Subgroups — C₂²×#11719222 #16769621

Order 3328.

8238 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719222	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88142	C₂	1	2	yes	maximal, complemented
3	C₂²×#88144	C₂	1	2	yes	maximal, complemented
4	C₂²×#88132	C₂	1	4	yes	maximal
5	C₂²×C₄×Dic₂₆	C₂	1	4	yes	maximal, complemented
6	C₂²×#88180	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂³×Dic₂₆	C₂²	1	4	yes	—
12	C₂×C₁₃×#117	C₂²	1	24	yes	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88180	C₂²	1	24	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
20	C₂²×#6101	C₂²	1	4	yes	—
21	C₂²×#6101	C₂²	1	4	yes	—
22	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×Dic₁₃	C₂²	1	24	yes	—
26	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
27	#11719222	C₂²	1	256	yes	direct product factor
28	C₂×#88144	C₂²	1	48	yes	complemented
29	C₂×#88142	C₂²	1	48	yes	complemented
30	C₂×#88132	C₂²	1	96	yes	—
31	C₂×C₄×Dic₂₆	C₂²	1	96	yes	complemented
32	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂³×Dic₁₃	—	2	16	no	—
39	C₂³×Dic₁₃	C₂³	1	24	yes	—
40	C₂³×Dic₁₃	C₂³	1	2	yes	—
41	C₂³×Dic₁₃	C₂³	1	4	yes	—
42	C₂³×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×Dic₂₆	—	2	32	no	—
44	C₂²×Dic₂₆	C₂³	1	48	yes	—
45	C₁₃×#117	C₂³	1	64	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
47	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	24	yes	—
49	C₂×C₂²⋊Dic₁₃	C₂³	1	24	yes	—
50	#88180	C₂³	1	64	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
53	C₂×#6101	C₂³	1	48	yes	—
54	C₂×#6101	C₂³	1	48	yes	—
55	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
56	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
57	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
58	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
59	C₂×C₄×Dic₁₃	C₂³	1	64	yes	—
60	#88144	C₂³	1	128	yes	complemented
61	#88142	C₂³	1	128	yes	complemented
62	#88132	C₂³	1	256	yes	—
63	C₄×Dic₂₆	C₂³	1	256	yes	complemented
64	C₂²×#533	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	—	2	8	no	—
66	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
67	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
68	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	—	2	16	no	—
70	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	96	yes	complemented
73	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
77	C₂²×Dic₁₃	—	2	16	no	—
78	C₂²×Dic₁₃	—	2	8	no	—
79	C₂²×Dic₁₃	—	2	96	no	—
80	C₂²×Dic₁₃	#41	1	16	yes	—
81	C₂²×Dic₁₃	C₂⁴	1	32	yes	—
82	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
83	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
84	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
85	C₂×Dic₂₆	—	2	192	no	—
86	C₂×Dic₂₆	C₂⁴	1	64	yes	—
87	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
88	C₁₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
89	C₄²×C₁₃	C₂⁴	1	32	yes	—
90	C₂²⋊Dic₁₃	C₂⁴	1	32	yes	—
91	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
92	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
93	#6101	C₂⁴	1	64	yes	—
94	#6101	C₂⁴	1	64	yes	—
95	C₄×Dic₁₃	C₂⁴	1	32	yes	—
96	C₄×Dic₁₃	C₂⁴	1	32	yes	—
97	C₄×Dic₁₃	C₂⁴	1	16	yes	—
98	C₄×Dic₁₃	C₂⁴	1	16	yes	—
99	C₂×#533	—	13	48	no	—
100	C₂²×C₄⋊Q₈	—	13	2	no	—
101	C₂²×#125	—	13	2	no	—
102	C₂²×C₂²⋊Q₈	—	13	4	no	—
