Subgroups — C₂²×#11718692 #16729721

Order 3328.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6107	C₂	1	2	yes	maximal
2	C₂²×#88123	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718692	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88050	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	C₂²	1	2	yes	—
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×#6107	—	2	16	no	—
16	C₂²×#6107	C₂²	1	24	yes	—
17	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
18	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
19	C₂³×#1055	C₂²	1	2	yes	—
20	C₂×#88123	—	2	6	no	—
21	C₂×#88123	C₂²	1	12	yes	—
22	#11718692	C₂²	1	64	yes	direct product factor
23	C₂×#88050	C₂²	1	24	yes	—
24	C₂×C₄⋊#1055	C₂²	1	12	yes	—
25	C₂²×Q₈×C₁₃	—	2	3	no	—
26	C₂²×Q₈×C₁₃	—	2	4	no	—
27	C₂²×Q₈×C₁₃	D₄	1	8	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	—	2	1	no	—
31	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×Dic₁₃	—	2	2	no	—
35	C₂²×Dic₂₆	—	2	8	no	—
36	C₂²×Dic₂₆	—	2	8	no	—
37	C₂²×Dic₂₆	—	2	6	no	—
38	C₂²×Dic₂₆	C₂³	1	12	yes	—
39	C₄×Q₈×C₁₃	—	2	24	no	—
40	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
41	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
44	C₂×C₄²×C₁₃	—	2	12	no	—
45	C₂×C₄²×C₁₃	D₄	1	6	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
47	C₂×#6107	—	2	96	no	—
48	C₂×#6107	C₂³	1	32	yes	—
49	C₂×C₄⋊Dic₁₃	—	2	12	no	—
50	C₂×C₄⋊Dic₁₃	—	2	12	no	—
51	C₂²×#1055	—	2	8	no	—
52	C₂²×#1055	C₂³	1	12	yes	—
53	#88123	—	2	24	no	—
54	#88123	C₂³	1	16	yes	—
55	#88050	C₂³	1	32	yes	—
56	C₄⋊#1055	C₂³	1	16	yes	—
57	C₂²×#462	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	24	no	—
60	C₂×Q₈×C₁₃	—	2	6	no	—
61	C₂×Q₈×C₁₃	—	2	12	no	—
62	C₂×Q₈×C₁₃	C₂×D₄	1	24	yes	—
63	C₂×Q₈×C₁₃	C₂⁴	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	8	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	6	no	—
67	C₂²×C₄×C₁₃	—	2	6	no	—
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	Q₁₆	1	8	yes	complemented
70	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
74	C₂²×C₄×C₁₃	#41	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
76	C₂²×Dic₁₃	—	2	12	no	—
77	C₂²×Dic₁₃	—	4	8	no	—
78	C₂×Dic₂₆	—	2	24	no	—
79	C₂×Dic₂₆	—	2	24	no	—
80	C₂×Dic₂₆	—	2	12	no	—
81	C₂×Dic₂₆	—	4	24	no	—
82	C₂×Dic₂₆	C₂⁴	1	8	yes	—
83	C₁₃×C₄⋊C₄	—	2	16	no	—
84	C₁₃×C₄⋊C₄	—	2	6	no	—
85	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
86	C₄²×C₁₃	—	2	16	no	—
87	C₄²×C₁₃	C₂×D₄	1	12	yes	—
88	C₄²×C₁₃	C₂⁴	1	4	yes	—
89	#6107	—	2	64	no	—
90	C₄⋊Dic₁₃	—	2	16	no	—
91	C₄⋊Dic₁₃	—	2	16	no	—
92	C₂×#1055	—	2	24	no	—
93	C₂×#1055	C₂⁴	1	8	yes	—
94	C₂³×Q₁₆	—	13	2	no	—
95	C₂²×C₄⋊Q₈	—	13	1	no	—
96	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
97	C₂×#462	—	13	24	no	—
98	C₂²×C₄⋊C₈	—	13	1	no	—
99	C₂²×#103	—	13	2	no	—
100	C₂³×C₁₃	#121	1	4	yes	—
