Subgroups — C₂×C₄×#88068 #16016186

Order 3328.

742 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88068	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
3	C₄×#88068	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718082	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718083	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₅₂	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	—	2	2	no	—
14	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	complemented
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#103	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88068	C₄	1	8	yes	direct product factor
19	C₂×#88068	C₂²	1	4	yes	—
20	C₂×#88061	—	2	1	no	—
21	C₂×#88058	C₂²	1	1	yes	characteristic
22	C₂²×#6090	C₂²	1	1	yes	characteristic
23	C₂²×#6090	C₂²	1	1	yes	characteristic
24	C₄×D₁₀₄	—	2	2	no	—
25	C₄×D₁₀₄	C₂²	1	4	yes	—
26	#11718156	C₂²	1	4	yes	complemented
27	#11718083	C₂²	1	4	yes	—
28	#11718082	C₂²	1	4	yes	complemented
29	C₄×#6090	C₂²	1	4	yes	complemented
30	C₂²×Q₈×C₁₃	—	2	1	no	—
31	C₂²×D₅₂	—	2	1	no	—
32	C₂²×C₄×D₁₃	—	4	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	2	no	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	—	2	4	no	—
40	C₄×Q₈×C₁₃	—	4	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
43	C₂×C₄²×C₁₃	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×#6103	—	4	1	no	—
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×D₁₀₄	—	2	1	no	—
48	C₂×D₁₀₄	—	2	2	no	—
49	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
50	C₂×D₁₀₄	C₂³	1	2	yes	—
51	C₄×D₅₂	—	2	4	no	—
52	C₄×D₅₂	—	4	2	no	—
53	C₁₃×#107	—	2	1	no	—
54	C₁₃×#107	C₂³	1	2	yes	—
55	C₁₃×#103	—	2	4	no	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	#88068	C₂×C₄	1	16	yes	direct product factor
59	#88061	—	2	4	no	—
60	#88058	—	2	1	no	—
61	#88058	C₂³	1	2	yes	—
62	C₂×#6090	—	2	2	no	—
63	C₂×#6090	C₂×C₄	1	4	yes	complemented
64	C₂×#6090	C₂³	1	2	yes	—
65	C₂×#6090	C₂³	1	2	yes	—
66	C₂×C₄×SD₃₂	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	2	4	no	—
69	C₂×Q₈×C₁₃	—	2	2	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	4	1	no	—
78	C₂×D₅₂	—	2	4	no	—
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	—	4	1	no	—
81	C₂×D₅₂	—	4	2	no	—
82	C₂×C₄×D₁₃	—	4	4	no	—
83	C₂×C₄×D₁₃	—	8	4	no	—
84	C₁₃×Q₁₆	—	2	4	no	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	complemented
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	4	1	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₄²×C₁₃	—	4	4	no	—
97	C₄²×C₁₃	D₈	1	2	yes	—
98	C₄²×C₁₃	C₂×D₄	1	2	yes	—
99	#6103	—	4	4	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	4	1	no	—
102	D₁₀₄	—	2	4	no	—
103	D₁₀₄	—	2	2	no	—
104	D₁₀₄	C₂²×C₄	1	4	yes	—
105	#6090	—	2	2	no	—
106	#6090	C₂²×C₄	1	4	yes	complemented
107	C₂²×SD₃₂	—	13	1	no	—
108	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
109	C₂×C₄×D₈	—	13	1	no	—
110	C₄×SD₃₂	—	13	8	no	—
111	C₂×#367	—	13	1	no	—
112	C₂×#358	—	13	1	no	—
113	C₂×#357	—	13	1	no	—
114	C₂×C₄×C₁₆	—	13	1	no	—
