Subgroups — #16777194

Order 3328.

282 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718257	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718254	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#1057	—	2	1	no	—
5	C₂²×#6092	—	2	1	no	—
6	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
7	C₂×#88058	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₁₃	—	2	1	no	—
10	#11718257	—	2	1	no	—
11	#11718257	C₄	1	2	yes	complemented
12	#11718254	—	2	1	no	—
13	#11718254	C₄	1	2	yes	complemented
14	#11718121	—	2	1	no	—
15	C₂²×#1057	—	4	1	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
18	C₂×C₄⋊Dic₁₃	—	2	1	no	—
19	C₂×C₄×Dic₁₃	—	4	1	no	—
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₄⋊#1057	—	2	2	no	—
23	C₄⋊#1057	—	4	1	no	—
24	C₂×#6092	—	2	2	no	—
25	C₂×#6092	—	2	2	no	—
26	C₂×#6092	—	2	2	no	—
27	C₁₃×#107	—	2	1	no	—
28	C₁₃×#107	C₈	1	2	yes	complemented
29	#88071	—	4	1	no	—
30	#88058	—	2	1	no	—
31	#88058	C₈	1	2	yes	complemented
32	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₃	D₄	1	1	yes	characteristic
35	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
36	#7171	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	4	1	no	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	4	1	no	—
40	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
41	C₂²×Dic₁₃	SD₁₆	1	1	yes	characteristic
42	C₂×#1057	—	4	2	no	—
43	C₂×#1057	—	8	1	no	—
44	C₂×#1057	—	8	1	no	—
45	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₄⋊Dic₁₃	—	2	2	no	—
54	C₄⋊Dic₁₃	—	4	1	no	—
55	C₄×Dic₁₃	—	8	1	no	—
56	C₄×Dic₁₃	—	8	1	no	—
57	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₁₃	SD₁₆	1	1	yes	characteristic
60	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
65	#6092	—	2	4	no	—
66	C₂×C₄×C₁₆	—	13	1	no	—
67	C₄×#107	—	13	1	no	—
68	C₂×#335	—	13	1	no	—
69	C₂³×C₁₃	#102	1	1	yes	characteristic
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	8	1	no	—
72	C₂×C₄×C₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#104	1	1	yes	characteristic
79	C₂×Dic₁₃	—	2	1	no	—
80	C₂×Dic₁₃	—	2	1	no	—
81	C₂×Dic₁₃	—	4	2	no	—
82	C₂×Dic₁₃	—	8	1	no	—
83	C₂×Dic₁₃	—	8	1	no	—
84	C₂×Dic₁₃	#102	1	1	yes	characteristic
85	C₂×Dic₁₃	#103	1	1	yes	characteristic
86	C₂×Dic₁₃	#106	1	1	yes	characteristic
87	C₂×Dic₁₃	#107	1	1	yes	characteristic
88	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
89	C₂×Dic₁₃	SD₃₂	1	1	yes	characteristic
90	C₂×Dic₁₃	SD₃₂	1	1	yes	characteristic
91	C₂×Dic₁₃	Q₃₂	1	1	yes	characteristic
92	#1057	—	8	2	no	—
93	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
94	C₈×C₁₃	#97	1	1	yes	characteristic
95	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
96	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
97	#1055	—	2	1	no	—
98	#1055	—	2	1	no	—
99	C₂²×C₁₆	—	26	1	no	—
100	C₂×#107	—	13	1	no	—
101	C₂×#107	#261	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	26	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₄×C₄⋊C₄	—	26	1	no	—
105	C₄×C₁₆	—	13	2	no	—
106	C₄×C₁₆	—	26	1	no	—
107	#340	—	26	1	no	—
108	#335	—	13	2	no	—
109	#335	—	26	1	no	—
110	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	D₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₁₃	#340	1	1	yes	characteristic
113	C₂²×C₁₃	#357	1	1	yes	characteristic
114	C₂²×C₁₃	#358	1	1	yes	characteristic
115	C₂²×C₁₃	#359	1	1	yes	characteristic
116	C₂²×C₁₃	#359	1	1	yes	characteristic
117	C₄×C₁₃	—	8	2	no	—
118	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
119	C₄×C₁₃	Q₈⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃	#336	1	1	yes	characteristic
121	C₄×C₁₃	#339	1	1	yes	characteristic
122	Dic₁₃	—	2	1	no	—
123	Dic₁₃	—	2	1	no	—
124	Dic₁₃	—	8	2	no	—
125	Dic₁₃	#357	1	1	yes	characteristic
