Subgroups — C₂×C₄⋊C₄×#1055 #15860849

Order 3328.

1438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1055	C₂	1	2	yes	maximal
3	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#1055	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718116	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₁₃×#95	C₄	1	2	yes	—
16	C₂×C₄⋊#1055	C₄	1	8	yes	complemented
17	C₂×C₄⋊#1055	C₂²	1	4	yes	—
18	C₂×C₄×#1055	—	2	8	no	—
19	C₂×C₄×#1055	C₄	1	8	yes	complemented
20	C₂×C₄×#1055	C₂²	1	8	yes	—
21	C₂×C₄×#1055	C₂²	1	4	yes	—
22	#11718116	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₃×C₄⋊C₄	C₈	1	16	yes	complemented
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₁₃	—	2	4	no	—
30	C₂×C₄²×C₁₃	—	2	4	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂²×#1055	—	2	4	no	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₂²×#1055	—	2	4	no	—
41	C₂²×#1055	C₂×C₄	1	4	yes	—
42	C₂²×#1055	C₂×C₄	1	8	yes	—
43	C₂²×#1055	C₂×C₄	1	4	yes	—
44	C₂²×#1055	D₄	1	4	yes	—
45	C₂²×#1055	Q₈	1	4	yes	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₂²×#1055	C₂³	1	2	yes	—
49	C₁₃×#95	C₂×C₄	1	8	yes	—
50	C₄⋊#1055	C₂×C₄	1	16	yes	complemented
51	C₄×#1055	—	2	16	no	—
52	C₄×#1055	C₂×C₄	1	16	yes	complemented
53	C₂×C₈×C₄⋊C₄	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	16	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	C₂×C₈	1	32	yes	complemented
74	C₄²×C₁₃	—	2	8	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	C₄²	1	8	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×#1055	—	2	8	no	—
81	C₂×#1055	—	2	16	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	C₄⋊C₄	1	16	yes	direct product factor
87	C₂×#1055	C₂²×C₄	1	4	yes	—
88	C₂×#1055	C₂²×C₄	1	8	yes	—
89	C₂×#1055	C₂²×C₄	1	4	yes	—
90	C₂×#1055	C₂×D₄	1	4	yes	—
91	C₂×#1055	C₂×Q₈	1	4	yes	—
92	C₂²×C₄⋊C₈	—	13	1	no	—
93	C₂²×C₄×C₈	—	13	1	no	—
94	C₂²×C₄×C₈	—	13	2	no	—
95	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
96	C₈×C₄⋊C₄	—	13	8	no	—
97	C₂×#336	—	13	2	no	—
98	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
99	C₂³×C₁₃	C₄×D₄	1	2	yes	—
100	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
101	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
102	C₂³×C₁₃	#131	1	4	yes	—
103	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	8	no	—
107	C₂×C₄×C₁₃	—	2	8	no	—
108	C₂×C₄×C₁₃	—	2	32	no	—
109	C₂×C₄×C₁₃	C₄×C₈	1	16	yes	complemented
110	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₁₃	#117	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	16	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	#1055	—	2	4	no	—
128	#1055	—	2	8	no	—
129	#1055	C₂×C₄⋊C₄	1	8	yes	direct product factor
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	—	13	2	no	—
132	C₂³×C₈	—	13	1	no	—
133	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
134	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
135	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	13	8	no	—
137	C₂×C₄⋊C₈	—	13	4	no	—
138	C₂×C₄×C₈	—	13	8	no	—
139	C₂×C₄×C₈	—	13	4	no	—
140	C₂×C₄×C₈	—	13	8	no	—
141	C₂×C₄×C₈	—	26	8	no	—
142	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
143	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
144	C₂×#95	Dic₁₃	1	2	yes	—
145	#336	—	13	8	no	—
146	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
147	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
148	C₂²×C₁₃	#405	1	4	yes	—
149	C₂²×C₁₃	C₈×D₄	1	8	yes	—
150	C₂²×C₁₃	C₈×Q₈	1	8	yes	—
151	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
155	C₂²×C₁₃	C₂×#131	1	2	yes	—
156	C₄×C₁₃	—	2	4	no	—
157	C₄×C₁₃	—	2	16	no	—
158	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
159	C₄×C₁₃	C₂×C₄×C₈	1	8	yes	complemented
160	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
165	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	4	no	—
170	C₂²×C₈	—	13	4	no	—
171	C₂²×C₈	—	13	2	no	—
172	C₂²×C₈	—	13	4	no	—
173	C₂²×C₈	—	13	8	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	—	13	4	no	—
176	C₂²×C₈	—	26	4	no	—
177	C₂²×C₈	—	26	2	no	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	—	26	4	no	—
180	C₂×C₄⋊C₄	#1055	1	16	yes	direct product factor
181	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	—	2	4	no	—
184	C₂×C₄²	C₄×D₁₃	1	4	yes	—
185	C₂×C₄²	C₄×D₁₃	1	4	yes	—
186	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
