Subgroups — #15772698

Order 3328.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13112806	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₄×C₁₃×Q₁₆	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	—	2	1	no	—
9	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
10	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#103	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	—	2	1	no	—
15	#11718586	—	2	1	no	—
16	#11718053	C₂²	1	2	yes	—
17	C₄×#6090	—	2	1	no	—
18	#11718050	—	2	1	no	—
19	#11718049	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂×C₁₃×Q₁₆	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	C₈	1	4	yes	complemented
24	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×Q₈×C₁₃	—	2	2	no	—
29	C₄×Q₈×C₁₃	—	4	1	no	—
30	C₄×Q₈×C₁₃	—	4	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₁₃	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×#107	—	2	1	no	—
36	C₁₃×#107	C₂×C₄	1	2	yes	—
37	C₁₃×#103	—	2	2	no	—
38	C₁₃×#103	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²⋊#1055	—	4	1	no	—
44	#88052	—	2	4	no	—
45	C₂×#6090	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄×#1055	—	4	1	no	—
48	#11600	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₃	—	4	1	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
58	C₁₃×Q₁₆	—	2	4	no	—
59	C₁₃×Q₁₆	—	2	2	no	—
60	C₁₃×Q₁₆	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	4	1	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	—	4	2	no	—
73	C₄²×C₁₃	—	4	2	no	—
74	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃	D₈	1	1	yes	characteristic
76	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×#1055	—	4	2	no	—
79	#6090	—	2	4	no	—
80	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
81	C₄×M₃₂	—	13	1	no	—
82	#1564	—	13	1	no	—
83	#1334	—	13	4	no	—
84	C₂³×C₁₃	#102	1	1	yes	characteristic
85	Q₈×C₁₃	—	2	4	no	—
86	Q₈×C₁₃	—	4	4	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	1	no	—
90	C₂×C₄×C₁₃	—	4	4	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	8	2	no	—
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#104	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
105	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	#1055	—	8	2	no	—
108	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
109	C₂×C₄×Q₈	—	2	1	no	—
110	C₂×M₃₂	—	26	1	no	—
111	C₄×Q₁₆	—	2	1	no	—
112	C₄×Q₁₆	—	2	1	no	—
113	C₄×Q₁₆	Dic₁₃	1	2	yes	—
114	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
115	#431	—	26	1	no	—
116	C₂×#107	Dic₁₃	1	1	yes	characteristic
117	C₂×#103	—	2	1	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	#346	—	13	2	no	—
120	C₄×C₁₆	—	26	1	no	—
121	#335	—	26	1	no	—
122	#334	—	13	2	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	D₄⋊C₈	1	2	yes	—
126	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	4	4	no	—
131	C₄×C₁₃	—	8	2	no	—
132	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
133	C₄×C₁₃	#362	1	2	yes	—
134	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
136	C₂²×Q₈	—	2	1	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	#1055	1	4	yes	complemented
140	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
141	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
151	M₃₂	—	26	4	no	—
152	C₂×C₁₆	—	26	2	no	—
153	#107	—	2	1	no	—
154	#107	C₂×Dic₁₃	1	2	yes	—
155	C₄⋊C₈	—	26	2	no	—
156	#103	—	2	2	no	—
157	#103	—	2	2	no	—
158	C₂²⋊C₈	—	52	1	no	—
159	C₄×C₈	—	52	1	no	—
160	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
163	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
166	C₂×C₁₃	#2145	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	#6104	1	1	yes	characteristic
175	C₂²×C₄	#6105	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
177	Q₁₆	—	2	4	no	—
178	Q₁₆	—	2	2	no	—
179	Q₁₆	C₂×#1055	1	4	yes	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	52	2	no	—
182	C₂×C₈	C₂×#1055	1	2	yes	—
183	C₂×C₈	#6099	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	#6104	1	1	yes	characteristic
195	C₄²	#6105	1	1	yes	characteristic
196	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₁₆	—	26	4	no	—
199	C₁₃	#11600	1	1	yes	characteristic, complemented, Sylow
200	C₂³	#88072	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂²⋊#1055	1	2	yes	—
211	C₂×C₄	#88072	1	1	yes	characteristic
212	C₂×C₄	#88072	1	1	yes	characteristic
213	C₂×C₄	#88072	1	1	yes	characteristic
214	C₂×C₄	#88077	1	2	yes	—
215	C₂×C₄	C₂×#6104	1	1	yes	characteristic
216	C₂×C₄	C₂×#6105	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	104	2	no	—
221	C₈	C₂²⋊#1055	1	2	yes	—
222	C₈	C₂²×#1055	1	1	yes	characteristic
223	C₈	C₂×#6099	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	D₄⋊#1055	1	2	yes	—
227	C₂²	C₂×#88072	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	8	2	no	—
233	C₄	D₄⋊#1055	1	2	yes	—
234	C₄	#11718157	1	2	yes	—
235	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
236	C₄	C₂×#88077	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×D₄⋊#1055	1	1	yes	characteristic
239	C₂	#13116572	1	1	yes	characteristic
240	C₂	C₂×#11718157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#11600
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13112806	C₂	semidirect
23. C₂×C₁₃×Q₁₆	C₈	semidirect
139. C₂×Q₁₆	#1055	semidirect
199. C₁₃	#11600	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
#13113584	2
#13112806	2
C₂×C₄×C₁₃×Q₁₆	2
#11600	13