Subgroups — #13128654

Order 1664.

1060 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719435	C₂	1	3	yes	maximal, complemented
3	#11719434	C₂	1	6	yes	maximal, complemented
4	#11719432	C₂	1	6	yes	maximal, complemented
5	D₁₃×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
6	#11719284	C₂	1	3	yes	maximal, complemented
7	#11719283	C₂	1	3	yes	maximal, complemented
8	#11719279	C₂	1	2	yes	maximal
9	C₄×#6131	C₂	1	1	yes	maximal, characteristic
10	#11719274	C₂	1	2	yes	maximal, complemented
11	Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6131	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₃	C₂²	1	6	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
19	#88200	C₂²	1	12	yes	—
20	#88200	C₂²	1	3	yes	—
21	Q₈×Dic₁₃	C₂²	1	2	yes	—
22	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	#88198	C₂²	1	6	yes	—
24	#88198	C₂²	1	1	yes	characteristic
25	#88151	C₂²	1	3	yes	—
26	#88150	C₂²	1	6	yes	—
27	#88150	C₂²	1	12	yes	—
28	#88147	C₂²	1	3	yes	—
29	#88147	C₂²	1	6	yes	—
30	#88146	C₂²	1	3	yes	—
31	C₄⋊C₄×D₁₃	C₂²	1	6	yes	—
32	C₄⋊C₄×D₁₃	C₂²	1	6	yes	—
33	#88144	C₂²	1	6	yes	—
34	#88143	C₂²	1	6	yes	—
35	#88142	C₂²	1	3	yes	—
36	#88142	C₂²	1	6	yes	—
37	#88142	C₂²	1	6	yes	—
38	C₄×D₅₂	C₂²	1	6	yes	—
39	C₄×D₅₂	C₂²	1	3	yes	—
40	#88126	C₂²	1	6	yes	—
41	C₄²×D₁₃	C₂²	1	3	yes	—
42	#88124	C₂²	1	6	yes	—
43	#88123	C₂²	1	3	yes	—
44	C₄×Dic₂₆	C₂²	1	6	yes	—
45	C₄×Dic₂₆	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	#6131	Q₈	1	8	yes	complemented
50	Q₈×D₁₃	—	4	12	no	—
51	C₂×D₅₂	C₂³	1	3	yes	—
52	C₂×C₄×D₁₃	C₂³	1	3	yes	—
53	C₂×C₄×D₁₃	C₂³	1	3	yes	—
54	C₂×C₄×D₁₃	C₂³	1	12	yes	—
55	C₂×C₄×D₁₃	C₂³	1	3	yes	—
56	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
57	C₂×Dic₂₆	C₂³	1	6	yes	—
58	C₂×Dic₂₆	C₂³	1	6	yes	—
59	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
60	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
61	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
63	C₄²×C₁₃	C₂³	1	3	yes	—
64	C₄²×C₁₃	C₂³	1	6	yes	—
65	#6103	C₂³	1	6	yes	—
66	#6103	C₂³	1	6	yes	—
67	#6103	C₂³	1	6	yes	—
68	C₄⋊Dic₁₃	C₂³	1	3	yes	—
69	C₄⋊Dic₁₃	C₂³	1	6	yes	—
70	C₄⋊Dic₁₃	C₂³	1	6	yes	—
71	C₄⋊Dic₁₃	C₂³	1	3	yes	—
72	#6101	C₂³	1	2	yes	—
73	#6101	C₂³	1	6	yes	—
74	#6101	C₂³	1	6	yes	—
75	#6101	C₂³	1	12	yes	—
76	#6101	C₂³	1	2	yes	—
77	#6101	C₂³	1	6	yes	—
78	C₄×Dic₁₃	C₂³	1	3	yes	—
79	C₄×Dic₁₃	C₂³	1	6	yes	—
80	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₃	C₂³	1	3	yes	—
82	#3503	—	13	1	no	maximal, Sylow
83	C₂²×D₁₃	C₂⁴	1	3	yes	—
84	Q₈×C₁₃	—	2	2	no	—
85	Q₈×C₁₃	—	4	6	no	—
86	Q₈×C₁₃	C₂×Q₈	1	4	yes	complemented