187	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
188	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
189	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
190	C₄⋊C₈	—	13	16	no	—
191	C₄×C₈	—	13	16	no	—
192	C₄×C₈	—	26	16	no	—
193	#95	C₂×Dic₁₃	1	8	yes	—
194	C₂×C₁₃	C₈×C₄⋊C₄	1	8	yes	complemented
195	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
198	C₂×C₁₃	C₂×C₈×D₄	1	2	yes	—
199	C₂×C₁₃	C₂×C₈×Q₈	1	2	yes	—
200	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	C₂×#1055	1	16	yes	—
205	C₂²×C₄	C₂×#1055	1	8	yes	—
206	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
207	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
210	C₂²×C₄	D₄×D₁₃	1	2	yes	—
211	C₂²×C₄	#6129	1	2	yes	—
212	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
213	C₂²×C₄	#6131	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
215	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
219	C₂×C₈	—	13	16	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	—	13	4	no	—
224	C₂×C₈	—	13	8	no	—
225	C₂×C₈	—	26	8	no	—
226	C₂×C₈	—	26	16	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	4	no	—
231	C₄⋊C₄	C₂×#1055	1	32	yes	direct product factor
232	C₄²	—	2	8	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	C₄×Dic₁₃	1	8	yes	—
236	C₄²	C₂×C₄×D₁₃	1	4	yes	—
237	C₄²	C₂×C₄×D₁₃	1	4	yes	—
238	C₁₃	C₂×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
239	C₂³	C₂²×#1055	1	4	yes	—
240	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
241	C₂³	D₄×Dic₁₃	1	2	yes	—
242	C₂³	Q₈×Dic₁₃	1	2	yes	—
243	C₂³	#88202	1	4	yes	—
244	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	32	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	C₄×#1055	1	16	yes	complemented
251	C₂×C₄	#88141	1	4	yes	—
252	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
253	C₂×C₄	#88146	1	4	yes	—
254	C₂×C₄	#88147	1	4	yes	—
255	C₂×C₄	C₂²×#1055	1	16	yes	—
256	C₂×C₄	C₂²×#1055	1	8	yes	—
257	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
258	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
259	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
260	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
261	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
262	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#6129	1	1	yes	characteristic
266	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#6131	1	1	yes	characteristic
268	C₈	—	13	8	no	—
269	C₈	—	26	4	no	—
270	C₈	—	26	8	no	—
271	C₂²	C₂×C₄×#1055	1	4	yes	—
272	C₂²	C₄⋊C₄×Dic₁₃	1	4	yes	—
273	C₂²	#11718658	1	4	yes	—
274	C₂²	D₄×#1055	1	8	yes	—
275	C₂²	Q₈×#1055	1	8	yes	—
276	C₂²	C₂³×#1055	1	2	yes	—
277	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
278	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
279	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
280	C₂²	C₂×#88202	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	2	16	no	—
283	C₄	C₂×C₄×#1055	1	8	yes	complemented
284	C₄	C₄⋊C₄×Dic₁₃	1	4	yes	—
285	C₄	C₂×#88141	1	1	yes	characteristic
286	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
287	C₄	C₂×#88146	1	1	yes	characteristic
288	C₄	C₂×#88147	1	1	yes	characteristic
289	C₂	C₄⋊C₄×#1055	1	8	yes	direct product factor
290	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
291	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
292	C₂	C₂×#11718658	1	1	yes	characteristic
293	C₂	C₂×D₄×#1055	1	2	yes	—
294	C₂	C₂×Q₈×#1055	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×#1055	C₂	direct (N × H)
16. C₂×C₄⋊#1055	C₄	semidirect
19. C₂×C₄×#1055	C₄	semidirect
27. C₂×C₁₃×C₄⋊C₄	C₈	semidirect
50. C₄⋊#1055	C₂×C₄	semidirect
52. C₄×#1055	C₂×C₄	semidirect
73. C₁₃×C₄⋊C₄	C₂×C₈	semidirect
86. C₂×#1055	C₄⋊C₄	direct (N × H)
109. C₂×C₄×C₁₃	C₄×C₈	semidirect
129. #1055	C₂×C₄⋊C₄	direct (N × H)
159. C₄×C₁₃	C₂×C₄×C₈	semidirect
180. C₂×C₄⋊C₄	#1055	direct (N × H)
194. C₂×C₁₃	C₈×C₄⋊C₄	semidirect
231. C₄⋊C₄	C₂×#1055	direct (N × H)
238. C₁₃	C₂×C₈×C₄⋊C₄	semidirect
250. C₂×C₄	C₄×#1055	semidirect
283. C₄	C₂×C₄×#1055	semidirect
289. C₂	C₄⋊C₄×#1055	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×C₄⋊#1055	2
C₂²×C₄×#1055	2
C₄⋊C₄×#1055	2
C₂×#11718116	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×C₈×C₄⋊C₄	13

Subgroups — C2×C4⋊C4×#1055 #15860849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×#1055 #15860849