Subgroups — Q₈×C₁₃×#97 #15706861

Order 3328.

954 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₁₃	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1770	C₂	1	6	yes	maximal
4	C₁₃×#1767	C₂	1	3	yes	maximal
5	C₄×C₁₃×#97	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₁₃	C₄	1	16	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	6	yes	—
10	C₂×C₁₃×#97	C₂²	1	3	yes	—
11	C₂×C₁₃×#97	C₂²	1	6	yes	—
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	6	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	6	yes	—
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
16	C₄³×C₁₃	C₂²	1	3	yes	—
17	C₁₃×#336	C₂²	1	6	yes	—
18	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
21	C₄×Q₈×C₁₃	—	2	8	no	—
22	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	C₂×C₄	1	24	yes	—
31	C₁₃×#97	—	2	12	no	—
32	C₁₃×#97	—	2	6	no	—
33	C₁₃×#97	Q₈	1	4	yes	direct product factor
34	C₄×C₈×C₁₃	C₂×C₄	1	24	yes	—
35	C₁₃×#95	C₂×C₄	1	6	yes	—
36	Q₈×#97	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
38	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	2	6	no	—
45	C₂×C₈×C₁₃	—	2	6	no	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
49	C₂×C₈×C₁₃	#41	1	4	yes	—
50	C₁₃×C₄⋊C₄	—	2	12	no	—
51	C₁₃×C₄⋊C₄	C₄²	1	12	yes	—
52	C₄²×C₁₃	—	2	12	no	—
53	C₄²×C₁₃	—	2	6	no	—
54	C₄²×C₁₃	—	4	6	no	—
55	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
56	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₁₃	#41	1	2	yes	—
58	C₂×C₈×Q₈	C₂×C₁₃	1	2	yes	—
59	C₄²×Q₈	C₂×C₁₃	1	1	yes	characteristic
60	#1770	C₂×C₁₃	1	6	yes	—
61	#1767	C₂×C₁₃	1	3	yes	—
62	C₄×#97	C₂×C₁₃	1	3	yes	—
63	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
64	Q₈×C₁₃	#97	1	16	yes	direct product factor
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	6	no	—
68	C₂×C₄×C₁₃	—	2	6	no	—
69	C₂×C₄×C₁₃	—	2	6	no	—
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
72	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
73	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
74	C₂×C₄×C₁₃	C₂×M₁₆	1	12	yes	—
75	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
79	C₈×C₁₃	—	2	12	no	—
80	C₈×C₁₃	C₄×Q₈	1	8	yes	complemented
81	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
82	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
83	C₈×Q₈	C₄×C₁₃	1	16	yes	complemented
84	C₂×C₄⋊C₈	C₂²×C₁₃	1	6	yes	—
85	C₂×#97	C₂²×C₁₃	1	3	yes	—
86	C₂×#97	C₂²×C₁₃	1	6	yes	—
87	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₁₃	1	6	yes	—
89	C₄×C₄⋊C₄	C₄×C₁₃	1	6	yes	—
90	C₄×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
91	C₄³	C₂²×C₁₃	1	3	yes	—
92	#336	C₂²×C₁₃	1	6	yes	—
93	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
94	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
96	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
97	C₂²×C₁₃	#568	1	2	yes	—
98	C₄×C₁₃	—	2	6	no	—
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	—	4	6	no	—
101	C₄×C₁₃	C₂×#97	1	12	yes	—
102	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
103	C₄×C₁₃	C₄×#41	1	2	yes	—
104	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
105	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
106	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
107	C₄×Q₈	—	2	8	no	—
108	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
109	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	6	yes	—
110	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	3	yes	—
111	C₂×C₄²	C₂×C₄×C₁₃	1	6	yes	—
112	C₂×C₄²	C₂×C₄×C₁₃	1	6	yes	—
113	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
114	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
115	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
116	C₄⋊C₈	C₂×C₄×C₁₃	1	24	yes	—
117	#97	—	2	12	no	—
118	#97	—	2	6	no	—
119	#97	Q₈×C₁₃	1	4	yes	direct product factor
120	C₄×C₈	C₂×C₄×C₁₃	1	24	yes	—
121	#95	C₂×C₄×C₁₃	1	6	yes	—
122	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
123	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
124	C₂×C₁₃	#2875	1	1	yes	characteristic
125	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
126	C₂×C₁₃	#2971	1	2	yes	—
127	C₂×Q₈	C₄²×C₁₃	1	4	yes	—
128	C₂×Q₈	C₁₃×M₁₆	1	8	yes	—
129	C₂²×C₄	C₂²×C₄×C₁₃	1	6	yes	—
130	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
131	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
132	C₂²×C₄	C₂²×C₄×C₁₃	1	3	yes	—
133	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
134	C₂×C₈	—	2	6	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	C₂²×C₄×C₁₃	1	12	yes	—
137	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
138	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
139	C₂×C₈	C₁₃×#41	1	4	yes	—
140	C₄⋊C₄	—	2	12	no	—
141	C₄⋊C₄	C₄²×C₁₃	1	12	yes	—
142	C₄²	—	2	12	no	—
143	C₄²	—	2	6	no	—
144	C₄²	—	4	6	no	—
145	C₄²	C₂²×C₄×C₁₃	1	12	yes	—
146	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
147	C₄²	C₁₃×#41	1	2	yes	—
148	C₁₃	Q₈×#97	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
150	Q₈	C₁₃×#97	1	16	yes	direct product factor
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	—	2	6	no	—
157	C₂×C₄	C₂×C₄²×C₁₃	1	6	yes	—
158	C₂×C₄	C₂×C₄²×C₁₃	1	6	yes	—
159	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
160	C₂×C₄	C₂×C₁₃×M₁₆	1	12	yes	—
161	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
162	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
164	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
165	C₈	—	2	12	no	—
166	C₈	C₄×Q₈×C₁₃	1	8	yes	complemented
167	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
168	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
169	C₂²	C₄×C₁₃×#41	1	1	yes	characteristic
170	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
171	C₂²	C₁₃×#568	1	2	yes	—
172	C₄	—	2	6	no	—
173	C₄	—	2	4	no	—
174	C₄	—	4	6	no	—
175	C₄	C₂×C₁₃×#97	1	12	yes	—
176	C₄	C₂×C₄×Q₈×C₁₃	1	2	yes	—
177	C₄	C₄×C₁₃×#41	1	2	yes	—
178	C₂	C₄²×Q₈×C₁₃	1	1	yes	characteristic
179	C₂	C₂²×C₁₃×#97	1	1	yes	characteristic
180	C₂	C₁₃×#2875	1	1	yes	characteristic
181	C₂	Q₈×C₁₃×M₁₆	1	2	yes	—
182	C₂	C₁₃×#2971	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₃	C₄	semidirect
33. C₁₃×#97	Q₈	direct (N × H)
36. Q₈×#97	C₁₃	direct (N × H)
64. Q₈×C₁₃	#97	direct (N × H)
80. C₈×C₁₃	C₄×Q₈	semidirect
83. C₈×Q₈	C₄×C₁₃	semidirect
119. #97	Q₈×C₁₃	direct (N × H)
148. C₁₃	Q₈×#97	direct (N × H)
150. Q₈	C₁₃×#97	direct (N × H)
166. C₈	C₄×Q₈×C₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₁₃	2
C₄²×Q₈×C₁₃	2
C₁₃×#1770	2
C₁₃×#1767	2
C₄×C₁₃×#97	2
Q₈×#97	13

Subgroups — Q8×C13×#97 #15706861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₃×#97 #15706861