Subgroups — C₂²×#11719436 #16769736

Order 3328.

7110 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×#11719436	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88152	C₂	1	4	yes	maximal
3	C₂²×#88124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88143	C₂	1	4	yes	maximal, complemented
5	C₂²×#88200	C₂	1	2	yes	maximal
6	C₂²×Q₈×Dic₁₃	C₂	1	2	yes	maximal, complemented
7	C₂²×#88126	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88200	C₂²	1	48	yes	—
15	C₂×Q₈×Dic₁₃	C₂²	1	48	yes	complemented
16	C₂²×#6103	C₂²	1	2	yes	—
17	C₂²×#6103	C₂²	1	4	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
20	C₂²×#6101	C₂²	1	2	yes	—
21	C₂²×#6101	C₂²	1	4	yes	—
22	C₂²×#6101	C₂²	1	4	yes	—
23	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
25	#11719436	C₂²	1	256	yes	direct product factor
26	C₂×#88152	C₂²	1	96	yes	—
27	C₂×#88143	C₂²	1	96	yes	complemented
28	C₂×#88126	C₂²	1	24	yes	—
29	C₂×#88124	C₂²	1	24	yes	complemented
30	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃	—	2	16	no	—
32	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂³×Dic₁₃	C₂³	1	2	yes	—
37	C₂³×Dic₁₃	C₂³	1	4	yes	—
38	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×D₁₃	—	2	8	no	—
40	C₂²×C₄×D₁₃	C₂³	1	12	yes	—
41	C₁₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
44	#88200	C₂³	1	128	yes	—
45	Q₈×Dic₁₃	C₂³	1	128	yes	complemented
46	C₂×#6103	C₂³	1	24	yes	—
47	C₂×#6103	C₂³	1	48	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
50	C₂×#6101	C₂³	1	48	yes	—
51	C₂×#6101	C₂³	1	24	yes	—
52	C₂×#6101	C₂³	1	48	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	48	yes	—
55	#88152	C₂³	1	256	yes	—
56	#88143	C₂³	1	256	yes	complemented
57	#88126	C₂³	1	64	yes	—
58	#88124	C₂³	1	64	yes	complemented
59	C₂²×#541	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂³×D₁₃	—	2	4	no	—
62	C₂³×D₁₃	—	2	8	no	—
63	C₂³×D₁₃	C₂⁴	1	6	yes	—
64	C₂×Q₈×C₁₃	—	2	96	no	—
65	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
66	C₂²×C₄×C₁₃	—	2	16	no	—
67	C₂²×C₄×C₁₃	—	2	8	no	—
68	C₂²×C₄×C₁₃	#41	1	8	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	2	8	no	—
74	C₂²×Dic₁₃	—	2	16	no	—
75	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
76	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
77	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
78	C₂×C₄×D₁₃	—	2	48	no	—
79	C₂×C₄×D₁₃	C₂⁴	1	16	yes	—
80	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
81	C₄²×C₁₃	C₂⁴	1	16	yes	—
82	#6103	C₂⁴	1	32	yes	—
83	#6103	C₂⁴	1	64	yes	—
84	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
85	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
86	#6101	C₂⁴	1	64	yes	—
87	#6101	C₂⁴	1	32	yes	—
88	#6101	C₂⁴	1	64	yes	—
89	C₄×Dic₁₃	C₂⁴	1	16	yes	—
90	C₄×Dic₁₃	C₂⁴	1	64	yes	—
91	C₂×#541	—	13	48	no	—
92	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
93	C₂²×#126	—	13	4	no	—
94	C₂²×#125	—	13	1	no	—
95	C₂²×#125	—	13	4	no	—
96	C₂²×C₂²⋊Q₈	—	13	2	no	—
97	C₂²×C₄×Q₈	—	13	2	no	—
98	C₂²×#117	—	13	1	no	—
99	C₂³×C₁₃	C₂×#41	1	4	yes	—
100	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	—
