Subgroups — #16807309

Order 3328.

548 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	Q₈×#6092	C₂	1	2	yes	maximal, complemented
2	#13131220	C₂	1	6	yes	maximal, complemented
3	#13131197	C₂	1	3	yes	maximal, complemented
4	C₄×#88106	C₂	1	3	yes	maximal, complemented
5	#41×#1055	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	6	yes	—
8	Q₈×#1055	C₄	1	1	yes	characteristic
9	Q₈×#1055	C₂²	1	1	yes	characteristic
10	D₄×#1055	C₄	1	3	yes	—
11	D₄×#1055	C₂²	1	3	yes	—
12	#11718586	C₄	1	3	yes	—
13	C₂×C₄×#1055	C₂²	1	3	yes	—
14	C₂²⋊#6092	C₂²	1	6	yes	—
15	#11718231	C₂²	1	1	yes	characteristic
16	#11718231	C₂²	1	3	yes	—
17	C₄⋊#6092	C₂²	1	6	yes	—
18	C₄×#6092	C₂²	1	6	yes	—
19	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₄×C₁₃	C₈	1	3	yes	—
23	C₄×D₄×C₁₃	C₂×C₄	1	3	yes	—
24	C₁₃×#117	C₈	1	3	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	C₂²×#1055	C₂×C₄	1	3	yes	—
27	C₂²×#1055	C₂³	1	3	yes	—
28	#88106	—	2	12	no	—
29	C₂×#6092	C₂³	1	6	yes	—
30	C₂×#6092	C₂³	1	2	yes	—
31	C₂²⋊#1055	C₂×C₄	1	6	yes	—
32	C₄⋊#1055	C₂×C₄	1	3	yes	—
33	C₄⋊#1055	C₂×C₄	1	3	yes	—
34	C₄×#1055	C₂×C₄	1	3	yes	—
35	C₄×#1055	C₂×C₄	1	3	yes	—
36	C₄×#1055	C₂³	1	3	yes	—
37	C₄×#1055	C₂³	1	1	yes	characteristic
38	#33485	—	13	1	no	maximal, Sylow
39	C₁₃×#41	—	2	4	no	—
40	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
47	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
48	C₄²×C₁₃	C₂×C₈	1	3	yes	—
49	C₄²×C₁₃	C₂×C₈	1	3	yes	—
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
52	C₂×#1055	—	2	6	no	—
53	C₂×#1055	—	2	6	no	—
54	C₂×#1055	C₂²×C₄	1	3	yes	—
55	C₂×#1055	C₂²×C₄	1	3	yes	—
56	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
57	C₂×#1055	C₂⁴	1	1	yes	characteristic
58	#6092	—	2	6	no	—
59	#6092	#41	1	4	yes	complemented
60	C₈×#41	—	13	1	no	—
61	Q₈×C₁₆	—	13	2	no	—
62	#2169	—	13	6	no	—
63	#2163	—	13	3	no	—
64	C₄×M₃₂	—	13	3	no	—
65	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
66	Q₈×C₁₃	M₃₂	1	4	yes	complemented
67	D₄×C₁₃	—	2	6	no	—
68	C₂×C₄×C₁₃	—	2	3	no	—
69	C₂×C₄×C₁₃	—	2	3	no	—
70	C₂×C₄×C₁₃	—	2	6	no	—
71	C₂×C₄×C₁₃	—	4	3	no	—
72	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	#1055	—	2	3	no	—
79	#1055	—	2	3	no	—
80	#1055	#131	1	2	yes	—
81	#1055	C₂×#41	1	2	yes	—
82	C₄×#41	#261	1	1	yes	characteristic
83	C₂×M₃₂	—	13	6	no	—
84	C₈×Q₈	—	13	1	no	—
85	C₈×Q₈	—	13	1	no	—
86	C₈×D₄	—	13	3	no	—
87	C₈×D₄	—	13	3	no	—
88	#431	—	13	3	no	—
89	C₂×C₄×C₈	—	13	3	no	—
90	C₄⋊C₁₆	—	13	6	no	—
91	C₂²⋊C₁₆	—	13	6	no	—
92	#346	—	13	1	no	—
93	#346	—	13	3	no	—
94	C₄×C₁₆	—	13	6	no	—
95	C₂²×C₁₃	—	2	3	no	—
96	C₂²×C₁₃	—	2	3	no	—
97	C₂²×C₁₃	—	2	3	no	—
98	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	3	no	—
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
