Subgroups — #16807003

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719162	C₂	1	2	yes	maximal
2	#13131171	C₂	1	6	yes	maximal, complemented
3	#13131168	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#6092	C₂	1	3	yes	maximal, complemented
5	C₄×#88202	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₂×#88202	C₄	1	1	yes	characteristic
8	C₂×#88202	C₂²	1	1	yes	characteristic
9	#11719162	—	2	8	no	—
10	C₂×#88106	C₂²	1	6	yes	—
11	C₂²×#6092	C₂²	1	6	yes	—
12	#11718658	C₄	1	3	yes	—
13	#11718658	C₂²	1	3	yes	—
14	C₄×#6099	C₄	1	3	yes	—
15	C₂×C₄×#1055	C₂²	1	3	yes	—
16	C₄⋊#6092	C₂²	1	1	yes	characteristic
17	C₄⋊#6092	C₂²	1	3	yes	—
18	C₄⋊#6092	C₂²	1	6	yes	—
19	C₄⋊#6092	C₂²	1	6	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
23	C₄×D₄×C₁₃	C₈	1	3	yes	—
24	C₄×D₄×C₁₃	C₈	1	3	yes	—
25	C₁₃×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
27	#88202	—	2	4	no	—
28	#88202	D₄	1	4	yes	—
29	#88202	Q₈	1	4	yes	—
30	C₂×#6099	C₂×C₄	1	3	yes	—
31	C₂×#6099	C₂×C₄	1	3	yes	—
32	C₂²×#1055	C₂×C₄	1	3	yes	—
33	C₂²×#1055	C₂³	1	3	yes	—
34	#88106	—	2	12	no	—
35	C₂×#6092	—	2	12	no	—
36	C₂×#6092	C₂³	1	2	yes	—
37	C₂×#6092	C₂³	1	6	yes	—
38	#88043	C₂×C₄	1	6	yes	—
39	C₄×#1055	C₂×C₄	1	3	yes	—
40	C₄×#1055	C₂×C₄	1	3	yes	—
41	C₄×#1055	C₂³	1	3	yes	—
42	C₄×#1055	C₂³	1	1	yes	characteristic
43	#33398	—	13	1	no	maximal, Sylow
44	C₁₃×#41	C₄⋊C₄	1	4	yes	—
45	C₁₃×#41	M₁₆	1	4	yes	—
46	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
52	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
54	C₄²×C₁₃	C₂×C₈	1	3	yes	—
55	C₄²×C₁₃	C₂×C₈	1	3	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
58	#6099	—	2	6	no	—
59	#6099	C₄⋊C₄	1	12	yes	—
60	C₂×#1055	—	2	6	no	—
61	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
62	C₂×#1055	C₂²×C₄	1	3	yes	—
63	C₂×#1055	C₂²×C₄	1	3	yes	—
64	C₂×#1055	C₂×D₄	1	6	yes	—
65	C₂×#1055	C₂×Q₈	1	6	yes	—
66	C₂×#1055	C₂⁴	1	1	yes	characteristic
67	#6092	—	2	2	no	—
68	#6092	—	2	6	no	—
69	C₂×#504	—	13	2	no	—
70	C₄×#131	—	13	1	no	—
71	#2151	—	13	6	no	—
72	#2150	—	13	3	no	—
73	C₂×C₄⋊C₁₆	—	13	3	no	—
74	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
75	Q₈×C₁₃	C₄⋊C₈	1	4	yes	—
76	D₄×C₁₃	C₄⋊C₈	1	12	yes	—
77	C₂×C₄×C₁₃	—	2	6	no	—
78	C₂×C₄×C₁₃	—	4	3	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	#1055	—	2	1	no	—
88	#1055	—	2	3	no	—
89	#1055	C₂×C₄⋊C₄	1	6	yes	—
90	#1055	C₂²×D₄	1	1	yes	characteristic
91	#1055	C₂²×Q₈	1	1	yes	characteristic
92	C₂×#131	—	13	1	no	—
93	C₂×#131	—	13	1	no	—
94	C₄×#41	#261	1	1	yes	characteristic
95	#504	—	26	8	no	—
96	C₂×M₃₂	—	13	6	no	—
97	C₂²×C₁₆	—	13	6	no	—
98	#405	—	13	3	no	—
99	#405	—	13	3	no	—
100	C₄×M₁₆	—	13	3	no	—
101	C₂×C₄×C₈	—	13	3	no	—
102	C₄⋊C₁₆	—	13	1	no	—
103	C₄⋊C₁₆	—	13	3	no	—
104	C₄⋊C₁₆	—	13	6	no	—
105	C₄⋊C₁₆	—	13	6	no	—
106	C₂²×C₁₃	—	2	3	no	—
107	C₂²×C₁₃	—	2	3	no	—
108	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
109	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
110	C₄×C₁₃	—	2	3	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	3	no	—
