Subgroups — #16854319

Order 3968.

440 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521815	C₂	1	2	yes	maximal
2	C₂×#11758478	C₂	1	1	yes	maximal, characteristic
3	#14521097	C₂	1	2	yes	maximal, complemented
4	#14521096	C₂	1	4	yes	maximal, complemented
5	C₂×#11758407	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₃₁	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#92495	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92474	C₂²	1	2	yes	—
12	C₂×#92474	C₂²	1	1	yes	characteristic
13	#11758484	—	2	4	no	—
14	M₁₆×D₃₁	C₂²	1	8	yes	—
15	#11758478	—	2	4	no	—
16	#11758478	C₄	1	4	yes	—
17	#11758478	C₄	1	4	yes	—
18	C₂×#92469	C₂²	1	2	yes	—
19	C₂×C₈×D₃₁	C₂²	1	2	yes	—
20	C₃₁×#108	C₂²	1	2	yes	—
21	C₃₁×#108	C₂²	1	2	yes	—
22	#11758410	C₂²	1	8	yes	—
23	#11758407	C₂²	1	2	yes	—
24	#11758407	C₂²	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₃₁	Q₈	1	2	yes	—
27	#92495	C₂×C₄	1	2	yes	—
28	#92495	C₂×C₄	1	2	yes	—
29	#92495	D₄	1	2	yes	—
30	#92495	D₄	1	2	yes	—
31	C₂×D₁₂₄	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₃₁	C₂³	1	2	yes	—
33	C₂×Dic₆₂	Q₈	1	1	yes	characteristic
34	M₁₆×C₃₁	—	2	2	no	—
35	M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×C₈×C₃₁	—	2	2	no	—
37	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	#92474	—	2	2	no	—
42	#92474	C₂×C₄	1	2	yes	—
43	#92474	C₂×C₄	1	2	yes	—
44	#92474	C₂³	1	4	yes	—
45	C₂×C₃₁⋊C₈	—	2	2	no	—
46	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
47	#92469	—	2	8	no	—
48	#92469	C₂×C₄	1	8	yes	—
49	C₈×D₃₁	—	2	8	no	—
50	C₈×D₃₁	C₂×C₄	1	8	yes	—
51	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₃₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₃₁	C₄⋊C₄	1	4	yes	—
60	C₂²⋊D₃₁	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₃₁	C₂×Q₈	1	2	yes	—
62	D₁₂₄	C₄⋊C₄	1	2	yes	—
63	D₁₂₄	C₄⋊C₄	1	2	yes	—
64	C₄×D₃₁	C₂²×C₄	1	4	yes	—
65	C₄×D₃₁	C₂×D₄	1	4	yes	—
66	Dic₆₂	C₄⋊C₄	1	2	yes	—
67	Dic₆₂	C₄⋊C₄	1	2	yes	—
68	C₈×C₃₁	—	2	4	no	—
69	C₈×C₃₁	C₂²×C₄	1	2	yes	—
70	C₈×C₃₁	C₂²×C₄	1	2	yes	—
71	C₃₁⋊C₈	—	2	4	no	—
72	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
73	#2916	—	31	1	no	maximal, Sylow
74	C₂²×C₃₁	—	4	1	no	—
75	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
78	C₂×D₃₁	—	4	2	no	—
79	C₂×D₃₁	C₂×C₄⋊C₄	1	4	yes	—
80	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	Dic₃₁	C₂×C₄⋊C₄	1	4	yes	—
85	#568	—	31	2	no	—
86	C₂×#131	—	31	1	no	—
87	#430	—	31	2	no	—
88	#430	—	31	4	no	—
89	C₂×#108	—	31	1	no	—
90	C₂×#108	—	31	4	no	—
91	C₂×#108	D₃₁	1	1	yes	characteristic, complemented
92	C₂×C₃₁	—	2	1	no	—
93	C₂×C₃₁	—	2	1	no	—
94	C₂×C₃₁	—	2	1	no	—
95	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	D₃₁	—	2	4	no	—
97	C₂×#41	—	31	1	no	—
98	#131	—	31	4	no	—
99	#131	—	31	4	no	—
100	#131	—	62	4	no	—
101	#131	—	62	4	no	—
102	C₂×M₁₆	—	31	1	no	—
103	C₂×M₁₆	—	31	2	no	—
104	C₂×M₁₆	—	31	2	no	—
105	C₂×M₁₆	—	31	8	no	—
106	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
107	C₂²×C₈	—	31	2	no	—
108	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
109	#108	—	31	8	no	—
110	#108	—	31	2	no	—
111	#108	—	31	2	no	—
112	#108	C₂×D₃₁	1	2	yes	—
113	#108	C₂×D₃₁	1	2	yes	—
114	C₃₁	#2916	1	1	yes	characteristic, complemented, Sylow
115	#41	—	31	2	no	—
116	#41	—	31	2	no	—
117	#41	—	31	2	no	—
118	#41	—	31	2	no	—
119	C₂×Q₈	—	31	1	no	—
120	C₂×D₄	—	31	2	no	—
121	C₂×D₄	—	31	1	no	—
122	C₂²×C₄	—	31	2	no	—
123	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	31	2	no	—
126	M₁₆	—	31	2	no	—
127	M₁₆	—	31	8	no	—
128	M₁₆	—	31	4	no	—
129	M₁₆	—	62	8	no	—
130	M₁₆	—	62	2	no	—
131	M₁₆	C₂²×D₃₁	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	31	2	no	—
134	C₂×C₈	—	31	8	no	—
135	C₂×C₈	—	62	8	no	—
136	C₂×C₈	—	62	2	no	—
137	C₂×C₈	C₄×D₃₁	1	2	yes	—
138	C₂×C₈	C₄×D₃₁	1	2	yes	—
139	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
141	C₂³	—	31	2	no	—
142	C₂³	Q₈×D₃₁	1	1	yes	characteristic
143	Q₈	—	31	2	no	—
144	Q₈	—	31	2	no	—
145	D₄	—	31	4	no	—
146	D₄	—	31	4	no	—
147	D₄	—	31	2	no	—
148	D₄	—	31	2	no	—
149	C₂×C₄	—	31	2	no	—
150	C₂×C₄	—	31	4	no	—
151	C₂×C₄	—	31	4	no	—
152	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
154	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
155	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	31	4	no	—
160	C₈	—	62	4	no	—
161	C₈	C₂×C₄×D₃₁	1	2	yes	—
162	C₈	C₂×C₄×D₃₁	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	31	4	no	—
165	C₂²	—	124	2	no	—
166	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₃₁	1	1	yes	characteristic
169	C₄	—	31	4	no	—
170	C₄	C₄⋊C₄×D₃₁	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₃₁	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
173	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	62	4	no	—
178	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#108	C₂
socle	C₂×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#108	D₃₁
hypocenter	C₃₁	#2916
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #14521097	C₂	semidirect
4. #14521096	C₂	semidirect
5. C₂×#11758407	C₂	semidirect
6. #108×D₃₁	C₂	semidirect
7. C₂×C₃₁×#108	C₂	semidirect
91. C₂×#108	D₃₁	semidirect
114. C₃₁	#2916	semidirect

Maximal subgroups

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

subgroup	index
#14521815	2
C₂×#11758478	2
#14521097	2
#14521096	2
C₂×#11758407	2
#108×D₃₁	2
C₂×C₃₁×#108	2
#2916	31