Subgroups — #13118671

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88141	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	2	yes	maximal
4	#11718890	C₂	1	2	yes	maximal
5	#11718646	C₂	1	1	yes	maximal, characteristic
6	#11718645	C₂	1	2	yes	maximal
7	#11718639	C₂	1	2	yes	maximal
8	#11718635	C₂	1	1	yes	maximal, characteristic
9	#11718634	C₂	1	1	yes	maximal, characteristic
10	C₄×#6101	C₂	1	1	yes	maximal, characteristic
11	#11718590	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	1	yes	characteristic
23	C₂×#6101	C₂²	1	2	yes	—
24	C₂×#6101	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	#88141	—	2	4	no	—
29	#88071	C₂²	1	2	yes	—
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	1	yes	characteristic
32	#88071	C₂²	1	1	yes	characteristic
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₃	—	2	4	no	—
38	C₂×Q₈×C₁₃	—	2	4	no	—
39	C₂×Q₈×C₁₃	—	4	2	no	—
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₆	—	2	2	no	—
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	Q₈	1	4	yes	complemented
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	#6101	—	2	4	no	—
61	#6101	—	2	2	no	—
62	#6101	—	4	2	no	—
63	#6101	Q₈	1	4	yes	complemented
64	C₄×Dic₁₃	—	2	4	no	—
65	C₄×Dic₁₃	—	2	2	no	—
66	C₄×Dic₁₃	—	2	2	no	—
67	#2451	—	13	1	no	maximal, Sylow
68	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
69	Q₈×C₁₃	—	4	8	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	#41	1	2	yes	—
84	C₂×C₄×C₁₃	#41	1	2	yes	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
101	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
102	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
103	Dic₂₆	—	2	8	no	—
104	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
105	C₂×C₄×Q₈	—	13	1	no	—
106	#398	—	13	2	no	—
107	#395	—	13	2	no	—
108	#391	—	13	2	no	—
109	#391	—	13	1	no	—
110	#391	—	13	1	no	—
111	#389	—	13	1	no	—
112	#389	—	13	1	no	—
113	#387	—	13	2	no	—
114	C₄×C₄⋊C₄	—	13	1	no	—
115	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	2	2	no	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	—	4	1	no	—
131	Dic₁₃	—	4	2	no	—
132	Dic₁₃	—	4	1	no	—
133	C₂²×Q₈	—	13	1	no	—
134	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
135	C₄⋊Q₈	—	2	4	no	—
136	C₄×Q₈	—	26	4	no	—
137	C₂×C₄⋊C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	2	no	—
139	C₂×C₄⋊C₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₄⋊C₄	—	13	2	no	—
142	C₂×C₄⋊C₄	—	13	2	no	—
143	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	—	13	2	no	—
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
150	#95	—	13	1	no	—
151	#95	—	13	1	no	—
152	#95	—	13	2	no	—
153	#95	—	13	2	no	—
154	#95	—	13	2	no	—
155	#95	—	13	2	no	—
156	#95	—	13	2	no	—
157	#95	—	13	2	no	—
158	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₁₃	#533	1	1	yes	characteristic
160	C₂×C₁₃	#542	1	1	yes	characteristic
161	C₂×C₁₃	#546	1	1	yes	characteristic
162	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
163	C₂×C₁₃	#557	1	1	yes	characteristic
164	C₂×C₁₃	Q₈²	1	1	yes	characteristic
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	13	4	no	—
169	C₂×Q₈	—	26	2	no	—
170	C₂×Q₈	—	26	2	no	—
171	C₂²×C₄	—	13	1	no	—
172	C₂²×C₄	—	13	2	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	2	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	13	4	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	—	26	4	no	—
188	C₄⋊C₄	—	52	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	26	2	no	—
193	C₄²	—	26	4	no	—
194	C₄²	—	26	2	no	—
195	C₁₃	#2451	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₂³×D₁₃	1	1	yes	characteristic
197	Q₈	—	4	8	no	—
198	Q₈	—	26	8	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	2	no	—
210	C₂×C₄	—	13	4	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	1	no	—
215	C₂×C₄	—	26	1	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	1	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	1	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	1	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	26	1	no	—
226	C₂×C₄	#6129	1	2	yes	—
227	C₂×C₄	Q₈×D₁₃	1	2	yes	—
228	C₂×C₄	Q₈×D₁₃	1	2	yes	—
229	C₂×C₄	#6131	1	2	yes	—
230	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
232	C₂²	C₂×#6129	1	1	yes	characteristic
233	C₂²	#88251	1	1	yes	characteristic
234	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
235	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
236	C₂²	C₂×#6131	1	1	yes	characteristic
237	C₂²	#88254	1	1	yes	characteristic
238	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	26	2	no	—
244	C₄	—	26	4	no	—
245	C₄	—	26	2	no	—
246	C₄	—	52	1	no	—
247	C₄	—	52	2	no	—
248	C₄	—	52	1	no	—
249	C₄	#88200	1	4	yes	—
250	C₂	#11719426	1	1	yes	characteristic
251	C₂	#11719427	1	1	yes	characteristic
252	C₂	#11719433	1	1	yes	characteristic
253	C₂	#11719438	1	1	yes	characteristic
254	C₂	#11719511	1	1	yes	characteristic
255	C₂	C₂×#88200	1	1	yes	characteristic
256	C₂	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2451
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
53. C₂×Dic₂₆	Q₈	semidirect
63. #6101	Q₈	semidirect
195. C₁₃	#2451	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88141	2
#11718891	2
#11718890	2
#11718646	2
#11718645	2
#11718639	2
#11718635	2
#11718634	2
C₄×#6101	2
#11718590	2
#2451	13