Subgroups — Q₈×C₁₃×#108 #15709071

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₃×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1950	C₂	1	6	yes	maximal
4	C₁₃×#1848	C₂	1	3	yes	maximal
5	C₄×C₁₃×#108	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#446	—	2	6	no	—
8	C₈×Q₈×C₁₃	—	2	2	no	—
9	C₈×Q₈×C₁₃	C₄	1	6	yes	—
10	C₈×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₁₃×#108	C₂²	1	3	yes	—
12	C₂×C₁₃×#108	C₂²	1	6	yes	—
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₁₃×#423	C₂²	1	6	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
16	C₄×C₁₃×M₁₆	C₂²	1	6	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
18	C₁₃×#341	C₂²	1	6	yes	—
19	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
21	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	D₄	1	3	yes	—
28	C₂×C₁₃×C₄⋊C₄	Q₈	1	3	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
30	C₁₃×#108	—	2	6	no	—
31	C₁₃×#108	—	4	6	no	—
32	C₁₃×#108	Q₈	1	4	yes	direct product factor
33	C₁₃×C₄⋊C₈	—	2	6	no	—
34	C₁₃×C₄⋊C₈	—	2	6	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
36	C₁₃×C₄⋊C₈	C₂×C₄	1	6	yes	—
37	C₁₃×#97	—	2	6	no	—
38	C₄×C₈×C₁₃	—	2	6	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
41	Q₈×#108	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×Q₈×C₁₃	—	2	4	no	—
43	C₂×Q₈×C₁₃	C₄⋊C₄	1	3	yes	—
44	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₁₃×M₁₆	—	2	2	no	—
49	C₁₃×M₁₆	—	2	6	no	—
50	C₁₃×M₁₆	—	2	6	no	—
51	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	3	no	—
54	C₂×C₈×C₁₃	—	2	3	no	—
55	C₂×C₈×C₁₃	—	2	6	no	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₁₃	#41	1	2	yes	—
60	C₁₃×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
61	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
62	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
65	C₄²×C₁₃	C₂×D₄	1	3	yes	—
66	C₄²×C₁₃	C₂×D₄	1	3	yes	—
67	Q₈×M₁₆	C₂×C₁₃	1	2	yes	—
68	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic
69	#1950	C₂×C₁₃	1	6	yes	—
70	#1848	C₂×C₁₃	1	3	yes	—
71	C₄×#108	C₂×C₁₃	1	3	yes	—
72	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	6	no	—
74	Q₈×C₁₃	#108	1	4	yes	direct product factor
75	C₂×C₄×C₁₃	—	2	6	no	—
76	C₂×C₄×C₁₃	—	4	3	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
86	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
87	C₈×C₁₃	—	2	6	no	—
88	C₈×C₁₃	—	2	4	no	—
89	C₈×C₁₃	—	4	6	no	—
90	C₈×C₁₃	C₄×Q₈	1	4	yes	—
91	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
92	#446	—	2	6	no	—
93	C₈×Q₈	—	2	2	no	—
94	C₈×Q₈	C₄×C₁₃	1	6	yes	—
95	C₈×Q₈	C₄×C₁₃	1	2	yes	—
96	C₂×#108	C₂²×C₁₃	1	3	yes	—
97	C₂×#108	C₂²×C₁₃	1	6	yes	—
98	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
99	#423	C₂²×C₁₃	1	6	yes	—
100	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
101	C₄×M₁₆	C₂²×C₁₃	1	6	yes	—
102	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
103	#341	C₂²×C₁₃	1	6	yes	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₃	#557	1	1	yes	characteristic
108	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
109	C₄×C₁₃	—	2	3	no	—
110	C₄×C₁₃	—	2	3	no	—
111	C₄×C₁₃	—	2	3	no	—
112	C₄×C₁₃	C₂×#108	1	6	yes	—
113	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₄×C₁₃	#520	1	1	yes	characteristic
115	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
