Subgroups — #16854300

Order 3968.

492 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×#92474	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
3	#14521092	C₂	1	4	yes	maximal, complemented
4	#14521089	C₂	1	4	yes	maximal, complemented
5	#14521085	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#117	—	2	1	no	—
9	C₃₁×#117	—	2	1	no	—
10	C₃₁×#117	C₂²	1	2	yes	—
11	C₃₁×#117	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
14	C₂×C₄²×C₃₁	—	2	1	no	—
15	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
16	#11758503	—	2	1	no	—
17	#11758503	—	2	1	no	—
18	#11758503	C₂²	1	2	yes	—
19	#11758503	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₃₁	—	2	1	no	—
22	C₂×#92474	C₄	1	8	yes	complemented
23	C₂×#92474	C₂²	1	2	yes	—
24	C₂×#92474	C₂²	1	2	yes	—
25	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
26	#11758396	C₂²	1	8	yes	complemented
27	#11758395	C₂²	1	8	yes	complemented
28	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	—	2	2	no	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	Q₈	1	4	yes	—
35	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₃₁	—	2	2	no	—
39	C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂²⋊C₄×C₃₁	—	2	2	no	—
43	C₄²×C₃₁	—	2	2	no	—
44	C₄²×C₃₁	—	2	2	no	—
45	C₂²⋊Dic₃₁	—	2	2	no	—
46	C₂²⋊Dic₃₁	—	2	2	no	—
47	C₄⋊Dic₃₁	—	2	2	no	—
48	C₄⋊Dic₃₁	C₂³	1	4	yes	—
49	C₄×Dic₃₁	—	2	2	no	—
50	C₄×Dic₃₁	—	2	2	no	—
51	#92474	C₂×C₄	1	8	yes	complemented
52	#92474	C₂×C₄	1	8	yes	complemented
53	C₂×C₃₁⋊C₈	C₂×C₄	1	8	yes	complemented
54	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
55	C₂³×C₃₁	—	4	2	no	—
56	C₂³×C₃₁	C₄⋊C₄	1	4	yes	—
57	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	—	2	4	no	—
61	C₂×C₄×C₃₁	—	4	4	no	—
62	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₃₁	—	2	4	no	—
76	C₂×Dic₃₁	—	4	4	no	—
77	C₃₁⋊C₈	C₂²×C₄	1	8	yes	—
78	#2911	—	31	1	no	maximal, Sylow
79	C₂²×C₃₁	—	2	2	no	—
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	—	2	2	no	—
82	C₂²×C₃₁	—	4	4	no	—
83	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₃₁	—	4	4	no	—
88	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
93	Dic₃₁	—	4	4	no	—
94	C₂²×M₁₆	—	31	1	no	—
95	C₂×#117	—	31	1	no	—
96	C₂×#117	D₃₁	1	1	yes	characteristic
97	#428	—	31	4	no	—
98	#428	—	31	4	no	—
99	#427	—	31	4	no	—
100	C₂×C₃₁	—	2	2	no	—
101	C₂×C₃₁	—	2	2	no	—
102	C₂×C₃₁	—	2	2	no	—
103	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₃₁	#575	1	2	yes	—
105	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
106	C₂×M₁₆	—	31	8	no	—
107	C₂×M₁₆	—	31	2	no	—
108	C₂×M₁₆	—	31	2	no	—
109	C₂²×C₈	—	31	2	no	—
110	#117	—	2	1	no	—
111	#117	—	2	1	no	—
112	#117	—	31	2	no	—
113	#117	—	31	2	no	—
114	#117	—	62	1	no	—
115	#117	—	62	1	no	—
116	#117	C₂×D₃₁	1	2	yes	—
117	#117	C₂×D₃₁	1	2	yes	—
118	C₂×C₄⋊C₄	—	31	2	no	—
119	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
120	C₂×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₂²⋊C₄	—	62	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	—	62	1	no	—
124	#107	—	31	8	no	—
125	#106	—	31	8	no	—
126	C₃₁	#2911	1	1	yes	characteristic, complemented, Sylow
127	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	62	2	no	—
130	C₂²×C₄	Dic₆₂	1	4	yes	—
131	C₂²×C₄	C₄×D₃₁	1	4	yes	—
132	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
138	M₁₆	—	31	8	no	—
139	M₁₆	—	31	8	no	—
140	C₂×C₈	—	31	8	no	—
141	C₂×C₈	—	31	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	31	4	no	—
144	C₄⋊C₄	—	62	2	no	—
145	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	62	2	no	—
149	C₂²⋊C₄	—	62	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	62	2	no	—
153	C₄²	—	62	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	Dic₃₁⋊C₄	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	62	4	no	—
162	C₂×C₄	—	124	4	no	—
163	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
164	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
165	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
167	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
168	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
169	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
170	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
176	C₈	—	31	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
183	C₂²	C₂×Dic₃₁⋊C₄	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	124	4	no	—
187	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
188	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
189	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
190	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
191	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×Dic₃₁⋊C₄	1	1	yes	characteristic
196	C₂	#14521847	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	#2911
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14521092	C₂	semidirect
4. #14521089	C₂	semidirect
5. #14521085	C₂	semidirect
22. C₂×#92474	C₄	semidirect
26. #11758396	C₂²	semidirect
27. #11758395	C₂²	semidirect
51. #92474	C₂×C₄	semidirect
52. #92474	C₂×C₄	semidirect
53. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
126. C₃₁	#2911	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#92474	2
C₂×#11758503	2
#14521092	2
#14521089	2
#14521085	2
C₂×C₃₁×#117	2
#2911	31