Subgroups — #16777178

Order 3328.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6092	—	2	1	no	—
5	C₂²×#6092	—	2	1	no	—
6	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	—	2	1	no	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₄⋊#6092	—	2	1	no	—
11	C₄⋊#6092	C₄	1	2	yes	complemented
12	C₄×#6092	—	2	1	no	—
13	C₄×#6092	C₄	1	2	yes	complemented
14	#11718116	—	2	1	no	—
15	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
16	C₂×C₄²×C₁₃	—	2	1	no	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×#1055	—	2	1	no	—
19	C₂²×#1055	D₄	1	1	yes	characteristic
20	C₂²×#1055	D₄	1	1	yes	characteristic
21	C₂×#6092	—	2	2	no	—
22	C₂×#6092	—	2	2	no	—
23	C₂×#6092	—	2	2	no	—
24	C₂×#6092	—	2	2	no	—
25	C₂×#6092	—	4	1	no	—
26	C₂×#6092	—	4	1	no	—
27	C₁₃×#95	—	2	1	no	—
28	C₄⋊#1055	—	2	1	no	—
29	C₄⋊#1055	C₈	1	2	yes	—
30	C₄×#1055	—	4	1	no	—
31	C₄×#1055	—	4	1	no	—
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	C₄×#1055	C₂×C₄	1	1	yes	characteristic
34	C₄×#1055	D₄	1	1	yes	characteristic
35	C₄×#1055	Q₈	1	1	yes	characteristic
36	#7167	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₁₃×C₄⋊C₄	C₁₆	1	4	yes	complemented
43	C₄²×C₁₃	—	4	1	no	—
44	C₄²×C₁₃	—	4	1	no	—
45	C₄²×C₁₃	C₄²	1	1	yes	characteristic
46	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
49	C₂×#1055	—	2	1	no	—
50	C₂×#1055	—	2	1	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	4	1	no	—
53	C₂×#1055	—	4	1	no	—
54	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#1055	D₈	1	1	yes	characteristic
59	C₂×#1055	SD₁₆	1	1	yes	characteristic
60	C₂×#1055	SD₁₆	1	1	yes	characteristic
61	C₂×#1055	Q₁₆	1	1	yes	characteristic
62	#6092	—	2	4	no	—
63	#6092	—	4	2	no	—
64	C₂×C₄⋊C₁₆	—	13	1	no	—
65	C₂×C₄×C₁₆	—	13	1	no	—
66	C₈×C₄⋊C₄	—	13	1	no	—
67	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
84	#1055	—	2	1	no	—
85	#1055	—	2	1	no	—
86	#1055	—	4	2	no	—
87	#1055	#102	1	1	yes	characteristic
88	#1055	#103	1	1	yes	characteristic
89	#1055	#106	1	1	yes	characteristic
90	#1055	#107	1	1	yes	characteristic
91	C₂²×C₁₆	—	26	1	no	—
92	C₂²×C₁₆	—	26	1	no	—
93	C₂×C₄⋊C₈	—	13	1	no	—
94	C₂×C₄×C₈	—	13	1	no	—
95	C₂×C₄×C₈	—	26	1	no	—
96	C₄×C₄⋊C₄	#261	1	1	yes	characteristic
97	C₄⋊C₁₆	—	13	2	no	—
98	C₄⋊C₁₆	—	26	1	no	—
99	C₄×C₁₆	—	13	2	no	—
100	C₄×C₁₆	—	26	1	no	—
101	#336	—	26	1	no	—
102	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₃	#336	1	1	yes	characteristic
105	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
106	C₂²×C₁₃	#350	1	2	yes	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	4	4	no	—
110	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
111	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
112	C₄×C₁₃	#339	1	1	yes	characteristic
113	C₄×C₁₃	#340	1	1	yes	characteristic
114	C₄×C₁₃	C₄×C₁₆	1	1	yes	characteristic
115	C₄×C₁₃	#346	1	1	yes	characteristic
116	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
117	C₂²×C₈	—	13	1	no	—
118	C₂²×C₈	—	13	1	no	—
119	C₂²×C₈	—	26	1	no	—
120	C₂×C₄⋊C₄	#1057	1	2	yes	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	C₂×#261	1	1	yes	characteristic
123	C₂×C₁₆	—	26	2	no	—
124	C₂×C₁₆	—	26	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₂×C₁₆	—	26	2	no	—
