Subgroups — C₂²×#11718681 #16730396

Order 3328.

2358 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	C₂²×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718681	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88048	C₂	1	2	yes	maximal
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88075	C₂	1	2	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₆	—	2	1	no	—
9	C₂×C₄×Q₈×C₁₃	—	2	6	no	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	12	yes	—
11	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	—	2	1	no	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂³×#1055	C₂²	1	2	yes	—
18	C₂×#88123	—	2	6	no	—
19	C₂×#88123	C₂²	1	12	yes	—
20	C₂×#88075	C₂²	1	24	yes	—
21	#11718681	C₂²	1	64	yes	direct product factor
22	C₂×#88048	C₂²	1	24	yes	—
23	C₂×C₄⋊#1055	C₂²	1	12	yes	—
24	C₂²×Q₈×C₁₃	—	2	3	no	—
25	C₂²×Q₈×C₁₃	—	2	4	no	—
26	C₂²×Q₈×C₁₃	Q₈	1	8	yes	complemented
27	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
28	C₂³×C₄×C₁₃	—	2	1	no	—
29	C₂³×C₄×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂³×Dic₁₃	—	2	2	no	—
34	C₂²×Dic₂₆	—	2	12	no	—
35	C₂²×Dic₂₆	—	4	8	no	—
36	C₄×Q₈×C₁₃	—	2	24	no	—
37	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
41	C₂×C₄²×C₁₃	—	2	12	no	—
42	C₂×C₄²×C₁₃	D₄	1	6	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
44	C₂×C₄⋊Dic₁₃	—	2	6	no	—
45	C₂×C₄⋊Dic₁₃	—	2	12	no	—
46	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
47	C₂²×#1055	—	2	8	no	—
48	C₂²×#1055	C₂³	1	12	yes	—
49	#88123	—	2	24	no	—
50	#88123	C₂³	1	16	yes	—
51	#88075	C₂³	1	32	yes	—
52	#88048	C₂³	1	32	yes	—
53	C₄⋊#1055	C₂³	1	16	yes	—
54	C₂²×#475	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₃	—	2	24	no	—
57	C₂×Q₈×C₁₃	—	2	6	no	—
58	C₂×Q₈×C₁₃	—	2	12	no	—
59	C₂×Q₈×C₁₃	C₂×Q₈	1	24	yes	complemented
60	C₂×Q₈×C₁₃	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₁₃	—	2	8	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	6	no	—
64	C₂²×C₄×C₁₃	—	2	6	no	—
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₃	#41	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
73	C₂²×Dic₁₃	—	2	12	no	—
74	C₂²×Dic₁₃	—	4	8	no	—
75	C₂×Dic₂₆	—	2	16	no	—
76	C₂×Dic₂₆	—	4	48	no	—
77	C₁₃×C₄⋊C₄	—	2	16	no	—
78	C₁₃×C₄⋊C₄	—	2	6	no	—
79	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
80	C₄²×C₁₃	—	2	16	no	—
81	C₄²×C₁₃	C₂×D₄	1	12	yes	—
82	C₄²×C₁₃	C₂⁴	1	4	yes	—
83	C₄⋊Dic₁₃	—	2	12	no	—
84	C₄⋊Dic₁₃	—	2	16	no	—
85	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
86	C₂×#1055	—	2	24	no	—
87	C₂×#1055	C₂⁴	1	8	yes	—
88	C₂²×C₄⋊Q₈	—	13	1	no	—
89	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
90	C₂×#475	—	13	24	no	—
91	C₂²×#106	—	13	2	no	—
92	C₂²×C₄⋊C₈	—	13	1	no	—
93	C₂²×#103	—	13	2	no	—
94	C₂³×C₁₃	C₂²⋊Q₈	1	4	yes	—
95	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
96	C₂³×C₁₃	#137	1	4	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
98	Q₈×C₁₃	—	2	24	no	—
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	C₂²×Q₈	1	8	yes	complemented
101	C₂×C₄×C₁₃	—	2	24	no	—
102	C₂×C₄×C₁₃	—	2	12	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	4	12	no	—
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	12	yes	—
107	C₂×C₄×C₁₃	C₂×SD₁₆	1	24	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
112	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
113	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
114	C₂×Dic₁₃	—	2	8	no	—
115	C₂×Dic₁₃	—	4	24	no	—
116	Dic₂₆	—	4	32	no	—
117	#1055	—	2	8	no	—
118	C₂³×Q₈	—	26	1	no	—
119	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂³×C₈	—	13	2	no	—
121	C₂×C₄⋊Q₈	—	13	12	no	—
122	C₂×C₄⋊Q₈	—	26	6	no	—
123	C₂×C₄×Q₈	—	2	6	no	—
124	C₂×C₄×Q₈	C₂×D₁₃	1	12	yes	—
125	C₂²×C₄⋊C₄	—	2	1	no	—
126	C₂²×C₄⋊C₄	—	13	2	no	—
127	C₂²×C₄⋊C₄	—	26	1	no	—
128	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
129	C₂²×C₄²	—	2	1	no	—
130	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
131	#475	—	13	64	no	—
132	C₂×#106	—	13	24	no	—
133	C₂×C₄⋊C₈	—	13	12	no	—
134	C₂×#103	—	13	24	no	—
135	C₂²×C₁₃	#475	1	16	yes	complemented
136	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	6	yes	—
137	C₂²×C₁₃	C₂²×SD₁₆	1	6	yes	—
138	C₂²×C₁₃	C₂×#137	1	6	yes	—
139	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₁₃	—	2	8	no	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	4	4	no	—
