Subgroups — C₂×C₄×#88048 #15860959

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88048	C₂	1	2	yes	maximal
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₄×#88048	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718121	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	—	2	1	no	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₁₃	—	2	1	no	—
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₂³×#1055	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₁₃×#95	—	2	1	no	—
17	C₂×#88071	—	2	1	no	—
18	C₂×#88048	C₄	1	16	yes	direct product factor
19	C₂×#88048	C₂²	1	8	yes	—
20	C₄×C₄⋊Dic₁₃	—	2	2	no	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
22	C₂×C₄×#1055	C₄	1	8	yes	complemented
23	C₂×C₄×#1055	C₂²	1	4	yes	—
24	#11718121	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×Dic₁₃	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
34	C₂×C₄²×C₁₃	—	2	4	no	—
35	C₂×C₄²×C₁₃	—	4	4	no	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	4	no	—
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
44	C₂×C₄×Dic₁₃	—	2	4	no	—
45	C₂×C₄×Dic₁₃	—	4	4	no	—
46	C₂²×#1055	—	2	4	no	—
47	C₂²×#1055	C₂×C₄	1	4	yes	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₂²×#1055	C₂×C₄	1	8	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#95	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88048	C₂×C₄	1	32	yes	direct product factor
55	C₄×#1055	C₂×C₄	1	16	yes	complemented
56	C₂×C₄×#106	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	8	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
66	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	8	no	—
76	C₂²×Dic₁₃	—	2	4	no	—
77	C₂²×Dic₁₃	—	4	4	no	—
78	C₂²×Dic₁₃	—	4	4	no	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
82	C₄²×C₁₃	—	4	8	no	—
83	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
86	C₄⋊Dic₁₃	—	2	8	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
89	C₄×Dic₁₃	—	4	8	no	—
90	C₂×#1055	—	2	8	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	C₄²	1	16	yes	complemented
93	C₂×#1055	C₂²×C₄	1	4	yes	—
94	C₂×#1055	C₂²×C₄	1	4	yes	—
95	C₂×#1055	C₂²×C₄	1	8	yes	—
96	C₂²×#106	—	13	2	no	—
97	C₂²×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₄⋊C₄	—	13	1	no	—
99	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
100	C₄×#106	—	13	8	no	—
101	C₂×#340	—	13	2	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₃	C₄×D₄	1	4	yes	—
104	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
105	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
106	C₂³×C₁₃	#135	1	4	yes	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	16	no	—
114	C₂×C₄×C₁₃	#106	1	16	yes	complemented
115	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₃	#117	1	4	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
122	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×Dic₁₃	—	2	8	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	4	16	no	—
132	#1055	—	2	4	no	—
133	#1055	C₂×C₄²	1	8	yes	complemented
134	C₂³×C₈	—	13	1	no	—
135	C₂³×C₈	—	13	1	no	—
136	C₂²×C₄⋊C₄	—	13	2	no	—
137	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
138	C₂²×C₄²	—	2	1	no	—
139	C₂²×C₄²	—	26	1	no	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#106	—	13	16	no	—
142	C₂×#106	—	13	8	no	—
143	C₂×C₄×C₈	—	13	8	no	—
144	C₂×C₄×C₈	—	13	4	no	—
145	C₄×C₄⋊C₄	—	2	2	no	—
146	C₄×C₄⋊C₄	—	13	4	no	—
147	C₄×C₄⋊C₄	—	26	2	no	—
148	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
149	C₂×#95	—	2	1	no	—
150	C₂×#95	—	26	1	no	—
151	#340	—	13	8	no	—
152	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
153	C₂²×C₁₃	C₂×#106	1	4	yes	—
154	C₂²×C₁₃	#427	1	4	yes	—
155	C₂²×C₁₃	C₄×SD₁₆	1	8	yes	—
156	C₂²×C₁₃	C₄×SD₁₆	1	8	yes	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
159	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#135	1	2	yes	—
162	C₄×C₁₃	—	2	4	no	—
163	C₄×C₁₃	—	4	8	no	—
164	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₄×C₁₃	C₂×#106	1	8	yes	complemented
166	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
169	Dic₁₃	—	4	8	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	26	2	no	—
172	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₈	—	13	4	no	—
176	C₂²×C₈	—	13	8	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	4	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	26	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	13	8	no	—