126	Dic₁₃	#358	1	1	yes	characteristic
127	Dic₁₃	#366	1	1	yes	characteristic
128	Dic₁₃	#367	1	1	yes	characteristic
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	52	1	no	—
131	C₂²×C₈	C₂×#261	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	26	1	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	—	52	1	no	—
136	C₂×C₁₆	—	26	2	no	—
137	C₂×C₁₆	—	26	2	no	—
138	C₂×C₁₆	—	26	2	no	—
139	#107	—	2	1	no	—
140	#107	—	13	2	no	—
141	#107	—	26	1	no	—
142	#107	#1057	1	2	yes	complemented
143	C₄⋊C₈	—	26	2	no	—
144	C₄⋊C₈	—	52	1	no	—
145	C₄×C₈	—	13	1	no	—
146	C₄×C₈	—	13	1	no	—
147	C₄×C₈	—	13	1	no	—
148	C₄×C₈	—	13	1	no	—
149	#95	—	52	1	no	—
150	C₂×C₁₃	#1279	1	1	yes	characteristic
151	C₂×C₁₃	#1331	1	1	yes	characteristic
152	C₂×C₁₃	#1331	1	1	yes	characteristic
153	C₂×C₁₃	#1332	1	1	yes	characteristic
154	C₂×C₁₃	#1332	1	1	yes	characteristic
155	C₂×C₁₃	#1380	1	1	yes	characteristic
156	C₂×C₁₃	#1382	1	1	yes	characteristic
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	52	1	no	—
161	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	26	1	no	—
166	C₂×C₈	—	52	2	no	—
167	C₂×C₈	—	104	1	no	—
168	C₂×C₈	—	104	1	no	—
169	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
170	C₂×C₈	#6118	1	1	yes	characteristic
171	C₂×C₈	C₄×#261	1	1	yes	characteristic
172	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
173	C₂×C₈	C₂×#1057	1	1	yes	characteristic
174	C₂×C₈	#6122	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	52	1	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	104	1	no	—
184	C₄²	—	104	1	no	—
185	C₁₆	—	26	4	no	—
186	C₁₃	#7171	1	1	yes	characteristic, complemented, Sylow
187	C₂³	#88115	1	1	yes	characteristic
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	—	104	1	no	—
203	C₂×C₄	—	104	1	no	—
204	C₂×C₄	#88111	1	1	yes	characteristic
205	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
206	C₂×C₄	#88116	1	1	yes	characteristic
207	C₂×C₄	#88117	1	1	yes	characteristic
208	C₂×C₄	#88118	1	1	yes	characteristic
209	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
210	C₈	—	26	1	no	—
211	C₈	—	26	1	no	—
212	C₈	—	104	2	no	—
213	C₈	C₄×#1057	1	1	yes	characteristic
214	C₈	C₄⋊#1057	1	1	yes	characteristic
215	C₈	#88110	1	1	yes	characteristic
216	C₈	#88112	1	1	yes	characteristic
217	C₂²	#11718243	1	1	yes	characteristic
218	C₂²	#11718246	1	1	yes	characteristic
219	C₂²	#11718276	1	1	yes	characteristic
220	C₂²	#11718277	1	1	yes	characteristic
221	C₂²	#11718280	1	1	yes	characteristic
222	C₂²	#11718281	1	1	yes	characteristic
223	C₂²	D₄⋊#1057	1	1	yes	characteristic
224	C₄	—	8	2	no	—
225	C₄	—	13	1	no	—
226	C₄	—	13	1	no	—
227	C₄	—	13	1	no	—
228	C₄	—	13	1	no	—
229	C₄	—	26	1	no	—
230	C₄	—	26	1	no	—
231	C₄	—	104	2	no	—
232	C₄	#11718244	1	1	yes	characteristic
233	C₄	#11718247	1	1	yes	characteristic
234	C₄	#11718290	1	1	yes	characteristic
235	C₄	Q₈⋊#1057	1	1	yes	characteristic
236	C₂	#13129672	1	1	yes	characteristic
237	C₂	#13129790	1	1	yes	characteristic
238	C₂	#13129791	1	1	yes	characteristic
239	C₂	#13129792	1	1	yes	characteristic
240	C₂	#13129793	1	1	yes	characteristic
241	C₂	#13129893	1	1	yes	characteristic
242	C₂	#13129897	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88115
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#261
Fitting	C₂×C₁₃×#107	C₄
socle, prime radical	C₂³×C₁₃	#102
cosocle	C₂×C₈×Dic₁₃	C₂²
hypercenter	C₂×#107	#261
hypocenter	C₁₃	#7171
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
11. #11718257	C₄	semidirect
13. #11718254	C₄	semidirect
28. C₁₃×#107	C₈	semidirect
31. #88058	C₈	semidirect
142. #107	#1057	semidirect
186. C₁₃	#7171	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718257	2
C₂×#11718254	2
#107×Dic₁₃	2
#7171	13