Subgroups — #16807401

Order 3328.

532 conjugacy classes of subgroups.

The subgroup lattice has 188 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	#13131193	C₂	1	3	yes	maximal, complemented
4	#13131168	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	C₄⋊#6092	C₂²	1	3	yes	—
16	C₄⋊#6092	C₂²	1	6	yes	—
17	C₄⋊#6092	C₂²	1	1	yes	characteristic
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	—	4	6	no	—
29	C₂×#6092	C₂³	1	2	yes	—
30	C₂×#6092	C₂³	1	6	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	#33505	—	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	3	no	—
53	C₂×#1055	—	2	3	no	—
54	C₂×#1055	—	4	3	no	—
55	C₂×#1055	C₂²×C₄	1	3	yes	—
56	C₂×#1055	C₂²×C₄	1	3	yes	—
57	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
58	C₂×#1055	C₂⁴	1	1	yes	characteristic
59	#6092	—	2	6	no	—
60	#6092	—	2	2	no	—
61	C₈×#41	—	13	1	no	—
62	Q₈×C₁₆	—	13	2	no	—
63	#2169	—	13	6	no	—
64	#2162	—	13	3	no	—
65	#2150	—	13	3	no	—
66	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
67	Q₈×C₁₃	M₃₂	1	4	yes	complemented
68	D₄×C₁₃	—	2	6	no	—
69	C₂×C₄×C₁₃	—	2	3	no	—
70	C₂×C₄×C₁₃	—	2	3	no	—
71	C₂×C₄×C₁₃	—	2	6	no	—
72	C₂×C₄×C₁₃	—	2	6	no	—
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	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
79	#1055	—	2	3	no	—
80	#1055	—	2	1	no	—
81	#1055	—	2	3	no	—
82	#1055	H(5,𝔽₂)	1	1	yes	characteristic
83	#1055	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×#41	#261	1	1	yes	characteristic
85	C₂×M₃₂	—	13	6	no	—
86	C₈×Q₈	—	13	1	no	—
87	C₈×Q₈	—	13	1	no	—
88	C₈×D₄	—	13	3	no	—
89	C₈×D₄	—	13	3	no	—
90	#431	—	13	3	no	—
91	C₂×C₄×C₈	—	13	3	no	—
92	C₄⋊C₁₆	—	13	3	no	—
93	C₄⋊C₁₆	—	13	6	no	—
94	C₄⋊C₁₆	—	13	1	no	—
95	C₂²⋊C₁₆	—	13	6	no	—
96	C₄×C₁₆	—	13	6	no	—
97	C₂²×C₁₃	—	2	3	no	—
98	C₂²×C₁₃	—	2	3	no	—
99	C₂²×C₁₃	—	2	3	no	—
100	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	3	no	—
103	C₄×C₁₃	—	2	3	no	—
104	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
105	C₄×C₁₃	#504	1	2	yes	—
106	C₄×C₁₃	#520	1	1	yes	characteristic
107	C₄×C₁₃	#568	1	1	yes	characteristic
108	C₂×#41	#1057	1	1	yes	characteristic
109	C₂²×C₈	—	13	3	no	—
110	C₂²×C₈	—	13	3	no	—
111	C₄×Q₈	#1057	1	1	yes	characteristic
112	C₄×Q₈	C₂×#261	1	1	yes	characteristic
113	C₄×D₄	#1057	1	3	yes	—
114	C₄×D₄	C₂×#261	1	3	yes	—
115	#117	#1057	1	3	yes	—
116	C₂×C₄²	C₂×#261	1	3	yes	—
117	M₃₂	—	52	6	no	—
118	C₂×C₁₆	—	13	2	no	—
119	C₂×C₁₆	—	13	6	no	—
120	C₄⋊C₈	—	13	3	no	—
121	C₄⋊C₈	—	13	3	no	—
122	C₂²⋊C₈	—	13	6	no	—
123	C₄×C₈	—	13	3	no	—
124	C₄×C₈	—	13	1	no	—
125	C₄×C₈	—	13	3	no	—
126	C₄×C₈	—	13	3	no	—
127	C₂×C₁₃	—	4	3	no	—
128	C₂×C₁₃	#2988	1	1	yes	characteristic
129	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
130	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
131	#41	—	2	4	no	—
132	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
133	C₂×D₄	C₂×#1057	1	3	yes	—
134	C₂²×C₄	C₂×#1057	1	3	yes	—
135	C₂²×C₄	C₂×#1057	1	3	yes	—
136	C₂²×C₄	C₂²×#261	1	3	yes	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	—	13	1	no	—
139	C₂×C₈	—	13	3	no	—
140	C₂×C₈	—	13	3	no	—
141	C₂×C₈	—	26	3	no	—
142	C₂×C₈	—	26	3	no	—
143	C₂×C₈	—	52	3	no	—
144	C₄⋊C₄	C₂×#1057	1	3	yes	—
145	C₄⋊C₄	C₂×#1057	1	3	yes	—
146	C₂²⋊C₄	C₂×#1057	1	6	yes	—
147	C₄²	C₂×#1057	1	3	yes	—
148	C₄²	C₂×#1057	1	3	yes	—
149	C₄²	C₂²×#261	1	1	yes	characteristic
150	C₄²	C₂²×#261	1	3	yes	—
151	C₁₆	—	26	6	no	—
152	C₁₆	—	26	2	no	—
153	C₁₃	#33505	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂²×#1057	1	3	yes	—
155	Q₈	#88106	1	4	yes	complemented
156	D₄	—	2	6	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	3	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
162	C₂×C₄	C₂²×#1057	1	3	yes	—
163	C₂×C₄	C₂²×#1057	1	3	yes	—
164	C₂×C₄	C₂²×#1057	1	3	yes	—
165	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
166	C₂×C₄	C₂³×#261	1	1	yes	characteristic
167	C₈	—	13	1	no	—
168	C₈	—	13	1	no	—
169	C₈	—	26	3	no	—
170	C₈	—	26	1	no	—
171	C₈	—	26	3	no	—
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	—	2	3	no	—
175	C₂²	C₂³×#1057	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	3	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂×#88106	1	6	yes	—
180	C₄	#11719162	1	2	yes	—
181	C₄	#11719629	1	1	yes	characteristic
182	C₄	#11719631	1	1	yes	characteristic
183	C₂	—	4	3	no	—
184	C₂	#13131751	1	1	yes	characteristic
185	C₂	C₂²×#88106	1	1	yes	characteristic
186	C₂	C₂×#11719162	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	#33505
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. #13131193	C₂	semidirect
4. #13131168	C₂	semidirect
67. Q₈×C₁₃	M₃₂	semidirect
153. C₁₃	#33505	semidirect
155. 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
#13131193	2
#13131168	2
#41×#1055	2
#33505	13