Subgroups — #15778971

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718101	C₂	1	2	yes	maximal
3	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112928	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88061	—	2	1	no	—
11	C₂×#88058	C₂²	1	2	yes	—
12	C₂×#88058	C₂²	1	1	yes	characteristic
13	C₂×#88057	—	2	1	no	—
14	#11718597	—	2	1	no	—
15	C₄×D₁₀₄	—	2	2	no	—
16	C₄×D₁₀₄	C₄	1	4	yes	complemented
17	C₁₃×C₄⋊C₁₆	—	2	2	no	—
18	#11718101	—	2	4	no	—
19	C₂²×D₅₂	—	2	1	no	—
20	C₂²×C₄×D₁₃	—	4	1	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×#6103	—	4	1	no	—
25	C₂×C₄⋊Dic₁₃	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×D₁₀₄	—	2	1	no	—
28	C₂×D₁₀₄	—	2	1	no	—
29	C₂×D₁₀₄	—	2	1	no	—
30	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
31	C₂×D₁₀₄	D₄	1	2	yes	—
32	C₂×D₁₀₄	Q₈	1	2	yes	—
33	C₄×D₅₂	—	2	4	no	—
34	C₄×D₅₂	—	4	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	—	2	2	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	#88071	—	4	1	no	—
41	#88071	—	4	1	no	—
42	#88061	—	2	4	no	—
43	#88058	—	2	1	no	—
44	#88058	—	2	2	no	—
45	#88058	—	2	2	no	—
46	#88058	C₂×C₄	1	2	yes	—
47	#88057	—	4	2	no	—
48	#12813	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	—	4	1	no	—
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂×D₅₂	—	2	2	no	—
56	C₂×D₅₂	—	2	2	no	—
57	C₂×D₅₂	—	2	2	no	—
58	C₂×D₅₂	—	4	1	no	—
59	C₂×D₅₂	—	4	1	no	—
60	C₂×D₅₂	—	4	1	no	—
61	C₂×C₄×D₁₃	—	4	4	no	—
62	C₂×C₄×D₁₃	—	8	4	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₃	SD₁₆	1	2	yes	—
74	#6103	—	4	4	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄⋊Dic₁₃	—	4	1	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	D₁₀₄	—	2	2	no	—
80	D₁₀₄	—	2	2	no	—
81	D₁₀₄	—	2	2	no	—
82	D₁₀₄	C₄⋊C₄	1	4	yes	—
83	C₁₃×C₁₆	—	4	2	no	—
84	C₂×C₄×D₈	—	13	1	no	—
85	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
86	C₂×#357	—	13	2	no	—
87	#1846	—	13	1	no	—
88	#1380	—	13	2	no	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₁₃	—	4	2	no	—
91	C₂²×D₁₃	—	4	2	no	—
92	C₂²×D₁₃	—	4	2	no	—
93	C₂²×D₁₃	—	8	2	no	—
94	C₂²×D₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	#102	1	2	yes	—
98	C₂×C₄×C₁₃	#102	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	C₂×Dic₁₃	—	8	2	no	—
108	C₂×Dic₁₃	—	8	2	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	D₅₂	—	2	4	no	—
111	D₅₂	—	4	2	no	—
112	D₅₂	—	4	2	no	—
113	D₅₂	—	4	2	no	—
114	C₄×D₁₃	—	8	8	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₈×C₁₃	#117	1	1	yes	characteristic
118	C₈×C₁₃	#121	1	1	yes	characteristic
119	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×D₈	—	13	1	no	—
121	C₂×C₄×D₄	—	26	1	no	—
122	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
123	C₄×D₈	—	13	4	no	—
124	C₄×D₈	—	26	2	no	—
125	C₂×#107	—	13	1	no	—
126	C₂×#107	—	13	2	no	—
127	C₂×#106	—	26	1	no	—
128	C₂×#102	—	26	1	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	#382	—	26	1	no	—
131	C₄⋊C₁₆	—	2	2	no	—
132	#357	—	26	4	no	—
133	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
134	C₂²×C₁₃	#459	1	1	yes	characteristic
135	C₂²×C₁₃	#474	1	1	yes	characteristic
136	C₂²×C₁₃	#508	1	2	yes	—
137	C₂²×C₁₃	#509	1	2	yes	—
138	C₂×D₁₃	—	4	4	no	—
139	C₂×D₁₃	—	8	2	no	—
140	C₂×D₁₃	—	8	2	no	—
141	C₂×D₁₃	—	8	4	no	—
142	C₂×D₁₃	—	8	4	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	#380	1	1	yes	characteristic
146	C₄×C₁₃	#417	1	1	yes	characteristic
147	C₄×C₁₃	#461	1	1	yes	characteristic
148	C₄×C₁₃	#476	1	1	yes	characteristic
149	Dic₁₃	—	8	2	no	—
150	Dic₁₃	—	16	2	no	—
151	C₂²×D₄	—	26	1	no	—
152	C₂³×C₄	—	52	1	no	—
153	C₂×D₈	—	13	2	no	—
154	C₂×D₈	—	13	2	no	—
155	C₂×D₈	—	13	2	no	—
156	C₂×D₈	—	26	1	no	—
157	C₂×D₈	—	26	1	no	—
158	C₂×D₈	—	26	1	no	—
