Subgroups — #31783878

Order 17536.

496 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984088	C₂	1	1	yes	maximal, characteristic
2	#22577307	C₂	1	4	yes	maximal
3	#22577301	C₂	1	2	yes	maximal
4	#22577299	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₃₇	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₁₃₇	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
10	#17984088	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#14555489	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
18	#17983995	C₂²	1	4	yes	—
19	#17983995	C₂²	1	2	yes	—
20	#17983995	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
25	#95×C₁₃₇	C₂²	1	2	yes	—
26	C₂×Dic₂₇₄	—	4	2	no	—
27	C₂×Dic₂₇₄	—	4	2	no	—
28	C₂×Dic₂₇₄	C₂×C₄	1	8	yes	—
29	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
30	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
31	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
32	C₄⋊Dic₁₃₇	—	2	2	no	—
33	C₄⋊Dic₁₃₇	C₂×C₄	1	8	yes	—
34	C₄⋊Dic₁₃₇	C₂×C₄	1	8	yes	—
35	C₄⋊Dic₁₃₇	Q₈	1	4	yes	complemented
36	#14555489	—	4	4	no	—
37	C₄×Dic₁₃₇	—	2	4	no	—
38	C₄×Dic₁₃₇	—	4	4	no	—
39	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₄⋊C₄×C₁₃₇	Q₈	1	4	yes	complemented
46	C₄²×C₁₃₇	—	2	4	no	—
47	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
48	Dic₂₇₄	—	4	8	no	—
49	C₂×Dic₁₃₇	—	2	2	no	—
50	C₂×Dic₁₃₇	—	2	2	no	—
51	C₂×Dic₁₃₇	—	2	4	no	—
52	C₂×Dic₁₃₇	—	2	4	no	—
53	C₂×Dic₁₃₇	—	2	2	no	—
54	C₂×Dic₁₃₇	—	2	2	no	—
55	C₂×Dic₁₃₇	—	2	2	no	—
56	C₂×Dic₁₃₇	C₂²×C₄	1	8	yes	—
57	C₂×Dic₁₃₇	C₂×Q₈	1	4	yes	—
58	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄×C₁₃₇	—	2	2	no	—
61	C₂×C₄×C₁₃₇	—	2	1	no	—
62	C₂×C₄×C₁₃₇	—	2	1	no	—
63	C₂×C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄×C₁₃₇	—	2	2	no	—
65	C₂×C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₁₃₇	#41	1	2	yes	—
73	C₂×C₄×C₁₃₇	#41	1	2	yes	—
74	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
78	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
81	C₄×C₁₃₇	—	2	2	no	—
82	C₄×C₁₃₇	—	2	4	no	—
83	C₄×C₁₃₇	—	4	2	no	—
84	C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
85	C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
86	Dic₁₃₇	—	2	8	no	—
87	Dic₁₃₇	—	2	4	no	—
88	Dic₁₃₇	—	4	2	no	—
89	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₁₃₇	#518	1	1	yes	characteristic
92	C₂×C₁₃₇	#554	1	1	yes	characteristic
93	C₂×C₁₃₇	#554	1	1	yes	characteristic
94	C₂×C₁₃₇	#555	1	1	yes	characteristic
95	C₂×C₁₃₇	Q₈²	1	1	yes	characteristic
96	C₁₃₇	#2369	1	1	yes	characteristic, complemented, Sylow
97	#2369	—	137	1	no	maximal, Sylow
98	C₂×C₄⋊Q₈	—	137	1	no	—
99	#391	—	137	4	no	—
100	#389	—	137	2	no	—
101	#389	—	137	2	no	—
102	C₄×C₄⋊C₄	—	137	2	no	—
103	C₄×C₄⋊C₄	—	137	2	no	—
104	C₄×C₄⋊C₄	—	137	1	no	—
105	C₄×C₄⋊C₄	D₁₃₇	1	1	yes	characteristic
106	C₂²×Q₈	—	137	2	no	—
107	C₄⋊Q₈	—	137	8	no	—
108	C₂×C₄⋊C₄	—	137	1	no	—
109	C₂×C₄⋊C₄	—	137	4	no	—
110	C₂×C₄⋊C₄	—	137	2	no	—
111	C₂×C₄⋊C₄	—	137	2	no	—