101	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
102	C₂³×C₁₃	#137	1	4	yes	—
103	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
104	Q₈×C₁₃	—	2	24	no	—
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	C₂²×D₄	1	8	yes	—
107	C₂×C₄×C₁₃	—	2	24	no	—
108	C₂×C₄×C₁₃	—	2	12	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	4	12	no	—
112	C₂×C₄×C₁₃	#121	1	12	yes	—
113	C₂×C₄×C₁₃	C₂×Q₁₆	1	24	yes	complemented
114	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
120	C₂×Dic₁₃	—	2	8	no	—
121	C₂×Dic₁₃	—	4	24	no	—
122	Dic₂₆	—	2	8	no	—
123	Dic₂₆	—	2	8	no	—
124	Dic₂₆	—	4	24	no	—
125	#1055	—	2	8	no	—
126	C₂³×Q₈	—	13	2	no	—
127	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×Q₁₆	—	13	24	no	—
129	C₂²×Q₁₆	—	26	16	no	—
130	C₂³×C₈	—	13	2	no	—
131	C₂×C₄⋊Q₈	—	13	12	no	—
132	C₂×C₄⋊Q₈	—	26	6	no	—
133	C₂×C₄×Q₈	—	2	6	no	—
134	C₂×C₄×Q₈	C₂×D₁₃	1	12	yes	—
135	C₂²×C₄⋊C₄	—	2	1	no	—
136	C₂²×C₄⋊C₄	—	26	1	no	—
137	C₂²×C₄⋊C₄	—	26	1	no	—
138	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
139	C₂²×C₄²	—	2	1	no	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	#462	—	13	64	no	—
142	C₂×C₄⋊C₈	—	13	12	no	—
143	C₂×#103	—	13	24	no	—
144	C₂²×C₁₃	#462	1	16	yes	complemented
145	C₂²×C₁₃	C₂×#121	1	6	yes	—
146	C₂²×C₁₃	C₂²×Q₁₆	1	6	yes	—
147	C₂²×C₁₃	C₂×#137	1	6	yes	—
148	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
149	C₄×C₁₃	—	2	8	no	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	C₂×#121	1	6	yes	—
153	C₄×C₁₃	C₂²×Q₁₆	1	8	yes	complemented
154	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
156	Dic₁₃	—	4	8	no	—
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	—	2	3	no	—
159	C₂²×Q₈	—	13	12	no	—
160	C₂²×Q₈	—	26	6	no	—
161	C₂²×Q₈	—	26	8	no	—
162	C₂²×Q₈	—	26	8	no	—
163	C₂²×Q₈	D₅₂	1	8	yes	—
164	C₂²×Q₈	C₂²×D₁₃	1	6	yes	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	—	26	2	no	—
168	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×Q₁₆	—	13	32	no	—
172	C₂×Q₁₆	—	26	96	no	—
173	C₂²×C₈	—	13	12	no	—
174	C₂²×C₈	—	26	8	no	—
175	C₄⋊Q₈	—	13	16	no	—
176	C₄⋊Q₈	—	26	24	no	—
177	C₄×Q₈	—	2	24	no	—
178	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	12	no	—
180	C₂×C₄⋊C₄	—	2	3	no	—
181	C₂×C₄⋊C₄	—	26	12	no	—
182	C₂×C₄⋊C₄	—	26	12	no	—
183	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
184	C₂×C₄²	—	2	12	no	—
185	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
186	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
187	C₄⋊C₈	—	13	16	no	—
188	#103	—	13	32	no	—
189	C₂×C₁₃	C₂×#462	1	12	yes	complemented
190	C₂×C₁₃	C₂²×#121	1	1	yes	characteristic
191	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×#137	1	1	yes	characteristic
193	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	12	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	13	8	no	—