103	C₂²×C₄×Q₈	—	13	4	no	—
104	C₂²×#117	—	13	1	no	—
105	C₂²×#117	D₁₃	1	1	yes	characteristic
106	C₂³×C₄²	—	13	1	no	—
107	C₂³×C₁₃	—	2	32	no	—
108	C₂³×C₁₃	—	2	24	no	—
109	C₂³×C₁₃	C₂²×Q₈	1	24	yes	—
110	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
111	C₂³×C₁₃	C₂×#41	1	8	yes	—
112	C₂³×C₁₃	C₂⁵	1	4	yes	—
113	C₂³×C₁₃	C₂⁵	1	3	yes	—
114	C₂×C₄×C₁₃	—	2	48	no	—
115	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	—
116	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	—
117	C₂×C₄×C₁₃	C₂²×Q₈	1	64	yes	complemented
118	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
120	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
121	C₂×Dic₁₃	—	2	48	no	—
122	C₂×Dic₁₃	—	2	24	no	—
123	C₂×Dic₁₃	—	2	64	no	—
124	C₂×Dic₁₃	C₂×#41	1	48	yes	—
125	C₂×Dic₁₃	C₂⁵	1	8	yes	—
126	C₂×Dic₁₃	C₂⁵	1	16	yes	—
127	C₂×Dic₁₃	C₂⁵	1	8	yes	—
128	Dic₂₆	—	2	128	no	—
129	C₂³×Q₈	—	13	4	no	—
130	C₂⁴×C₄	—	13	2	no	—
131	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
132	#533	—	13	256	no	—
133	C₂×C₄⋊Q₈	—	13	48	no	—
134	C₂×#125	—	13	48	no	—
135	C₂×C₂²⋊Q₈	—	13	96	no	—
136	C₂×C₄×Q₈	—	13	96	no	—
137	C₂×#117	—	13	24	no	—
138	C₂×#117	C₂×D₁₃	1	24	yes	—
139	C₂²×C₄⋊C₄	—	13	4	no	—
140	C₂²×C₄⋊C₄	—	13	4	no	—
141	C₂²×C₄⋊C₄	—	13	2	no	—
142	C₂²×C₄⋊C₄	—	13	4	no	—
143	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂²×C₂²⋊C₄	—	13	2	no	—
145	C₂²×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
146	C₂²×C₄²	—	13	1	no	—
147	C₂²×C₄²	—	13	1	no	—
148	C₂²×C₄²	—	13	24	no	—
149	C₂²×C₄²	—	13	2	no	—
150	C₂²×C₄²	—	13	2	no	—
151	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
152	C₂²×C₁₃	—	2	8	no	—
153	C₂²×C₁₃	—	2	48	no	—
154	C₂²×C₁₃	#533	1	16	yes	complemented
155	C₂²×C₁₃	C₂³×Q₈	1	8	yes	—
156	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
157	C₂²×C₁₃	C₂²×#41	1	12	yes	—
158	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
159	C₄×C₁₃	—	2	16	no	—
160	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
161	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
162	Dic₁₃	—	2	16	no	—
163	Dic₁₃	—	2	8	no	—
164	Dic₁₃	C₂²×#41	1	16	yes	—
165	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
166	C₂²×Q₈	—	13	48	no	—
167	C₂²×Q₈	—	26	32	no	—
168	C₂³×C₄	—	13	2	no	—
169	C₂³×C₄	—	13	4	no	—
170	C₂³×C₄	—	13	2	no	—
171	C₂³×C₄	—	13	24	no	—
172	C₂³×C₄	—	26	16	no	—
173	C₂³×C₄	Dic₂₆	1	16	yes	complemented
174	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
175	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
177	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	13	128	no	—
179	#125	—	13	128	no	—
180	C₂²⋊Q₈	—	13	256	no	—
181	C₄×Q₈	—	13	256	no	—
182	#117	—	13	64	no	—
183	#117	C₂²×D₁₃	1	64	yes	—
184	C₂×C₄⋊C₄	—	13	24	no	—
185	C₂×C₄⋊C₄	—	13	48	no	—
186	C₂×C₄⋊C₄	—	13	48	no	—
187	C₂×C₄⋊C₄	—	13	48	no	—
188	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	—
189	C₂×C₂²⋊C₄	—	13	24	no	—
190	C₂×C₂²⋊C₄	C₂²×D₁₃	1	24	yes	—
191	C₂×C₄²	—	13	24	no	—
192	C₂×C₄²	—	13	24	no	—
193	C₂×C₄²	—	13	12	no	—
194	C₂×C₄²	—	13	12	no	—
195	C₂×C₄²	—	13	64	no	—
196	C₂×C₄²	C₂²×D₁₃	1	24	yes	—
197	C₂×C₁₃	—	2	8	no	—
198	C₂×C₁₃	C₂×#533	1	12	yes	complemented
199	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
200	C₂×C₁₃	C₂³×#41	1	2	yes	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	C₂×Dic₂₆	1	8	yes	—
203	C₂⁴	C₂³×D₁₃	1	6	yes	—