87	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
88	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
89	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
90	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂⁴	1	2	yes	—
93	C₂×Dic₁₃	C₂⁴	1	3	yes	—
94	C₂×Dic₁₃	C₂⁴	1	6	yes	—
95	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
97	D₅₂	C₂×Q₈	1	12	yes	complemented
98	C₄×D₁₃	—	2	6	no	—
99	C₄×D₁₃	—	2	6	no	—
100	C₄×D₁₃	—	4	12	no	—
101	C₄×D₁₃	C₂×Q₈	1	12	yes	complemented
102	Dic₂₆	—	2	2	no	—
103	Dic₂₆	—	2	12	no	—
104	Dic₂₆	—	4	6	no	—
105	Q₈²	—	13	1	no	—
106	Q₈²	D₁₃	1	1	yes	characteristic, complemented
107	#557	—	13	2	no	—
108	#554	—	13	3	no	—
109	#554	—	13	3	no	—
110	D₄×Q₈	—	13	6	no	—
111	#544	—	13	6	no	—
112	#533	—	13	3	no	—
113	C₂×C₄⋊Q₈	—	13	3	no	—
114	#520	—	13	2	no	—
115	C₄×#41	—	13	1	no	—
116	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₁₃	—	2	3	no	—
118	C₂×D₁₃	—	2	3	no	—
119	C₂×D₁₃	C₂²×Q₈	1	6	yes	—
120	C₄×C₁₃	—	2	6	no	—
121	C₄×C₁₃	—	2	3	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	C₂²×Q₈	1	6	yes	—
124	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
125	Dic₁₃	—	2	6	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	3	no	—
128	Dic₁₃	C₂²×Q₈	1	2	yes	—
129	Dic₁₃	H(5,𝔽₂)	1	2	yes	—
130	C₂×#41	—	13	1	no	—
131	C₂²×Q₈	—	13	6	no	—
132	C₄⋊Q₈	—	13	6	no	—
133	C₄⋊Q₈	—	13	1	no	—
134	C₄⋊Q₈	—	13	3	no	—
135	C₄⋊Q₈	—	13	6	no	—
136	C₄⋊Q₈	—	13	6	no	—
137	C₄⋊Q₈	—	13	3	no	—
138	C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
139	C₄⋊Q₈	C₂×D₁₃	1	6	yes	—
140	#125	—	13	6	no	—
141	#125	—	13	6	no	—
142	#125	—	13	6	no	—
143	C₂²⋊Q₈	—	13	3	no	—
144	C₂²⋊Q₈	—	13	12	no	—
145	C₂²⋊Q₈	—	13	6	no	—
146	C₂²⋊Q₈	—	13	3	no	—
147	C₂²⋊Q₈	—	13	12	no	—
148	C₄×Q₈	—	13	1	no	—
149	C₄×Q₈	—	13	2	no	—
150	C₄×Q₈	—	13	6	no	—
151	C₄×Q₈	—	13	3	no	—
152	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
153	C₄×Q₈	C₂×D₁₃	1	2	yes	—
154	C₄×Q₈	C₂×D₁₃	1	3	yes	—
155	C₄×D₄	—	13	3	no	—
156	C₄×D₄	—	13	6	no	—
157	C₄×D₄	—	13	3	no	—
158	C₄×D₄	—	13	6	no	—
159	#117	—	13	3	no	—
160	#117	—	13	6	no	—
161	C₂×C₄⋊C₄	—	13	6	no	—
162	C₂×C₄⋊C₄	—	13	6	no	—
163	C₂×C₄²	—	13	3	no	—
164	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
165	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
167	D₁₃	—	2	6	no	—
168	#41	—	13	8	no	—
169	C₂×Q₈	—	13	1	no	—
170	C₂×Q₈	—	13	6	no	—
171	C₂×Q₈	—	13	6	no	—
172	C₂×Q₈	—	52	12	no	—
173	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×Q₈	C₂²×D₁₃	1	6	yes	—
175	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×D₄	—	13	3	no	—
177	C₂²×C₄	—	13	3	no	—
178	C₂²×C₄	—	13	3	no	—