198	C₂×Q₈	—	26	12	no	—
199	C₂×Q₈	—	26	24	no	—
200	C₂×Q₈	—	26	24	no	—
201	C₂×Q₈	—	52	24	no	—
202	C₂×Q₈	C₂×D₅₂	1	24	yes	—
203	C₂×Q₈	C₂³×D₁₃	1	4	yes	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	26	12	no	—
210	C₂²×C₄	—	52	8	no	—
211	C₂²×C₄	#6107	1	8	yes	complemented
212	C₂²×C₄	C₂×D₅₂	1	4	yes	—
213	C₂²×C₄	#6127	1	4	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
217	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
218	Q₁₆	—	26	64	no	—
219	C₂×C₈	—	13	8	no	—
220	C₂×C₈	—	26	24	no	—
221	C₄⋊C₄	—	2	16	no	—
222	C₄⋊C₄	—	2	6	no	—
223	C₄⋊C₄	—	26	16	no	—
224	C₄⋊C₄	—	26	16	no	—
225	C₄⋊C₄	C₂³×D₁₃	1	4	yes	—
226	C₄²	—	2	16	no	—
227	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
228	C₄²	C₂³×D₁₃	1	4	yes	—
229	C₁₃	C₂²×#462	1	1	yes	characteristic, complemented, Sylow
230	C₂³	#88184	1	4	yes	—
231	C₂³	C₂×#6107	1	4	yes	—
232	C₂³	#88205	1	4	yes	—
233	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
234	Q₈	—	2	24	no	—
235	Q₈	—	2	4	no	—
236	Q₈	—	26	8	no	—
237	Q₈	—	26	8	no	—
238	Q₈	—	52	24	no	—
239	Q₈	C₂²×D₅₂	1	8	yes	—
240	C₂×C₄	—	2	24	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	26	8	no	—
246	C₂×C₄	—	52	24	no	—
247	C₂×C₄	#88184	1	12	yes	—
248	C₂×C₄	C₂×#6107	1	24	yes	complemented
249	C₂×C₄	C₂²×D₅₂	1	6	yes	—
250	C₂×C₄	C₂×#6127	1	6	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
252	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
254	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
255	C₈	—	26	8	no	—
256	C₂²	#11718692	1	16	yes	direct product factor
257	C₂²	C₂×#88184	1	6	yes	—
258	C₂²	C₂²×#6107	1	6	yes	—
259	C₂²	C₂×#88205	1	6	yes	—
260	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	4	no	—
264	C₄	—	52	8	no	—
265	C₄	C₂×#88184	1	6	yes	—
266	C₄	C₂²×#6107	1	8	yes	complemented
267	C₄	C₂³×D₅₂	1	1	yes	characteristic
268	C₄	C₂²×#6127	1	1	yes	characteristic
269	C₂	C₂×#11718692	1	12	yes	direct product factor
270	C₂	C₂²×#88184	1	1	yes	characteristic
271	C₂	C₂³×#6107	1	1	yes	characteristic
272	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×#462
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
3. C₂×#11718692	C₂	direct (N × H)
22. #11718692	C₂²	direct (N × H)
69. C₂²×C₄×C₁₃	Q₁₆	semidirect
113. C₂×C₄×C₁₃	C₂×Q₁₆	semidirect
144. C₂²×C₁₃	#462	semidirect
153. C₄×C₁₃	C₂²×Q₁₆	semidirect
189. C₂×C₁₃	C₂×#462	semidirect
211. C₂²×C₄	#6107	semidirect
229. C₁₃	C₂²×#462	semidirect
248. C₂×C₄	C₂×#6107	semidirect
256. C₂²	#11718692	direct (N × H)
266. C₄	C₂²×#6107	semidirect
269. C₂	C₂×#11718692	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6107	2
C₂²×#88123	2
C₂×#11718692	2
C₂²×C₄⋊#1055	2
C₂²×#88050	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×#462	13

Subgroups — C22×#11718692 #16729721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718692 #16729721