112	C₂×C₄⋊C₄	—	137	1	no	—
113	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
115	C₂×C₄²	—	137	2	no	—
116	C₂×C₄²	—	137	4	no	—
117	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
118	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
119	#95	—	137	4	no	—
120	#95	—	137	4	no	—
121	#95	—	137	2	no	—
122	#95	C₂×D₁₃₇	1	2	yes	—
123	C₂×Q₈	—	548	2	no	—
124	C₂×Q₈	—	548	2	no	—
125	C₂×Q₈	—	137	8	no	—
126	C₂²×C₄	—	137	2	no	—
127	C₂²×C₄	—	137	2	no	—
128	C₂²×C₄	—	137	4	no	—
129	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
130	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
132	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	274	2	no	—
136	C₄⋊C₄	—	548	4	no	—
137	C₄⋊C₄	—	137	8	no	—
138	C₄⋊C₄	—	137	8	no	—
139	C₄⋊C₄	—	137	4	no	—
140	C₄⋊C₄	Dic₂₇₄	1	4	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	274	4	no	—
143	C₄²	—	548	4	no	—
144	C₄²	C₄×D₁₃₇	1	4	yes	—
145	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
146	Q₈	—	548	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	274	2	no	—
155	C₂×C₄	—	274	2	no	—
156	C₂×C₄	—	274	4	no	—
157	C₂×C₄	—	274	4	no	—
158	C₂×C₄	—	274	2	no	—
159	C₂×C₄	—	274	2	no	—
160	C₂×C₄	—	274	2	no	—
161	C₂×C₄	—	137	8	no	—
162	C₂×C₄	—	137	4	no	—
163	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
164	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
166	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
167	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
168	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
169	C₂×C₄	#14555509	1	2	yes	—
170	C₂×C₄	#14555510	1	2	yes	—
171	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
172	C₂²	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
173	C₂²	C₂²×Dic₂₇₄	1	1	yes	characteristic
174	C₂²	C₂×#14555509	1	1	yes	characteristic
175	C₂²	C₂×#14555510	1	1	yes	characteristic
176	C₂²	#17984134	1	1	yes	characteristic
177	C₂²	#17984134	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	274	4	no	—
182	C₄	—	274	8	no	—
183	C₄	—	548	2	no	—
184	C₄	#17984053	1	4	yes	—
185	C₄	C₄×Dic₂₇₄	1	4	yes	—
186	C₂	C₂×#17984053	1	1	yes	characteristic
187	C₂	C₂×C₄×Dic₂₇₄	1	1	yes	characteristic
188	C₂	#22577877	1	1	yes	characteristic
189	C₂	#22578110	1	1	yes	characteristic
190	C₂	#22578113	1	1	yes	characteristic
191	C₂	#22578118	1	1	yes	characteristic
192	C₂	Q₈×Dic₂₇₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₇
commutator	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₃₇	C₂
socle, prime radical, cosocle	C₂³×C₁₃₇	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₃₇
hypocenter	C₁₃₇	#2369
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #17984088	C₄	semidirect
35. C₄⋊Dic₁₃₇	Q₈	semidirect
45. C₄⋊C₄×C₁₃₇	Q₈	semidirect
96. C₁₃₇	#2369	semidirect
140. C₄⋊C₄	Dic₂₇₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984088	2
#22577307	2
#22577301	2
#22577299	2
C₄⋊C₄×Dic₁₃₇	2
C₄×C₄⋊Dic₁₃₇	2
C₄×C₄⋊C₄×C₁₃₇	2
#2369	137