179	C₂²×C₄	—	13	3	no	—
180	C₂²×C₄	—	13	12	no	—
181	C₄⋊C₄	—	13	6	no	—
182	C₄⋊C₄	—	13	3	no	—
183	C₄⋊C₄	—	13	6	no	—
184	C₄⋊C₄	—	13	12	no	—
185	C₄⋊C₄	—	13	2	no	—
186	C₄⋊C₄	—	13	3	no	—
187	C₄⋊C₄	—	13	2	no	—
188	C₄⋊C₄	—	13	6	no	—
189	C₄⋊C₄	—	13	6	no	—
190	C₄⋊C₄	—	13	6	no	—
191	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
192	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
193	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
194	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
195	C₂²⋊C₄	—	13	6	no	—
196	C₂²⋊C₄	—	13	6	no	—
197	C₂²⋊C₄	—	13	6	no	—
198	C₄²	—	13	1	no	—
199	C₄²	—	13	3	no	—
200	C₄²	—	13	3	no	—
201	C₄²	—	13	6	no	—
202	C₄²	C₂²×D₁₃	1	3	yes	—
203	C₄²	C₂²×D₁₃	1	6	yes	—
204	C₁₃	#3503	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	13	3	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	6	no	—
208	Q₈	—	26	12	no	—
209	Q₈	—	26	2	no	—
210	Q₈	—	52	6	no	—
211	Q₈	Q₈×D₁₃	1	4	yes	complemented
212	D₄	—	13	12	no	—
213	C₂×C₄	—	13	1	no	—
214	C₂×C₄	—	13	1	no	—
215	C₂×C₄	—	13	12	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	3	no	—
218	C₂×C₄	—	13	6	no	—
219	C₂×C₄	—	26	6	no	—
220	C₂×C₄	—	26	6	no	—
221	C₂×C₄	—	52	12	no	—
222	C₂×C₄	C₂³×D₁₃	1	3	yes	—
223	C₂×C₄	C₂³×D₁₃	1	2	yes	—
224	C₂×C₄	C₂³×D₁₃	1	6	yes	—
225	C₂×C₄	C₂³×D₁₃	1	3	yes	—
226	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
227	C₂²	—	13	6	no	—
228	C₂²	—	26	3	no	—
229	C₂²	—	26	3	no	—
230	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	3	no	—
233	C₄	—	2	6	no	—
234	C₄	—	13	2	no	—
235	C₄	—	13	2	no	—
236	C₄	—	26	3	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	6	no	—
239	C₄	C₂×Q₈×D₁₃	1	6	yes	—
240	C₄	#88254	1	2	yes	—
241	C₂	—	26	6	no	—
242	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
243	C₂	C₂×#88254	1	1	yes	characteristic
244	C₂	D₁₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	Q₈²×C₁₃	C₂
prime radical	C₂×#6131	C₂²
hypercenter	Q₈²	D₁₃
hypocenter	C₁₃	#3503
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₁₃	C₂	semidirect
2. #11719435	C₂	semidirect
3. #11719434	C₂	semidirect
4. #11719432	C₂	semidirect
5. D₁₃×C₄⋊Q₈	C₂	semidirect
6. #11719284	C₂	semidirect
7. #11719283	C₂	semidirect
10. #11719274	C₂	semidirect
11. Q₈×Dic₂₆	C₂	semidirect
49. #6131	Q₈	semidirect
86. Q₈×C₁₃	C₂×Q₈	semidirect
97. D₅₂	C₂×Q₈	semidirect
101. C₄×D₁₃	C₂×Q₈	semidirect
106. Q₈²	D₁₃	semidirect
204. C₁₃	#3503	semidirect
211. Q₈	Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₁₃	2
#11719435	2
#11719434	2
#11719432	2
D₁₃×C₄⋊Q₈	2
#11719284	2
#11719283	2
#11719279	2
C₄×#6131	2
#11719274	2
Q₈×Dic₂₆	2
#3503	13