⁴	C₂²×D₁₃₇	1	2	yes	—
143	C₂×Q₈	—	274	4	no	—
144	C₂×Q₈	—	548	4	no	—
145	C₂×Q₈	—	137	4	no	—
146	C₂×Q₈	—	137	8	no	—
147	C₂×D₄	Dic₂₇₄	1	8	yes	complemented
148	C₂×D₄	C₂²×D₁₃₇	1	4	yes	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	548	8	no	—
151	C₂²×C₄	—	137	2	no	—
152	C₂²×C₄	—	137	2	no	—
153	C₂²×C₄	—	137	4	no	—
154	C₂²×C₄	—	137	16	no	—
155	C₂²×C₄	C₂²×D₁₃₇	1	8	yes	—
156	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
160	C₄⋊C₄	—	137	8	no	—
161	C₄⋊C₄	—	137	16	no	—
162	C₄⋊C₄	—	137	4	no	—
163	C₄⋊C₄	—	137	16	no	—
164	C₄⋊C₄	—	137	8	no	—
165	C₄⋊C₄	—	137	8	no	—
166	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
167	C₂²⋊C₄	—	137	16	no	—
168	C₂²⋊C₄	C₂²×D₁₃₇	1	8	yes	—
169	C₄²	—	137	8	no	—
170	C₄²	C₂²×D₁₃₇	1	4	yes	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₂×Dic₂₇₄	1	8	yes	—
174	C₂³	C₂³×D₁₃₇	1	4	yes	—
175	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
176	Q₈	—	274	8	no	—
177	Q₈	—	548	8	no	—
178	D₄	C₂×Dic₂₇₄	1	16	yes	complemented
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	274	4	no	—
183	C₂×C₄	—	274	4	no	—
184	C₂×C₄	—	274	8	no	—
185	C₂×C₄	—	548	16	no	—
186	C₂×C₄	—	137	4	no	—
187	C₂×C₄	—	137	4	no	—
188	C₂×C₄	—	137	8	no	—
189	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
190	C₂×C₄	#14555509	1	4	yes	—
191	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
192	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
193	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
194	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	16	no	—
198	C₂²	C₂²×Dic₂₇₄	1	8	yes	—
199	C₂²	C₂²×Dic₂₇₄	1	2	yes	—
200	C₂²	C₂×#14555509	1	2	yes	—
201	C₂²	#17984134	1	2	yes	—
202	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	274	4	no	—
206	C₄	—	274	4	no	—
207	C₄	—	274	8	no	—
208	C₄	C₂²×Dic₂₇₄	1	4	yes	—
209	C₄	C₂×#14555509	1	4	yes	—
210	C₂	—	2	8	no	—
211	C₂	#22578110	1	4	yes	direct product factor
212	C₂	C₂³×Dic₂₇₄	1	1	yes	characteristic
213	C₂	C₂²×#14555509	1	1	yes	characteristic
214	C₂	C₂×#17984134	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×C₄×D₄×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂²×D₄×C₁₃₇	C₂²
hypercenter	C₂×C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	C₂×#554
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #22578110	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
3. C₂×#17984079	C₂	semidirect
4. C₂×#17984067	C₂	semidirect
5. C₂×#17984065	C₂	semidirect
6. C₂×C₄×Dic₂₇₄	C₂	semidirect
8. C₂²×C₄⋊Dic₁₃₇	C₂	semidirect
13. #17984088	C₂²	semidirect
14. #17984079	C₂²	semidirect
15. #17984067	C₂²	semidirect
16. #17984065	C₂²	semidirect
17. C₄×Dic₂₇₄	C₂²	semidirect
23. C₂×C₄⋊Dic₁₃₇	C₂²	semidirect
52. C₂×D₄×C₁₃₇	Q₈	semidirect
77. D₄×C₁₃₇	C₂×Q₈	semidirect
103. C₂×C₁₃₇	#554	semidirect
107. C₁₃₇	C₂×#554	semidirect
147. C₂×D₄	Dic₂₇₄	semidirect
178. D₄	C₂×Dic₂₇₄	semidirect
211. C₂	#22578110	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578110	2
C₂×#17984088	2
C₂×#17984079	2
C₂×#17984067	2
C₂×#17984065	2
C₂×C₄×Dic₂₇₄	2
C₂×D₄×Dic₁₃₇	2
C₂²×C₄⋊Dic₁₃₇	2
C₂×C₄×D₄×C₁₃₇	2
C₂×#554	137

Subgroups — C2×#22578110 #31794320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578110 #31794320