113	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
114	C₄×C₁₃	#504	1	2	yes	complemented
115	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×#41	C₂×#261	1	1	yes	characteristic
118	#131	—	13	4	no	—
119	#131	—	13	4	no	—
120	#131	—	26	4	no	—
121	C₂×M₁₆	—	13	3	no	—
122	C₂×M₁₆	—	13	3	no	—
123	C₂²×C₈	—	13	3	no	—
124	C₂²×C₈	—	13	3	no	—
125	C₄×Q₈	#1057	1	1	yes	characteristic
126	C₄×Q₈	#1057	1	1	yes	characteristic
127	C₄×D₄	#1057	1	3	yes	—
128	C₄×D₄	#1057	1	3	yes	—
129	#117	C₂×#261	1	3	yes	—
130	C₂×C₄²	C₂×#261	1	3	yes	—
131	M₃₂	—	26	12	no	—
132	C₂×C₁₆	—	13	2	no	—
133	C₂×C₁₆	—	13	6	no	—
134	C₂×C₁₆	—	26	12	no	—
135	#97	—	13	6	no	—
136	C₄×C₈	—	13	1	no	—
137	C₄×C₈	—	13	3	no	—
138	C₄×C₈	—	13	3	no	—
139	C₄×C₈	—	13	3	no	—
140	C₂×C₁₃	—	2	6	no	—
141	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
143	C₂×C₁₃	#3407	1	1	yes	characteristic
144	#41	#6118	1	4	yes	—
145	#41	C₄⋊#261	1	4	yes	—
146	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
147	C₂×D₄	C₂×#1057	1	3	yes	—
148	C₂²×C₄	C₂×#1057	1	3	yes	—
149	C₂²×C₄	C₂×#1057	1	3	yes	—
150	C₂²×C₄	C₂²×#261	1	3	yes	—
151	M₁₆	—	13	12	no	—
152	M₁₆	—	26	6	no	—
153	C₂×C₈	—	13	6	no	—
154	C₂×C₈	—	13	6	no	—
155	C₂×C₈	—	13	1	no	—
156	C₂×C₈	—	13	3	no	—
157	C₂×C₈	—	13	1	no	—
158	C₂×C₈	—	13	3	no	—
159	C₂×C₈	—	26	6	no	—
160	C₄⋊C₄	C₂×#1057	1	3	yes	—
161	C₄⋊C₄	C₂×#1057	1	3	yes	—
162	C₂²⋊C₄	C₂×#1057	1	6	yes	—
163	C₄²	C₂×#1057	1	3	yes	—
164	C₄²	C₂×#1057	1	3	yes	—
165	C₄²	C₂²×#261	1	1	yes	characteristic
166	C₄²	C₂²×#261	1	3	yes	—
167	C₁₆	—	26	2	no	—
168	C₁₆	—	26	6	no	—
169	C₁₃	#33398	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂²×#1057	1	3	yes	—
171	Q₈	C₄⋊#1057	1	4	yes	—
172	D₄	C₄⋊#1057	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×#6118	1	6	yes	—
176	C₂×C₄	C₂×C₄⋊#261	1	6	yes	—
177	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
178	C₂×C₄	C₂²×#1057	1	3	yes	—
179	C₂×C₄	C₂²×#1057	1	3	yes	—
180	C₂×C₄	C₂²×#1057	1	3	yes	—
181	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
182	C₂×C₄	C₂³×#261	1	1	yes	characteristic
183	C₈	—	13	1	no	—
184	C₈	—	13	6	no	—
185	C₈	—	13	1	no	—
186	C₈	—	26	1	no	—
187	C₈	—	26	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₄⋊#1057	1	6	yes	—
191	C₂²	C₂³×#1057	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₄⋊#1057	1	6	yes	—
196	C₄	#11719162	1	2	yes	complemented
197	C₄	C₂²×#6118	1	1	yes	characteristic
198	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₄⋊#1057	1	1	yes	characteristic
201	C₂	C₂×#11719162	1	1	yes	characteristic
202	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	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#33398
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
2. #13131171	C₂	semidirect
3. #13131168	C₂	semidirect
4. C₂×C₄⋊#6092	C₂	semidirect
114. C₄×C₁₃	#504	semidirect
169. C₁₃	#33398	semidirect
196. C₄	#11719162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719162	2
#13131171	2
#13131168	2
C₂×C₄⋊#6092	2
C₄×#88202	2
#33398	13