101	C₂³×C₁₃	C₂⁵	1	3	yes	—
102	C₂²×D₁₃	—	2	12	no	—
103	C₂²×D₁₃	—	2	24	no	—
104	C₂²×D₁₃	—	4	16	no	—
105	C₂²×D₁₃	C₂⁵	1	4	yes	—
106	Q₈×C₁₃	—	2	64	no	—
107	C₂×C₄×C₁₃	—	2	48	no	—
108	C₂×C₄×C₁₃	—	2	24	no	—
109	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
110	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
111	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
112	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
113	C₂×Dic₁₃	—	2	12	no	—
114	C₂×Dic₁₃	—	2	24	no	—
115	C₂×Dic₁₃	—	2	48	no	—
116	C₂×Dic₁₃	C₂⁵	1	8	yes	—
117	C₂×Dic₁₃	C₂⁵	1	16	yes	—
118	C₂×Dic₁₃	C₂⁵	1	4	yes	—
119	C₄×D₁₃	—	2	32	no	—
120	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
121	C₂⁴×C₄	—	13	1	no	—
122	#541	—	13	256	no	—
123	C₂×C₄⋊Q₈	C₂×D₁₃	1	24	yes	complemented
124	C₂×#126	—	13	96	no	—
125	C₂×#125	—	13	96	no	—
126	C₂×#125	—	13	24	no	—
127	C₂×C₂²⋊Q₈	—	13	48	no	—
128	C₂×C₄×Q₈	—	13	48	no	—
129	C₂×#117	—	13	24	no	—
130	C₂²×C₄⋊C₄	—	13	2	no	—
131	C₂²×C₄⋊C₄	—	13	4	no	—
132	C₂²×C₄⋊C₄	—	13	4	no	—
133	C₂²×C₄⋊C₄	—	13	2	no	—
134	C₂²×C₄⋊C₄	—	13	4	no	—
135	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
136	C₂²×C₂²⋊C₄	—	13	2	no	—
137	C₂²×C₂²⋊C₄	—	13	4	no	—
138	C₂²×C₄²	—	13	1	no	—
139	C₂²×C₄²	—	13	4	no	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	C₂²×C₁₃	#541	1	16	yes	complemented
142	C₂²×C₁₃	C₂²×#41	1	6	yes	—
143	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	12	yes	—
144	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
145	C₂×D₁₃	—	2	4	no	—
146	C₂×D₁₃	—	2	8	no	—
147	C₂×D₁₃	—	4	24	no	—
148	C₄×C₁₃	—	2	16	no	—
149	C₄×C₁₃	—	2	8	no	—
150	C₄×C₁₃	C₂²×#41	1	8	yes	—
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	2	8	no	—
153	Dic₁₃	—	2	16	no	—
154	C₂⁵	—	13	1	no	—
155	C₂²×Q₈	—	2	16	no	—
156	C₂²×Q₈	C₂²×D₁₃	1	24	yes	—
157	C₂³×C₄	—	13	4	no	—
158	C₂³×C₄	—	13	2	no	—
159	C₂³×C₄	—	13	1	no	—
160	C₂³×C₄	—	13	12	no	—
161	C₂³×C₄	—	26	8	no	—
162	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
163	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
164	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
165	C₄⋊Q₈	C₂²×D₁₃	1	64	yes	complemented
166	#126	—	13	256	no	—
167	#125	—	13	256	no	—
168	#125	—	13	64	no	—
169	C₂²⋊Q₈	—	13	128	no	—
170	C₄×Q₈	—	13	128	no	—
171	#117	—	13	64	no	—
172	C₂×C₄⋊C₄	—	13	24	no	—
173	C₂×C₄⋊C₄	—	13	24	no	—
174	C₂×C₄⋊C₄	—	13	48	no	—
175	C₂×C₄⋊C₄	—	13	48	no	—
176	C₂×C₄⋊C₄	—	13	48	no	—
177	C₂×C₄⋊C₄	C₂²×D₁₃	1	48	yes	—
178	C₂×C₂²⋊C₄	—	13	24	no	—
179	C₂×C₂²⋊C₄	—	13	48	no	—
180	C₂×C₄²	—	13	12	no	—
181	C₂×C₄²	—	13	48	no	—
182	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
183	C₂×C₁₃	C₂×#541	1	12	yes	complemented
184	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
185	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	2	yes	—
186	D₁₃	—	4	4	no	—
187	C₂⁴	—	13	6	no	—
188	C₂⁴	—	26	4	no	—
189	C₂⁴	—	26	8	no	—
190	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
191	C₂×Q₈	—	2	96	no	—
192	C₂×Q₈	C₂³×D₁₃	1	32	yes	—
193	C₂²×C₄	—	2	16	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	13	12	no	—