115	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	—	4	4	no	—
118	C₂²×D₁₃	—	8	2	no	—
119	C₂²×D₁₃	—	8	2	no	—
120	Q₈×C₁₃	—	2	4	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	#135	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃	—	4	2	no	—
136	C₂×Dic₁₃	—	8	2	no	—
137	D₅₂	—	2	4	no	—
138	D₅₂	—	4	2	no	—
139	D₅₂	—	4	4	no	—
140	C₄×D₁₃	—	8	8	no	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	C₄×D₄	1	2	yes	—
143	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₂²×D₈	—	13	1	no	—
147	C₂×C₄×Q₈	—	2	1	no	—
148	C₂×C₄×D₄	—	26	1	no	—
149	C₂×SD₃₂	—	13	8	no	—
150	C₂×SD₃₂	—	13	4	no	—
151	C₂²×C₁₆	—	13	1	no	—
152	C₂²×C₁₆	—	13	1	no	—
153	C₄×Q₁₆	—	2	2	no	—
154	C₄×Q₁₆	C₂×D₁₃	1	4	yes	complemented
155	C₄×D₈	—	13	4	no	—
156	C₄×D₈	—	26	2	no	—
157	C₂×#107	—	13	1	no	—
158	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
159	C₂×#103	—	2	1	no	—
160	C₂×#102	—	26	1	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	#367	—	13	4	no	—
163	#358	—	13	4	no	—
164	#357	—	13	4	no	—
165	C₄×C₁₆	—	13	4	no	—
166	C₂²×C₁₃	C₄×D₈	1	2	yes	—
167	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
168	C₂²×C₁₃	#508	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₁₃	—	4	4	no	—
171	C₂×D₁₃	—	8	2	no	—
172	C₂×D₁₃	—	8	2	no	—
173	C₂×D₁₃	—	8	8	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	C₄×D₈	1	2	yes	—
178	C₄×C₁₃	C₂×SD₃₂	1	4	yes	complemented
179	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
181	Dic₁₃	—	8	2	no	—
182	C₂²×Q₈	—	2	1	no	—
183	C₂²×D₄	—	26	1	no	—
184	C₂³×C₄	—	52	1	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	C₄×D₁₃	1	4	yes	complemented
188	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
189	C₂×D₈	—	13	4	no	—
190	C₂×D₈	—	13	2	no	—
191	C₂×D₈	—	26	2	no	—
192	C₂×D₈	—	26	1	no	—
193	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	4	2	no	—
197	C₄×D₄	—	26	4	no	—
198	C₄×D₄	—	52	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄⋊C₄	—	26	1	no	—
202	C₂×C₂²⋊C₄	—	52	1	no	—
203	C₂×C₄²	—	4	1	no	—
204	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
205	SD₃₂	—	13	16	no	—
206	C₂×C₁₆	—	13	4	no	—
207	C₂×C₁₆	—	13	2	no	—
208	C₂×C₁₆	—	13	2	no	—
209	C₂×C₁₆	—	26	2	no	—
210	#107	—	2	1	no	—
211	#107	—	13	2	no	—
212	#107	—	26	1	no	—
213	#107	C₂²×D₁₃	1	2	yes	—
214	#103	—	2	4	no	—
215	#102	—	26	4	no	—
216	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
217	C₄×C₈	C₂²×D₁₃	1	2	yes	—
218	C₂×C₁₃	C₄×SD₃₂	1	4	yes	complemented
219	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
220	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
222	D₁₃	—	8	4	no	—
223	C₂⁴	—	52	1	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×D₄	—	26	4	no	—
229	C₂×D₄	—	26	2	no	—
230	C₂×D₄	—	52	1	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	52	1	no	—
235	C₂²×C₄	—	52	4	no	—
236	C₂²×C₄	—	104	4	no	—
237	C₂²×C₄	#6104	1	1	yes	characteristic
238	C₂²×C₄	#6104	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	Q₁₆	—	2	4	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	C₂×C₄×D₁₃	1	4	yes	complemented
243	D₈	—	13	4	no	—