204	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
205	C₂×Q₈	—	13	64	no	—
206	C₂×Q₈	—	26	192	no	—
207	C₂²×C₄	—	2	16	no	—
208	C₂²×C₄	—	13	16	no	—
209	C₂²×C₄	—	13	12	no	—
210	C₂²×C₄	—	13	12	no	—
211	C₂²×C₄	—	13	32	no	—
212	C₂²×C₄	—	13	24	no	—
213	C₂²×C₄	—	26	96	no	—
214	C₂²×C₄	—	26	8	no	—
215	C₂²×C₄	—	26	16	no	—
216	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
217	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
218	C₂²×C₄	C₂×Dic₂₆	1	96	yes	complemented
219	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
220	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
221	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
223	C₄⋊C₄	—	13	32	no	—
224	C₄⋊C₄	—	13	64	no	—
225	C₄⋊C₄	—	13	64	no	—
226	C₄⋊C₄	—	13	64	no	—
227	C₄⋊C₄	C₂³×D₁₃	1	32	yes	—
228	C₂²⋊C₄	—	13	32	no	—
229	C₂²⋊C₄	C₂³×D₁₃	1	32	yes	—
230	C₄²	—	13	32	no	—
231	C₄²	—	13	16	no	—
232	C₄²	—	13	16	no	—
233	C₄²	—	13	32	no	—
234	C₄²	C₂³×D₁₃	1	32	yes	—
235	C₁₃	C₂²×#533	1	1	yes	characteristic, complemented, Sylow
236	C₂³	—	2	32	no	—
237	C₂³	—	2	24	no	—
238	C₂³	C₂²×Dic₂₆	1	24	yes	—
239	C₂³	C₂²×Dic₂₆	1	4	yes	—
240	C₂³	#41×D₁₃	1	8	yes	—
241	C₂³	C₂⁴×D₁₃	1	4	yes	—
242	C₂³	C₂⁴×D₁₃	1	3	yes	—
243	Q₈	—	26	128	no	—
244	C₂×C₄	—	2	48	no	—
245	C₂×C₄	—	13	48	no	—
246	C₂×C₄	—	13	8	no	—
247	C₂×C₄	—	13	8	no	—
248	C₂×C₄	—	13	16	no	—
249	C₂×C₄	—	26	64	no	—
250	C₂×C₄	—	26	24	no	—
251	C₂×C₄	—	26	48	no	—
252	C₂×C₄	C₂²×Dic₂₆	1	24	yes	—
253	C₂×C₄	C₂²×Dic₂₆	1	24	yes	—
254	C₂×C₄	C₂²×Dic₂₆	1	64	yes	complemented
255	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
256	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
257	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	48	no	—
260	C₂²	#11719222	1	16	yes	direct product factor
261	C₂²	C₂³×Dic₂₆	1	8	yes	—
262	C₂²	C₂³×Dic₂₆	1	6	yes	—
263	C₂²	C₂×#41×D₁₃	1	12	yes	—
264	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
265	C₄	—	2	16	no	—
266	C₄	—	13	16	no	—
267	C₄	—	26	8	no	—
268	C₄	—	26	16	no	—
269	C₄	C₂³×Dic₂₆	1	8	yes	—
270	C₄	C₂³×Dic₂₆	1	8	yes	—
271	C₂	—	2	8	no	—
272	C₂	C₂×#11719222	1	12	yes	direct product factor
273	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
274	C₂	C₂²×#41×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂²×#533
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#11719222	C₂	direct (N × H)
2. C₂²×#88142	C₂	semidirect
3. C₂²×#88144	C₂	semidirect
5. C₂²×C₄×Dic₂₆	C₂	semidirect
27. #11719222	C₂²	direct (N × H)
28. C₂×#88144	C₂²	semidirect
29. C₂×#88142	C₂²	semidirect
31. C₂×C₄×Dic₂₆	C₂²	semidirect
33. C₂³×C₄×C₁₃	Q₈	semidirect
60. #88144	C₂³	semidirect
61. #88142	C₂³	semidirect
63. C₄×Dic₂₆	C₂³	semidirect
72. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
117. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
154. C₂²×C₁₃	#533	semidirect
173. C₂³×C₄	Dic₂₆	semidirect
198. C₂×C₁₃	C₂×#533	semidirect
218. C₂²×C₄	C₂×Dic₂₆	semidirect
235. C₁₃	C₂²×#533	semidirect
254. C₂×C₄	C₂²×Dic₂₆	semidirect
260. C₂²	#11719222	direct (N × H)
272. C₂	C₂×#11719222	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×#11719222	2
C₂²×#88142	2
C₂²×#88144	2
C₂²×#88132	2
C₂²×C₄×Dic₂₆	2
C₂²×#88180	2
C₂³×C₄×Dic₁₃	2
C₂²×C₁₃×#117	2
C₂²×#533	13

Subgroups — C22×#11719222 #16769621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719222 #16769621