184	C₂×C₄⋊C₄	—	13	4	no	—
185	C₂×C₄⋊C₄	—	26	4	no	—
186	C₂×C₄⋊C₄	—	26	2	no	—
187	C₂×C₄⋊C₄	C₄×D₁₃	1	8	yes	—
188	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	—	4	4	no	—
191	C₂×C₄²	—	26	4	no	—
192	C₂×C₄²	—	52	4	no	—
193	C₂×C₄²	Dic₂₆	1	4	yes	—
194	C₂×C₄²	C₄×D₁₃	1	4	yes	—
195	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
196	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
197	#106	—	13	32	no	—
198	C₄×C₈	—	13	16	no	—
199	#95	—	2	4	no	—
200	#95	—	26	4	no	—
201	C₂×C₁₃	C₄×#106	1	8	yes	complemented
202	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
205	C₂×C₁₃	C₂×C₄×SD₁₆	1	2	yes	—
206	C₂×C₁₃	C₂×C₄×SD₁₆	1	2	yes	—
207	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	26	8	no	—
214	C₂²×C₄	—	26	4	no	—
215	C₂²×C₄	—	52	4	no	—
216	C₂²×C₄	—	52	4	no	—
217	C₂²×C₄	#6101	1	4	yes	—
218	C₂²×C₄	#6101	1	4	yes	—
219	C₂²×C₄	#6105	1	4	yes	—
220	C₂²×C₄	#6106	1	4	yes	—
221	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
223	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
224	C₂²×C₄	#6127	1	4	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
228	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
229	C₂×C₈	—	13	16	no	—
230	C₂×C₈	—	13	4	no	—
231	C₂×C₈	—	13	8	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	26	8	no	—
234	C₂×C₈	—	26	4	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	13	8	no	—
238	C₄⋊C₄	—	26	8	no	—
239	C₄⋊C₄	—	26	4	no	—
240	C₄⋊C₄	C₂×C₄×D₁₃	1	8	yes	—
241	C₄²	—	4	8	no	—
242	C₄²	—	52	8	no	—
243	C₄²	#6101	1	8	yes	—
244	C₄²	C₂×Dic₂₆	1	4	yes	—
245	C₄²	C₂×C₄×D₁₃	1	4	yes	—
246	C₁₃	C₂×C₄×#106	1	1	yes	characteristic, complemented, Sylow
247	C₂³	C₂×#6101	1	2	yes	—
248	C₂³	C₄×C₂²⋊D₁₃	1	4	yes	—
249	C₂³	C₂×#6105	1	2	yes	—
250	C₂³	C₂×#6106	1	2	yes	—
251	C₂³	#88204	1	4	yes	—
252	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	16	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	26	8	no	—
260	C₂×C₄	—	52	16	no	—
261	C₂×C₄	—	52	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	#88048	1	16	yes	direct product factor
264	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
265	C₂×C₄	C₄²×D₁₃	1	4	yes	—
266	C₂×C₄	#88126	1	4	yes	—
267	C₂×C₄	C₂×#6101	1	4	yes	—
268	C₂×C₄	C₂×#6101	1	2	yes	—
269	C₂×C₄	C₂×#6101	1	4	yes	—
270	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
271	C₂×C₄	C₂×#6105	1	4	yes	—
272	C₂×C₄	C₂×#6106	1	4	yes	—
273	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#6127	1	2	yes	—
277	C₈	—	13	8	no	—
278	C₈	—	26	4	no	—
279	C₂²	C₄×#6101	1	4	yes	—
280	C₂²	C₂×#88048	1	4	yes	—
281	C₂²	#11718659	1	4	yes	—
282	C₂²	C₄×#6105	1	8	yes	—
283	C₂²	C₄×#6106	1	8	yes	—
284	C₂²	C₂²×#6101	1	1	yes	characteristic
285	C₂²	C₂×C₄×C₂²⋊D₁₃	1	2	yes	—
286	C₂²	C₂²×#6105	1	1	yes	characteristic
287	C₂²	C₂²×#6106	1	1	yes	characteristic
288	C₂²	C₂×#88204	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	4	8	no	—
291	C₄	—	52	8	no	—
292	C₄	C₄×#6101	1	4	yes	—
293	C₄	C₂×#88048	1	8	yes	direct product factor
294	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
295	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
296	C₄	C₂×#88126	1	1	yes	characteristic
297	C₂	C₄×#88048	1	8	yes	direct product factor
298	C₂	C₂×C₄×#6101	1	1	yes	characteristic
299	C₂	C₂²×#88048	1	1	yes	characteristic
300	C₂	C₂×#11718659	1	1	yes	characteristic
301	C₂	C₂×C₄×#6105	1	2	yes	—
302	C₂	C₂×C₄×#6106	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂×C₄×#106
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄×#88048	C₂	direct (N × H)
18. C₂×#88048	C₄	direct (N × H)
22. C₂×C₄×#1055	C₄	semidirect
54. #88048	C₂×C₄	direct (N × H)
55. C₄×#1055	C₂×C₄	semidirect
92. C₂×#1055	C₄²	semidirect
114. C₂×C₄×C₁₃	#106	semidirect
133. #1055	C₂×C₄²	semidirect
165. C₄×C₁₃	C₂×#106	semidirect
201. C₂×C₁₃	C₄×#106	semidirect
246. C₁₃	C₂×C₄×#106	semidirect
263. C₂×C₄	#88048	direct (N × H)
293. C₄	C₂×#88048	direct (N × H)
297. C₂	C₄×#88048	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×#88048	2
C₂²×C₄×#1055	2
C₂×C₄×C₄⋊Dic₁₃	2
C₄×#88048	2
C₂×#11718121	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×C₄×#106	13

Subgroups — C2×C4×#88048 #15860959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88048 #15860959