196	C₂²×C₄	—	13	24	no	—
197	C₂²×C₄	—	13	6	no	—
198	C₂²×C₄	—	13	16	no	—
199	C₂²×C₄	—	26	4	no	—
200	C₂²×C₄	—	26	16	no	—
201	C₂²×C₄	—	26	8	no	—
202	C₂²×C₄	—	26	48	no	—
203	C₂²×C₄	#6129	1	8	yes	—
204	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
205	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
206	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
207	C₄⋊C₄	—	13	32	no	—
208	C₄⋊C₄	—	13	32	no	—
209	C₄⋊C₄	—	13	64	no	—
210	C₄⋊C₄	—	13	64	no	—
211	C₄⋊C₄	—	13	64	no	—
212	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
213	C₂²⋊C₄	—	13	32	no	—
214	C₂²⋊C₄	—	13	64	no	—
215	C₄²	—	13	16	no	—
216	C₄²	—	13	64	no	—
217	C₄²	C₂³×D₁₃	1	16	yes	—
218	C₁₃	C₂²×#541	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	13	4	no	—
220	C₂³	—	26	12	no	—
221	C₂³	—	26	24	no	—
222	C₂³	—	52	16	no	—
223	C₂³	C₂×#6129	1	4	yes	—
224	C₂³	#88254	1	8	yes	—
225	C₂³	C₂⁴×D₁₃	1	3	yes	—
226	Q₈	—	2	64	no	—
227	C₂×C₄	—	2	48	no	—
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	13	8	no	—
230	C₂×C₄	—	13	16	no	—
231	C₂×C₄	—	13	4	no	—
232	C₂×C₄	—	26	12	no	—
233	C₂×C₄	—	26	24	no	—
234	C₂×C₄	—	26	48	no	—
235	C₂×C₄	—	26	32	no	—
236	C₂×C₄	C₂×#6129	1	24	yes	—
237	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
238	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
239	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
240	C₂²	—	26	4	no	—
241	C₂²	—	26	8	no	—
242	C₂²	—	52	24	no	—
243	C₂²	#11719436	1	16	yes	direct product factor
244	C₂²	C₂²×#6129	1	6	yes	—
245	C₂²	C₂×#88254	1	12	yes	—
246	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
247	C₄	—	2	16	no	—
248	C₄	—	2	8	no	—
249	C₄	—	26	4	no	—
250	C₄	—	26	8	no	—
251	C₄	—	26	16	no	—
252	C₄	C₂²×#6129	1	8	yes	—
253	C₂	—	52	4	no	—
254	C₂	C₂×#11719436	1	12	yes	direct product factor
255	C₂	C₂³×#6129	1	1	yes	characteristic
256	C₂	C₂²×#88254	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁴×D₁₃	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂²×#541
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11719436	C₂	direct (N × H)
3. C₂²×#88124	C₂	semidirect
4. C₂²×#88143	C₂	semidirect
6. C₂²×Q₈×Dic₁₃	C₂	semidirect
8. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
11. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
15. C₂×Q₈×Dic₁₃	C₂²	semidirect
25. #11719436	C₂²	direct (N × H)
27. C₂×#88143	C₂²	semidirect
29. C₂×#88124	C₂²	semidirect
41. C₁₃×C₄⋊Q₈	C₂³	semidirect
45. Q₈×Dic₁₃	C₂³	semidirect
56. #88143	C₂³	semidirect
58. #88124	C₂³	semidirect
92. C₂²×C₄⋊Q₈	D₁₃	semidirect
123. C₂×C₄⋊Q₈	C₂×D₁₃	semidirect
141. C₂²×C₁₃	#541	semidirect
165. C₄⋊Q₈	C₂²×D₁₃	semidirect
183. C₂×C₁₃	C₂×#541	semidirect
218. C₁₃	C₂²×#541	semidirect
243. C₂²	#11719436	direct (N × H)
254. C₂	C₂×#11719436	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719436	2
C₂²×#88152	2
C₂²×#88124	2
C₂²×#88143	2
C₂²×#88200	2
C₂²×Q₈×Dic₁₃	2
C₂²×#88126	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×#541	13

Subgroups — C22×#11719436 #16769736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719436 #16769736