104	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
105	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
106	C₂×#41	#1057	1	1	yes	characteristic
107	C₂²×C₈	—	13	3	no	—
108	C₂²×C₈	—	13	3	no	—
109	C₄×Q₈	#1057	1	1	yes	characteristic
110	C₄×Q₈	C₂×#261	1	1	yes	characteristic
111	C₄×D₄	#1057	1	3	yes	—
112	C₄×D₄	C₂×#261	1	3	yes	—
113	#117	#1057	1	3	yes	—
114	C₂×C₄²	C₂×#261	1	3	yes	—
115	M₃₂	—	26	12	no	—
116	C₂×C₁₆	—	13	2	no	—
117	C₂×C₁₆	—	13	6	no	—
118	C₄⋊C₈	—	13	3	no	—
119	C₄⋊C₈	—	13	3	no	—
120	C₂²⋊C₈	—	13	6	no	—
121	C₄×C₈	—	13	3	no	—
122	C₄×C₈	—	13	1	no	—
123	C₄×C₈	—	13	3	no	—
124	C₄×C₈	—	13	3	no	—
125	C₂×C₁₃	—	4	3	no	—
126	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
127	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
128	C₂×C₁₃	#3407	1	1	yes	characteristic
129	#41	—	2	4	no	—
130	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
131	C₂×D₄	C₂×#1057	1	3	yes	—
132	C₂²×C₄	C₂×#1057	1	3	yes	—
133	C₂²×C₄	C₂×#1057	1	3	yes	—
134	C₂²×C₄	C₂²×#261	1	3	yes	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	1	no	—
137	C₂×C₈	—	13	3	no	—
138	C₂×C₈	—	13	3	no	—
139	C₂×C₈	—	26	6	no	—
140	C₂×C₈	—	26	6	no	—
141	C₄⋊C₄	C₂×#1057	1	3	yes	—
142	C₄⋊C₄	C₂×#1057	1	3	yes	—
143	C₂²⋊C₄	C₂×#1057	1	6	yes	—
144	C₄²	C₂×#1057	1	3	yes	—
145	C₄²	C₂×#1057	1	3	yes	—
146	C₄²	C₂²×#261	1	1	yes	characteristic
147	C₄²	C₂²×#261	1	3	yes	—
148	C₁₆	—	13	4	no	—
149	C₁₆	—	26	6	no	—
150	C₁₃	#33485	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂²×#1057	1	3	yes	—
152	Q₈	#88106	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
159	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
160	C₂×C₄	C₂²×#1057	1	3	yes	—
161	C₂×C₄	C₂²×#1057	1	3	yes	—
162	C₂×C₄	C₂²×#1057	1	3	yes	—
163	C₂×C₄	C₂³×#261	1	1	yes	characteristic
164	C₈	—	13	2	no	—
165	C₈	—	13	2	no	—
166	C₈	—	26	3	no	—
167	C₈	—	26	3	no	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×#1057	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	3	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#88106	1	6	yes	—
177	C₄	D₁₃⋊#131	1	1	yes	characteristic
178	C₄	#41×#261	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×#1057	1	1	yes	characteristic
181	C₂	C₂²×#88106	1	1	yes	characteristic
182	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×#41	C₄
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#33485
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#6092	C₂	semidirect
2. #13131220	C₂	semidirect
3. #13131197	C₂	semidirect
4. C₄×#88106	C₂	semidirect
59. #6092	#41	semidirect
66. Q₈×C₁₃	M₃₂	semidirect
150. C₁₃	#33485	semidirect
152. Q₈	#88106	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6092	2
#13131220	2
#13131197	2
C₄×#88106	2
#41×#1055	2
#33485	13