Subgroups — #31783701

Order 17536.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984053	C₂	1	2	yes	maximal
3	#22577299	C₂	1	2	yes	maximal
4	#22577298	C₂	1	4	yes	maximal
5	#22577272	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₁₃₇	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#384×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₇₄	C₄	1	8	yes	complemented
11	#17984053	—	2	8	no	—
12	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#14555489	C₂²	1	4	yes	—
16	C₂×#14555489	C₂²	1	4	yes	—
17	C₂×#14555489	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
20	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
21	#17983995	C₂²	1	2	yes	—
22	#17983995	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×Dic₂₇₄	C₂×C₄	1	4	yes	—
28	C₂×Dic₂₇₄	D₄	1	4	yes	—
29	C₂×Dic₂₇₄	Q₈	1	4	yes	—
30	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
31	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
32	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
33	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
34	C₄⋊Dic₁₃₇	—	2	2	no	—
35	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₁₃₇	Q₈	1	4	yes	complemented
37	#14555489	—	2	8	no	—
38	#14555489	—	4	4	no	—
39	#14555489	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₃₇	—	2	8	no	—
41	C₄×Dic₁₃₇	—	2	4	no	—
42	C₄×Dic₁₃₇	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₇	—	2	4	no	—
48	C₄⋊C₄×C₁₃₇	—	2	4	no	—
49	C₄⋊C₄×C₁₃₇	—	4	2	no	—
50	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
51	Dic₂₇₄	C₄⋊C₄	1	16	yes	complemented
52	C₂×Dic₁₃₇	—	2	2	no	—
53	C₂×Dic₁₃₇	—	2	2	no	—
54	C₂×Dic₁₃₇	—	2	2	no	—
55	C₂×Dic₁₃₇	—	2	2	no	—
56	C₂×Dic₁₃₇	—	2	2	no	—
57	C₂×Dic₁₃₇	—	2	2	no	—
58	C₂×Dic₁₃₇	—	2	2	no	—
59	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
61	C₂×Dic₁₃₇	C₂×D₄	1	4	yes	—
62	C₂×Dic₁₃₇	C₂×Q₈	1	4	yes	—
63	C₂×Dic₁₃₇	C₂×Q₈	1	4	yes	—
64	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	—	2	1	no	—
66	C₂×C₄×C₁₃₇	—	2	1	no	—
67	C₂×C₄×C₁₃₇	—	2	4	no	—
68	C₂×C₄×C₁₃₇	—	2	4	no	—
69	C₂×C₄×C₁₃₇	—	2	4	no	—
70	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₁₃₇	#41	1	2	yes	—
77	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
78	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
82	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₁₃₇	—	2	2	no	—
85	C₄×C₁₃₇	—	4	4	no	—
86	C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₄×C₁₃₇	C₄×Q₈	1	4	yes	complemented
88	Dic₁₃₇	—	2	4	no	—
89	Dic₁₃₇	—	2	4	no	—
90	Dic₁₃₇	—	4	2	no	—
91	Dic₁₃₇	C₂×C₄⋊C₄	1	8	yes	—
92	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₂×C₁₃₇	#520	1	1	yes	characteristic
95	C₂×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
96	C₂×C₁₃₇	#550	1	1	yes	characteristic
97	C₂×C₁₃₇	#557	1	1	yes	characteristic
98	C₂×C₁₃₇	Q₈²	1	1	yes	characteristic
99	C₁₃₇	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
100	Q₈×C₄⋊C₄	—	137	1	no	maximal, Sylow
101	C₂×C₄×Q₈	—	137	2	no	—
102	C₂×C₄×Q₈	—	137	1	no	—
103	#389	—	137	4	no	—
104	#389	—	137	2	no	—
