Subgroups — #15778148

Order 3328.

442 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×#11718159	C₂	1	2	yes	maximal
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112931	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×#41	—	2	1	no	—
8	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
9	C₄×C₁₃×SD₁₆	—	2	2	no	—
10	C₄×C₁₃×D₈	C₄	1	2	yes	—
11	C₁₃×#427	C₂²	1	1	yes	characteristic
12	C₁₃×#416	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88059	C₂²	1	2	yes	—
15	C₂²×#6090	C₂²	1	2	yes	—
16	C₄×#6099	—	2	1	no	—
17	#11718159	C₄	1	8	yes	complemented
18	#11718144	—	2	1	no	—
19	#11718053	—	2	2	no	—
20	C₂×C₁₃×#41	—	2	1	no	—
21	C₁₃×#135	—	2	2	no	—
22	C₁₃×#135	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×SD₁₆	—	2	1	no	—
25	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	4	2	no	—
31	C₁₃×#117	—	2	1	no	—
32	C₁₃×#117	—	4	1	no	—
33	C₂×C₄²×C₁₃	—	4	1	no	—
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×#6099	—	2	2	no	—
36	C₁₃×#107	C₂×C₄	1	2	yes	—
37	C₁₃×#106	—	2	1	no	—
38	C₁₃×#103	—	2	2	no	—
39	C₁₃×#102	—	2	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88059	—	2	2	no	—
43	#88059	C₂×C₄	1	4	yes	complemented
44	C₂×#6090	—	2	2	no	—
45	C₂×#6090	—	2	2	no	—
46	C₂×#6090	C₂×C₄	1	4	yes	complemented
47	#88043	—	4	1	no	—
48	C₄×#1055	—	4	1	no	—
49	#12640	—	13	1	no	maximal, Sylow
50	C₁₃×#41	—	2	4	no	—
51	C₁₃×#41	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	—	2	1	no	—
54	C₂×D₄×C₁₃	—	4	1	no	—
55	C₂²×C₄×C₁₃	—	4	1	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₁₃×Q₁₆	—	2	1	no	—
60	C₁₃×Q₁₆	C₄²	1	2	yes	—
61	C₁₃×SD₁₆	—	2	4	no	—
62	C₁₃×D₈	—	2	1	no	—
63	C₁₃×D₈	C₄²	1	2	yes	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₂²⋊C₄	—	4	1	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₄²×C₁₃	—	4	2	no	—
75	C₄²×C₁₃	—	4	1	no	—
76	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₃	D₈	1	1	yes	characteristic
78	C₄²×C₁₃	D₈	1	1	yes	characteristic
79	#6099	—	2	4	no	—
80	#6099	—	4	2	no	—
81	C₂×#1055	—	4	2	no	—
82	#6090	—	2	4	no	—
83	C₄×#135	D₁₃	1	1	yes	characteristic
84	C₂×#359	—	13	2	no	—
85	C₂×#346	—	13	1	no	—
86	C₄×#108	—	13	1	no	—
87	#1382	—	13	2	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	2	no	—
91	Q₈×C₁₃	—	4	1	no	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	4	4	no	—
94	D₄×C₁₃	—	4	1	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	1	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	#102	1	2	yes	—
101	C₂×C₄×C₁₃	#102	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₈×C₁₃	#117	1	1	yes	characteristic
109	C₈×C₁₃	C₄×D₄	1	2	yes	—
110	#1055	—	4	4	no	—
111	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
112	C₄×#41	—	2	1	no	—
113	C₂²×C₁₆	—	13	2	no	—
114	C₄×Q₁₆	Dic₁₃	1	2	yes	—
115	C₄×SD₁₆	—	2	2	no	—
116	C₄×D₈	Dic₁₃	1	2	yes	—
117	C₂×#108	—	13	2	no	—
118	#427	C₂×D₁₃	1	1	yes	characteristic
119	#416	—	2	1	no	—
120	C₄×M₁₆	—	26	1	no	—
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	#359	—	13	8	no	—
123	#346	—	26	2	no	—
124	#341	—	26	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	8	2	no	—
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	—	4	2	no	—
133	C₄×C₁₃	—	8	2	no	—
134	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₁₃	#416	1	1	yes	characteristic
136	C₄×C₁₃	C₄×D₈	1	2	yes	—
137	C₂×#41	—	2	1	no	—
138	#135	—	2	2	no	—
139	#135	C₄×D₁₃	1	4	yes	—
140	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂×SD₁₆	—	2	1	no	—
142	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
143	C₂×M₁₆	—	26	2	no	—
144	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
146	C₄×Q₈	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₄×D₄	—	4	2	no	—
149	#117	—	2	1	no	—
150	#117	—	4	1	no	—
151	C₂×C₄²	—	4	1	no	—
152	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₁₆	—	13	4	no	—
154	C₂×C₁₆	—	26	2	no	—
155	C₂×C₁₆	—	26	2	no	—
156	#108	—	13	4	no	—
157	#108	—	26	2	no	—
158	#107	C₂×Dic₁₃	1	2	yes	—
159	#106	—	2	1	no	—
160	#103	—	2	2	no	—
161	#102	—	2	2	no	—
162	#97	—	52	1	no	—
163	C₄×C₈	—	52	1	no	—
164	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
165	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
166	C₂×C₁₃	—	2	2	no	—
167	C₂×C₁₃	—	8	2	no	—
168	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
169	C₂×C₁₃	#2143	1	2	yes	—
170	#41	—	2	4	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	4	1	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	#6105	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
179	Q₁₆	—	2	1	no	—
180	Q₁₆	C₄×Dic₁₃	1	2	yes	—
181	SD₁₆	—	2	4	no	—
182	D₈	—	2	1	no	—
183	D₈	C₄×Dic₁₃	1	2	yes	—
184	M₁₆	—	26	4	no	—
185	M₁₆	—	52	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
189	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
190	C₂×C₈	#6127	1	2	yes	—
191	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	#6104	1	1	yes	characteristic
200	C₄²	#6104	1	1	yes	characteristic
201	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
202	C₁₆	—	26	4	no	—
203	C₁₃	#12640	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	4	1	no	—
205	C₂³	C₂×#6105	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#88072	1	2	yes	—
217	C₂×C₄	#88072	1	2	yes	—
218	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₄	C₂×#6104	1	1	yes	characteristic
220	C₂×C₄	#88204	1	2	yes	—
221	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	52	4	no	—
224	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
225	C₈	#88180	1	1	yes	characteristic
226	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₄×#6105	1	2	yes	—
231	C₂²	C₂×#88072	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	C₄×#6104	1	2	yes	—
237	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₄	#11718894	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×#88072	1	1	yes	characteristic
242	C₂	#13116564	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂×C₁₃×#135	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#12640
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#11718053	C₂	semidirect
3. C₄×#88059	C₂	semidirect
4. #13112931	C₂	semidirect
17. #11718159	C₄	semidirect
43. #88059	C₂×C₄	semidirect
46. C₂×#6090	C₂×C₄	semidirect
203. C₁₃	#12640	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718159	2
C₂×#11718053	2
C₄×#88059	2
#13112931	2
C₄×C₁₃×#135	2
#12640	13