Subgroups — C₂×#108×D₃₁ #16854312

Order 3968.

480 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#108	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×D₃₁	Q₈
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#108	D₃₁
hypocenter	C₃₁	C₂²×#108
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#11758407	C₂	semidirect
4. C₂×#11758410	C₂	semidirect
5. #108×D₃₁	C₂	direct (N × H)
6. C₂×C₃₁×#108	C₂	semidirect
20. C₃₁×#108	C₂²	semidirect
21. #11758410	C₂²	semidirect
22. #11758407	C₂²	semidirect
86. C₂×D₃₁	#108	direct (N × H)
100. C₂×#108	D₃₁	direct (N × H)
103. C₂×C₃₁	C₂×#108	semidirect
106. D₃₁	C₂×#108	direct (N × H)
122. #108	C₂×D₃₁	direct (N × H)
123. C₃₁	C₂²×#108	semidirect
199. C₂	#108×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₃₁	2
C₂²×C₈×D₃₁	2
C₂×#11758407	2
C₂×#11758410	2
#108×D₃₁	2
C₂×C₃₁×#108	2
C₂²×#108	31

Subgroups — C2×#108×D31 #16854312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#108×D₃₁ #16854312