143	C₄×C₁₃	C₂×C₂²⋊Q₈	1	6	yes	—
144	C₄×C₁₃	C₂²×SD₁₆	1	8	yes	—
145	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
147	Dic₁₃	—	4	8	no	—
148	C₂²×Q₈	—	2	4	no	—
149	C₂²×Q₈	—	2	3	no	—
150	C₂²×Q₈	—	26	12	no	—
151	C₂²×Q₈	—	52	8	no	—
152	C₂²×Q₈	Dic₂₆	1	8	yes	complemented
153	C₂²×Q₈	C₂²×D₁₃	1	6	yes	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	—	2	1	no	—
156	C₂³×C₄	—	26	2	no	—
157	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	13	12	no	—
161	C₂²×C₈	—	26	8	no	—
162	C₄⋊Q₈	—	13	16	no	—
163	C₄⋊Q₈	—	26	24	no	—
164	C₄×Q₈	—	2	24	no	—
165	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
166	C₂×C₄⋊C₄	—	2	12	no	—
167	C₂×C₄⋊C₄	—	2	3	no	—
168	C₂×C₄⋊C₄	—	13	12	no	—
169	C₂×C₄⋊C₄	—	26	6	no	—
170	C₂×C₄⋊C₄	—	26	12	no	—
171	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
172	C₂×C₄²	—	2	12	no	—
173	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
175	#106	—	13	32	no	—
176	C₄⋊C₈	—	13	16	no	—
177	#103	—	13	32	no	—
178	C₂×C₁₃	C₂×#475	1	12	yes	complemented
179	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
180	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
181	C₂×C₁₃	C₂²×#137	1	1	yes	characteristic
182	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂×Q₈	—	2	24	no	—
184	C₂×Q₈	—	2	12	no	—
185	C₂×Q₈	—	2	6	no	—
186	C₂×Q₈	—	26	16	no	—
187	C₂×Q₈	—	52	48	no	—
188	C₂×Q₈	C₂×Dic₂₆	1	24	yes	complemented
189	C₂×Q₈	C₂³×D₁₃	1	4	yes	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	6	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	26	12	no	—
196	C₂²×C₄	—	52	8	no	—
197	C₂²×C₄	#6106	1	8	yes	—
198	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
199	C₂²×C₄	#6127	1	4	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
203	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
204	C₂×C₈	—	13	8	no	—
205	C₂×C₈	—	26	24	no	—
206	C₄⋊C₄	—	2	16	no	—
207	C₄⋊C₄	—	2	6	no	—
208	C₄⋊C₄	—	13	8	no	—
209	C₄⋊C₄	—	26	12	no	—
210	C₄⋊C₄	—	26	16	no	—
211	C₄⋊C₄	C₂³×D₁₃	1	4	yes	—
212	C₄²	—	2	16	no	—
213	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
214	C₄²	C₂³×D₁₃	1	4	yes	—
215	C₁₃	C₂²×#475	1	1	yes	characteristic, complemented, Sylow
216	C₂³	#88178	1	4	yes	—
217	C₂³	C₂×#6106	1	4	yes	—
218	C₂³	#88205	1	4	yes	—
219	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
220	Q₈	—	2	24	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	52	32	no	—
223	Q₈	C₂²×Dic₂₆	1	8	yes	complemented
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	12	no	—
229	C₂×C₄	—	26	8	no	—
230	C₂×C₄	—	52	24	no	—
231	C₂×C₄	#88178	1	12	yes	—
232	C₂×C₄	C₂×#6106	1	24	yes	—
233	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
234	C₂×C₄	C₂×#6127	1	6	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
236	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
237	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
238	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
239	C₈	—	26	8	no	—
240	C₂²	#11718681	1	16	yes	direct product factor
241	C₂²	C₂×#88178	1	6	yes	—
242	C₂²	C₂²×#6106	1	6	yes	—
243	C₂²	C₂×#88205	1	6	yes	—
244	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	8	no	—
247	C₄	—	4	4	no	—
248	C₄	—	52	8	no	—
249	C₄	C₂×#88178	1	6	yes	—
250	C₄	C₂²×#6106	1	8	yes	—
251	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
252	C₄	C₂²×#6127	1	1	yes	characteristic
253	C₂	C₂×#11718681	1	12	yes	direct product factor
254	C₂	C₂²×#88178	1	1	yes	characteristic
255	C₂	C₂³×#6106	1	1	yes	characteristic
256	C₂	C₂²×#88205	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂²×#475
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718681	C₂	direct (N × H)
21. #11718681	C₂²	direct (N × H)
26. C₂²×Q₈×C₁₃	Q₈	semidirect
59. C₂×Q₈×C₁₃	C₂×Q₈	semidirect
100. Q₈×C₁₃	C₂²×Q₈	semidirect
135. C₂²×C₁₃	#475	semidirect
152. C₂²×Q₈	Dic₂₆	semidirect
178. C₂×C₁₃	C₂×#475	semidirect
188. C₂×Q₈	C₂×Dic₂₆	semidirect
215. C₁₃	C₂²×#475	semidirect
223. Q₈	C₂²×Dic₂₆	semidirect
240. C₂²	#11718681	direct (N × H)
253. C₂	C₂×#11718681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88123	2
C₂×#11718681	2
C₂²×#88048	2
C₂²×C₄⋊#1055	2
C₂²×#88075	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×#475	13

Subgroups — C22×#11718681 #16730396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718681 #16730396