Subgroups — #108×#1055 #15754213

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#1055	C₂	1	2	yes	maximal
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	#13112674	C₂	1	2	yes	maximal
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	2	yes	—
7	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	#11718586	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#341	C₄	1	2	yes	—
13	#11718050	C₄	1	2	yes	—
14	#11718049	C₄	1	2	yes	—
15	#11718047	—	2	2	no	—
16	C₈×#1055	—	2	2	no	—
17	C₈×#1055	C₄	1	2	yes	—
18	C₈×#1055	C₄	1	2	yes	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	1	no	—
24	C₂²×#1055	Q₈	1	2	yes	—
25	C₁₃×#108	C₈	1	8	yes	complemented
26	C₁₃×#97	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊#1055	—	2	2	no	—
33	C₄⋊#1055	C₂×C₄	1	4	yes	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	C₂×C₄	1	2	yes	—
37	C₄×#1055	C₂×C₄	1	2	yes	—
38	C₄×#1055	D₄	1	1	yes	characteristic
39	C₄×#1055	D₄	1	1	yes	characteristic
40	C₈×#108	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₁₃×M₁₆	—	2	4	no	—
45	C₁₃×M₁₆	C₂×C₈	1	8	yes	—
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₃	#41	1	1	yes	characteristic
57	C₂×#1055	—	2	4	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	2	4	no	—
61	C₂×#1055	C₄⋊C₄	1	2	yes	—
62	C₂×#1055	C₄⋊C₄	1	2	yes	—
63	C₄×#108	D₁₃	1	1	yes	characteristic
64	#1564	—	13	1	no	—
65	C₈×M₁₆	—	13	2	no	—
66	C₂×C₈²	—	13	1	no	—
67	#1278	—	13	2	no	—
68	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₈×C₁₃	—	2	1	no	—
80	C₈×C₁₃	—	2	1	no	—
81	C₈×C₁₃	—	2	8	no	—
82	C₈×C₁₃	C₄×C₈	1	2	yes	—
83	C₈×C₁₃	C₄×C₈	1	2	yes	—
84	#1055	—	2	4	no	—
85	#1055	—	2	2	no	—
86	#1055	—	4	4	no	—
87	#1055	#108	1	4	yes	direct product factor
88	#431	—	13	2	no	—
89	C₂×#108	Dic₁₃	1	2	yes	—
90	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	#341	Dic₁₃	1	2	yes	—
95	#335	—	13	2	no	—
96	#334	—	13	2	no	—
97	#322	—	26	2	no	—
98	C₈²	—	13	2	no	—
99	C₈²	—	13	2	no	—
100	C₈²	—	26	2	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₁₃	#405	1	1	yes	characteristic
109	C₄×C₁₃	C₂×#108	1	2	yes	—
110	C₄×C₁₃	C₈×D₄	1	2	yes	—
111	C₄×C₁₃	#443	1	2	yes	—
112	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
113	C₂²×C₈	—	13	2	no	—
114	C₂²×C₈	—	26	1	no	—
115	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
118	#108	#1055	1	8	yes	direct product factor
119	C₄⋊C₈	—	13	4	no	—
120	C₂²⋊C₈	—	26	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	13	2	no	—
124	C₄×C₈	—	13	2	no	—
125	C₄×C₈	—	13	1	no	—
126	C₄×C₈	—	13	1	no	—
127	C₄×C₈	—	26	2	no	—
128	C₄×C₈	—	26	1	no	—
129	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
130	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
133	C₂×C₁₃	—	2	2	no	—
134	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₁₃	#1772	1	1	yes	characteristic
136	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
137	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
138	C₂²×C₄	#6131	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	M₁₆	—	2	4	no	—
141	M₁₆	C₂×#1055	1	8	yes	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	26	4	no	—
147	C₂×C₈	—	26	4	no	—
148	C₂×C₈	—	26	2	no	—
149	C₂×C₈	—	26	4	no	—
150	C₂×C₈	C₂×#1055	1	4	yes	—
151	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
157	C₄²	D₄×D₁₃	1	1	yes	characteristic
158	C₄²	#6129	1	1	yes	characteristic
159	C₁₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
160	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	#88141	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
165	C₂×C₄	#88146	1	1	yes	characteristic
166	C₂×C₄	#88147	1	1	yes	characteristic
167	C₂×C₄	C₂²×#1055	1	2	yes	—
168	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
169	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
170	C₂×C₄	#88202	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	8	no	—
174	C₈	—	13	4	no	—
175	C₈	—	26	4	no	—
176	C₈	—	26	2	no	—
177	C₈	—	52	4	no	—
178	C₈	C₄×#1055	1	2	yes	—
179	C₈	C₄×#1055	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
183	C₂²	Q₈×#1055	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	D₁₃×#108	1	2	yes	—
187	C₄	#11718556	1	2	yes	—
188	C₄	C₂×C₄×#1055	1	1	yes	characteristic
189	C₄	#11718658	1	1	yes	characteristic
190	C₄	D₄×#1055	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	C₄⋊C₄×#1055	1	1	yes	characteristic
193	C₂	#13114718	1	1	yes	characteristic
194	C₂	#108×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	C₈×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
25. C₁₃×#108	C₈	semidirect
87. #1055	#108	direct (N × H)
118. #108	#1055	direct (N × H)
159. C₁₃	C₈×#108	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
M₁₆×#1055	2
C₂×C₈×#1055	2
#13112674	2
C₄×C₁₃×#108	2
C₈×#108	13