244	D₈	—	26	4	no	—
245	D₈	—	26	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
248	C₂×C₈	#6127	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	26	2	no	—
256	C₄⋊C₄	—	52	1	no	—
257	C₂²⋊C₄	—	52	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	#6104	1	2	yes	—
260	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
261	C₁₆	—	13	4	no	—
262	C₁₆	—	26	2	no	—
263	C₁₃	C₂×C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	52	4	no	—
265	C₂³	—	52	2	no	—
266	C₂³	—	104	2	no	—
267	C₂³	—	104	2	no	—
268	C₂³	C₂×#6104	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	26	4	no	—
273	D₄	—	52	4	no	—
274	D₄	—	52	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	52	2	no	—
281	C₂×C₄	—	104	2	no	—
282	C₂×C₄	—	104	8	no	—
283	C₂×C₄	#88068	1	4	yes	direct product factor
284	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
285	C₂×C₄	C₂×#6104	1	2	yes	—
286	C₂×C₄	C₂×#6104	1	1	yes	characteristic
287	C₂×C₄	C₂×#6104	1	2	yes	—
288	C₂×C₄	#88204	1	2	yes	—
289	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
292	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
293	C₈	C₂×#6127	1	1	yes	characteristic
294	C₂²	—	52	4	no	—
295	C₂²	—	104	2	no	—
296	C₂²	—	104	8	no	—
297	C₂²	—	104	2	no	—
298	C₂²	C₄×#6104	1	2	yes	—
299	C₂²	C₂×#88068	1	2	yes	—
300	C₂²	#11718855	1	2	yes	—
301	C₂²	C₂²×#6104	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	2	no	—
305	C₄	—	104	2	no	—
306	C₄	C₄×#6104	1	2	yes	—
307	C₄	C₂×#88068	1	4	yes	direct product factor
308	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
309	C₄	C₂×#88204	1	1	yes	characteristic
310	C₂	—	104	4	no	—
311	C₂	C₄×#88068	1	4	yes	direct product factor
312	C₂	C₂×C₄×#6104	1	1	yes	characteristic
313	C₂	C₂²×#88068	1	1	yes	characteristic
314	C₂	C₂×#11718855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6104
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	C₂×C₄×SD₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#88068	C₂	direct (N × H)
4. C₂×#11718082	C₂	semidirect
5. C₂×#11718156	C₂	semidirect
7. C₂×C₄×#6090	C₂	semidirect
8. C₂×C₄×C₁₃×Q₁₆	C₂	semidirect
14. C₄×C₁₃×Q₁₆	C₂²	semidirect
18. C₂×#88068	C₄	direct (N × H)
26. #11718156	C₂²	semidirect
28. #11718082	C₂²	semidirect
29. C₄×#6090	C₂²	semidirect
35. C₂×C₁₃×Q₁₆	C₂×C₄	semidirect
58. #88068	C₂×C₄	direct (N × H)
63. C₂×#6090	C₂×C₄	semidirect
86. C₁₃×Q₁₆	C₂²×C₄	semidirect
106. #6090	C₂²×C₄	semidirect
108. C₂×C₄×Q₁₆	D₁₃	semidirect
128. C₂×C₄×C₁₃	SD₃₂	semidirect
154. C₄×Q₁₆	C₂×D₁₃	semidirect
178. C₄×C₁₃	C₂×SD₃₂	semidirect
187. C₂×Q₁₆	C₄×D₁₃	semidirect
218. C₂×C₁₃	C₄×SD₃₂	semidirect
242. Q₁₆	C₂×C₄×D₁₃	semidirect
263. C₁₃	C₂×C₄×SD₃₂	semidirect
283. C₂×C₄	#88068	direct (N × H)
307. C₄	C₂×#88068	direct (N × H)
311. C₂	C₄×#88068	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×#88068	2
C₂×C₄×D₁₀₄	2
C₄×#88068	2
C₂×#11718082	2
C₂×#11718156	2
C₂×#11718083	2
C₂×C₄×#6090	2
C₂×C₄×C₁₃×Q₁₆	2
C₂×C₄×SD₃₂	13

Subgroups — C2×C4×#88068 #16016186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88068 #16016186