116	C₄×C₁₃	#550	1	1	yes	characteristic
117	C₂²×Q₈	Q₈×C₁₃	1	1	yes	characteristic
118	C₂×M₁₆	C₂³×C₁₃	1	6	yes	—
119	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
120	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
121	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
122	C₄×Q₈	C₂×C₄×C₁₃	1	3	yes	—
123	C₄×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
124	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
125	C₄×Q₈	D₄×C₁₃	1	3	yes	—
126	C₂×C₄⋊C₄	Q₈×C₁₃	1	3	yes	—
127	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
128	#108	—	2	6	no	—
129	#108	—	4	6	no	—
130	#108	Q₈×C₁₃	1	4	yes	direct product factor
131	C₄⋊C₈	—	2	6	no	—
132	C₄⋊C₈	—	2	6	no	—
133	C₄⋊C₈	C₂×C₄×C₁₃	1	6	yes	—
134	C₄⋊C₈	C₂×C₄×C₁₃	1	6	yes	—
135	#97	—	2	6	no	—
136	C₄×C₈	—	2	6	no	—
137	C₄×C₈	C₂×C₄×C₁₃	1	6	yes	—
138	C₄×C₈	C₂×C₄×C₁₃	1	6	yes	—
139	C₂×C₁₃	—	2	1	no	—
140	C₂×C₁₃	—	2	1	no	—
141	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
143	C₂×C₁₃	#2916	1	1	yes	characteristic
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₁₃×C₄⋊C₄	1	3	yes	—
146	C₂×Q₈	C₁₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×Q₈×C₁₃	1	3	yes	—
148	C₂²×C₄	C₂×Q₈×C₁₃	1	3	yes	—
149	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	6	no	—
152	M₁₆	—	2	6	no	—
153	M₁₆	C₂×Q₈×C₁₃	1	4	yes	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
159	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
160	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
161	C₂×C₈	C₁₃×#41	1	2	yes	—
162	C₄⋊C₄	C₁₃×C₄⋊C₄	1	6	yes	—
163	C₄⋊C₄	C₁₃×C₄⋊C₄	1	3	yes	—
164	C₄⋊C₄	C₁₃×C₄⋊C₄	1	3	yes	—
165	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
166	C₄²	C₂²×C₄×C₁₃	1	3	yes	—
167	C₄²	C₂×D₄×C₁₃	1	3	yes	—
168	C₄²	C₂×D₄×C₁₃	1	3	yes	—
169	C₁₃	Q₈×#108	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	C₁₃×#108	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	3	yes	—
176	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	3	yes	—
177	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	3	yes	—
178	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	3	yes	—
179	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
183	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	2	4	no	—
187	C₈	—	4	6	no	—
188	C₈	C₄×Q₈×C₁₃	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
192	C₂²	C₁₃×#557	1	1	yes	characteristic
193	C₂²	Q₈²×C₁₃	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×C₁₃×#108	1	6	yes	—
198	C₄	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
199	C₄	C₁₃×#520	1	1	yes	characteristic
200	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
201	C₄	C₁₃×#550	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₁₃×C₄⋊C₄	1	1	yes	characteristic
205	C₂	C₂²×C₁₃×#108	1	1	yes	characteristic
206	C₂	C₁₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
32. C₁₃×#108	Q₈	direct (N × H)
41. Q₈×#108	C₁₃	direct (N × H)
74. Q₈×C₁₃	#108	direct (N × H)
130. #108	Q₈×C₁₃	direct (N × H)
169. C₁₃	Q₈×#108	direct (N × H)
172. Q₈	C₁₃×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₁₃×M₁₆	2
C₂×C₈×Q₈×C₁₃	2
C₁₃×#1950	2
C₁₃×#1848	2
C₄×C₁₃×#108	2
Q₈×#108	13

Subgroups — Q8×C13×#108 #15709071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₃×#108 #15709071