159	C₂²×C₈	D₅₂	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₄×D₄	—	26	4	no	—
162	C₄×D₄	—	52	2	no	—
163	C₂×C₄⋊C₄	—	26	1	no	—
164	C₂×C₄⋊C₄	—	26	2	no	—
165	C₂×C₂²⋊C₄	—	52	1	no	—
166	C₂×C₄²	D₅₂	1	1	yes	characteristic
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	#107	—	13	2	no	—
171	#107	—	26	1	no	—
172	#107	—	26	2	no	—
173	#107	—	26	2	no	—
174	#106	—	52	2	no	—
175	#102	—	26	4	no	—
176	C₄×C₈	C₄×D₁₃	1	2	yes	—
177	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
178	#95	—	52	1	no	—
179	#95	—	52	1	no	—
180	C₂×C₁₃	#1802	1	1	yes	characteristic
181	C₂×C₁₃	#2138	1	1	yes	characteristic
182	C₂×C₁₃	#2140	1	1	yes	characteristic
183	C₂×C₁₃	#2210	1	2	yes	—
184	C₂×C₁₃	#2228	1	2	yes	—
185	D₁₃	—	8	4	no	—
186	C₂⁴	—	52	1	no	—
187	C₂×D₄	—	26	2	no	—
188	C₂×D₄	—	26	2	no	—
189	C₂×D₄	—	26	2	no	—
190	C₂×D₄	—	52	1	no	—
191	C₂×D₄	—	52	1	no	—
192	C₂×D₄	—	52	1	no	—
193	C₂²×C₄	—	52	1	no	—
194	C₂²×C₄	—	52	4	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	—	104	4	no	—
197	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
198	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
200	D₈	—	13	4	no	—
201	D₈	—	26	2	no	—
202	D₈	—	26	2	no	—
203	D₈	—	26	2	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	#6103	1	2	yes	—
207	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
209	C₂×C₈	#6129	1	1	yes	characteristic
210	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
211	C₂×C₈	#6131	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	26	2	no	—
214	C₄⋊C₄	—	52	1	no	—
215	C₄⋊C₄	—	52	2	no	—
216	C₄⋊C₄	—	52	2	no	—
217	C₂²⋊C₄	—	52	4	no	—
218	C₄²	#6095	1	2	yes	—
219	C₄²	#6103	1	2	yes	—
220	C₁₆	—	4	2	no	—
221	C₁₃	#12813	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	52	2	no	—
223	C₂³	—	52	2	no	—
224	C₂³	—	52	2	no	—
225	C₂³	—	104	2	no	—
226	C₂³	—	104	2	no	—
227	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
228	D₄	—	26	4	no	—
229	D₄	—	52	2	no	—
230	D₄	—	52	2	no	—
231	D₄	—	52	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	52	2	no	—
235	C₂×C₄	—	104	8	no	—
236	C₂×C₄	—	104	2	no	—
237	C₂×C₄	—	104	2	no	—
238	C₂×C₄	—	104	2	no	—
239	C₂×C₄	—	104	2	no	—
240	C₂×C₄	#88061	1	2	yes	—
241	C₂×C₄	#88061	1	2	yes	—
242	C₂×C₄	#88149	1	1	yes	characteristic
243	C₂×C₄	#88151	1	1	yes	characteristic
244	C₂×C₄	C₂×#6095	1	1	yes	characteristic
245	C₂×C₄	#88162	1	1	yes	characteristic
246	C₂×C₄	#88163	1	1	yes	characteristic
247	C₂×C₄	C₂×#6103	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
250	C₈	#88146	1	1	yes	characteristic
251	C₈	#88192	1	1	yes	characteristic
252	C₈	#88200	1	1	yes	characteristic
253	C₂²	—	52	4	no	—
254	C₂²	—	104	4	no	—
255	C₂²	—	104	2	no	—
256	C₂²	—	104	2	no	—
257	C₂²	—	104	4	no	—
258	C₂²	#11718504	1	1	yes	characteristic
259	C₂²	#11718507	1	1	yes	characteristic
260	C₂²	#11718565	1	2	yes	—
261	C₂²	#11718569	1	2	yes	—
262	C₂²	C₂×#88061	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	104	2	no	—
266	C₄	—	208	2	no	—
267	C₄	#11718502	1	1	yes	characteristic
268	C₄	#11718503	1	1	yes	characteristic
269	C₄	#11718648	1	1	yes	characteristic
270	C₄	#11718794	1	1	yes	characteristic
271	C₂	—	104	4	no	—
272	C₂	#13114837	1	1	yes	characteristic
273	C₂	#13116544	1	1	yes	characteristic
274	C₂	#13116553	1	1	yes	characteristic
275	C₂	#13116892	1	2	yes	—
276	C₂	#13116994	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#12813
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #13115000	C₂	semidirect
4. #13112928	C₂	semidirect
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
16. C₄×D₁₀₄	C₄	semidirect
85. C₂×C₄⋊C₁₆	D₁₃	semidirect
221. C₁₃	#12813	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₀₄	2
C₂×#11718101	2
#13115000	2
#13112928	2
C₂×C₁₃×C₄⋊C₁₆	2
#12813	13