127	C₂×C₁₆	—	52	1	no	—
128	C₂×C₁₆	—	52	1	no	—
129	C₄⋊C₈	—	13	2	no	—
130	C₄⋊C₈	—	26	1	no	—
131	C₄×C₈	—	13	1	no	—
132	C₄×C₈	—	13	1	no	—
133	C₄×C₈	—	13	1	no	—
134	C₄×C₈	—	13	1	no	—
135	C₄×C₈	—	52	1	no	—
136	C₄×C₈	—	52	1	no	—
137	#95	—	2	1	no	—
138	C₂×C₁₃	#1279	1	1	yes	characteristic
139	C₂×C₁₃	D₄⋊C₁₆	1	2	yes	—
140	C₂×C₁₃	Q₈⋊C₁₆	1	2	yes	—
141	C₂×C₁₃	#1374	1	1	yes	characteristic
142	C₂×C₁₃	#1379	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
145	C₂²×C₄	#6122	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
147	C₂×C₈	—	13	1	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	13	1	no	—
150	C₂×C₈	—	13	1	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	1	no	—
154	C₂×C₈	—	13	1	no	—
155	C₂×C₈	—	26	1	no	—
156	C₂×C₈	—	26	1	no	—
157	C₂×C₈	—	26	2	no	—
158	C₂×C₈	—	52	1	no	—
159	C₂×C₈	—	52	1	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	#6092	1	4	yes	complemented
162	C₄²	—	4	1	no	—
163	C₄²	—	4	1	no	—
164	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
165	C₄²	#6118	1	1	yes	characteristic
166	C₄²	C₄×#261	1	1	yes	characteristic
167	C₄²	C₄⋊#261	1	1	yes	characteristic
168	C₁₆	—	26	4	no	—
169	C₁₆	—	52	2	no	—
170	C₁₃	#7167	1	1	yes	characteristic, complemented, Sylow
171	C₂³	C₂²⋊#1057	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂×#6092	1	2	yes	—
177	C₂×C₄	#88106	1	2	yes	—
178	C₂×C₄	C₄×#1057	1	1	yes	characteristic
179	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
180	C₂×C₄	#88110	1	1	yes	characteristic
181	C₂×C₄	#88111	1	1	yes	characteristic
182	C₂×C₄	#88112	1	1	yes	characteristic
183	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#88115	1	1	yes	characteristic
185	C₂×C₄	#88116	1	1	yes	characteristic
186	C₂×C₄	#88117	1	1	yes	characteristic
187	C₂×C₄	#88118	1	1	yes	characteristic
188	C₈	—	13	1	no	—
189	C₈	—	13	1	no	—
190	C₈	—	13	1	no	—
191	C₈	—	13	1	no	—
192	C₈	—	26	1	no	—
193	C₈	—	26	1	no	—
194	C₈	—	52	2	no	—
195	C₂²	C₂²⋊#6092	1	2	yes	—
196	C₂²	#11718290	1	1	yes	characteristic
197	C₂²	D₄⋊#1057	1	1	yes	characteristic
198	C₂²	Q₈⋊#1057	1	1	yes	characteristic
199	C₂²	#11718304	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	C₄×#6092	1	1	yes	characteristic
204	C₄	C₄⋊#6092	1	2	yes	—
205	C₄	#11718231	1	1	yes	characteristic
206	C₄	#11718243	1	1	yes	characteristic
207	C₄	#11718244	1	1	yes	characteristic
208	C₄	#11718246	1	1	yes	characteristic
209	C₄	#11718247	1	1	yes	characteristic
210	C₂	#13129672	1	1	yes	characteristic
211	C₂	D₄⋊#6092	1	2	yes	—
212	C₂	Q₈⋊#6092	1	2	yes	—
213	C₂	#13129874	1	1	yes	characteristic
214	C₂	#13129890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#261
commutator	C₄×C₁₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#261
Fitting	C₄×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂²⋊C₈
cosocle	C₂×C₄×#1055	C₂²
hypercenter	C₄×C₄⋊C₄	#261
hypocenter	C₁₃	#7167
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄⋊#6092	C₄	semidirect
13. C₄×#6092	C₄	semidirect
42. C₁₃×C₄⋊C₄	C₁₆	semidirect
161. C₄⋊C₄	#6092	semidirect
170. C₁₃	#7167	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×C₄×#6092	2
C₄⋊C₄×#1055	2
#7167	13