105	#384	—	137	2	no	—
106	#384	D₁₃₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	137	2	no	—
108	C₄×C₄⋊C₄	—	137	1	no	—
109	C₂²×Q₈	—	137	1	no	—
110	C₄×Q₈	—	274	8	no	—
111	C₄×Q₈	—	137	8	no	—
112	C₂×C₄⋊C₄	—	137	4	no	—
113	C₂×C₄⋊C₄	—	137	1	no	—
114	C₂×C₄⋊C₄	—	137	4	no	—
115	C₂×C₄⋊C₄	—	137	2	no	—
116	C₂×C₄⋊C₄	—	137	1	no	—
117	C₂×C₄⋊C₄	—	137	1	no	—
118	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
121	C₂×C₄²	—	137	4	no	—
122	C₂×C₄²	—	137	2	no	—
123	C₂×C₄²	—	137	2	no	—
124	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
125	#95	—	137	2	no	—
126	#95	—	137	4	no	—
127	C₂×Q₈	—	137	4	no	—
128	C₂×Q₈	—	137	4	no	—
129	C₂×Q₈	—	137	4	no	—
130	C₂²×C₄	—	137	2	no	—
131	C₂²×C₄	—	137	2	no	—
132	C₂²×C₄	—	137	2	no	—
133	C₂²×C₄	—	137	2	no	—
134	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	274	8	no	—
142	C₄⋊C₄	—	274	2	no	—
143	C₄⋊C₄	—	548	4	no	—
144	C₄⋊C₄	—	137	8	no	—
145	C₄⋊C₄	—	137	4	no	—
146	C₄⋊C₄	—	137	4	no	—
147	C₄²	—	274	8	no	—
148	C₄²	—	274	4	no	—
149	C₄²	—	137	8	no	—
150	C₄²	C₄×D₁₃₇	1	4	yes	—
151	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
152	Q₈	—	137	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	274	2	no	—
159	C₂×C₄	—	274	2	no	—
160	C₂×C₄	—	274	2	no	—
161	C₂×C₄	—	274	2	no	—
162	C₂×C₄	—	274	2	no	—
163	C₂×C₄	—	274	2	no	—
164	C₂×C₄	—	274	2	no	—
165	C₂×C₄	—	137	4	no	—
166	C₂×C₄	—	137	4	no	—
167	C₂×C₄	—	137	4	no	—
168	C₂×C₄	—	137	4	no	—
169	C₂×C₄	—	137	4	no	—
170	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
173	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
174	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
175	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
176	C₂×C₄	#14555509	1	2	yes	—
177	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
180	C₂²	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
181	C₂²	C₂×#14555509	1	1	yes	characteristic
182	C₂²	#17984133	1	1	yes	characteristic
183	C₂²	#17984136	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	274	4	no	—
187	C₄	—	274	4	no	—
188	C₄	—	548	2	no	—
189	C₄	—	137	8	no	—
190	C₄	C₄⋊C₄×D₁₃₇	1	4	yes	—
191	C₄	#17984053	1	4	yes	complemented
192	C₂	C₂×C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
193	C₂	C₂×#17984053	1	1	yes	characteristic
194	C₂	#22577884	1	1	yes	characteristic
195	C₂	#22578068	1	1	yes	characteristic
196	C₂	#22578073	1	1	yes	characteristic
197	C₂	#22578128	1	1	yes	characteristic
198	C₂	#22578133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₇
commutator	C₂²×C₁₃₇	C₂³×C₄
Fitting	#384×C₁₃₇	C₂
socle, prime radical, cosocle	C₂³×C₁₃₇	C₂⁴
hypercenter	#384	D₁₃₇
hypocenter	C₁₃₇	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₇₄	C₄	semidirect
36. C₄⋊Dic₁₃₇	Q₈	semidirect
51. Dic₂₇₄	C₄⋊C₄	semidirect
87. C₄×C₁₃₇	C₄×Q₈	semidirect
99. C₁₃₇	Q₈×C₄⋊C₄	semidirect
191. C₄	#17984053	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₇₄	2
C₂×#17984053	2
#22577299	2
#22577298	2
#22577272	2
C₄⋊C₄×Dic₁₃₇	2
C₄×C₄⋊Dic₁₃₇	2
#384×C₁₃₇	2
Q₈×C₄⋊C₄	137