Subgroups — C₂×C₄×#88066 #16016178

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88066	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#88066	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718154	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	—	2	1	no	—
11	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₅₂	—	2	1	no	—
13	C₄×C₁₃×D₈	—	2	2	no	—
14	C₄×C₁₃×D₈	C₂²	1	4	yes	complemented
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88066	C₄	1	8	yes	direct product factor
19	C₂×#88066	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	complemented
26	#11718154	C₂²	1	4	yes	complemented
27	#11718083	C₂²	1	4	yes	complemented
28	#11718081	C₂²	1	4	yes	complemented
29	C₄×#6090	C₂²	1	4	yes	complemented
30	C₂²×D₄×C₁₃	—	2	1	no	—
31	C₂³×C₄×C₁₃	—	4	1	no	—
32	C₂²×D₅₂	—	2	1	no	—
33	C₂²×C₄×D₁₃	—	4	1	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	complemented
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₄×D₄×C₁₃	—	2	4	no	—
41	C₄×D₄×C₁₃	—	4	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	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	complemented
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×#102	—	2	4	no	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	#88066	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×D₁₆	—	13	1	no	maximal, Sylow
67	C₂⁴×C₁₃	—	4	1	no	—
68	C₂³×D₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	—	4	4	no	—
75	C₂²×C₄×C₁₃	—	8	4	no	—
76	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	4	1	no	—
80	C₂×D₅₂	—	2	4	no	—
81	C₂×D₅₂	—	2	2	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×C₄×D₁₃	—	4	4	no	—
85	C₂×C₄×D₁₃	—	8	4	no	—
86	C₁₃×D₈	—	2	4	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	C₂²×C₄	1	4	yes	complemented
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₂²⋊C₄	—	4	4	no	—
98	C₄²×C₁₃	D₈	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	2	yes	—
100	#6103	—	4	4	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	—	4	1	no	—
103	D₁₀₄	—	2	4	no	—
104	D₁₀₄	—	2	2	no	—
105	D₁₀₄	C₂²×C₄	1	4	yes	complemented
106	#6090	—	2	2	no	—
107	#6090	C₂²×C₄	1	4	yes	complemented
108	C₂²×D₁₆	—	13	1	no	—
109	C₂×C₄×D₈	—	13	1	no	—
110	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic, complemented
111	C₄×D₁₆	—	13	8	no	—
112	C₂×#366	—	13	1	no	—
113	C₂×#357	—	13	1	no	—
114	C₂×#357	—	13	1	no	—
115	C₂×C₄×C₁₆	—	13	1	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	4	4	no	—
118	C₂³×C₁₃	—	8	2	no	—
119	C₂³×C₁₃	—	8	2	no	—
120	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂²×D₁₃	—	4	2	no	—
122	C₂²×D₁₃	—	4	4	no	—
123	C₂²×D₁₃	—	8	2	no	—
124	C₂²×D₁₃	—	8	2	no	—
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	—	8	8	no	—
132	C₂×C₄×C₁₃	D₁₆	1	4	yes	complemented
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃	#135	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	8	2	no	—
141	D₅₂	—	2	4	no	—
142	D₅₂	—	4	2	no	—
143	D₅₂	—	4	4	no	—
144	C₄×D₁₃	—	8	8	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	C₄×D₄	1	2	yes	—
147	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	C₂²×D₈	—	13	1	no	—
150	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×D₄	—	2	1	no	—
152	C₂×C₄×D₄	—	26	1	no	—
153	C₂×D₁₆	—	13	8	no	—
154	C₂×D₁₆	—	13	4	no	—
155	C₂²×C₁₆	—	13	1	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	C₄×D₈	—	2	2	no	—
158	C₄×D₈	—	13	4	no	—
159	C₄×D₈	—	26	2	no	—
160	C₄×D₈	C₂×D₁₃	1	4	yes	complemented
161	C₂×#107	—	13	1	no	—
162	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
163	C₂×#102	—	2	1	no	—
164	C₂×#102	—	26	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#366	—	13	4	no	—
167	#357	—	13	4	no	—
168	#357	—	13	4	no	—
169	C₄×C₁₆	—	13	4	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	8	2	no	—
173	C₂²×C₁₃	—	8	8	no	—
174	C₂²×C₁₃	C₄×D₈	1	2	yes	—
175	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
176	C₂²×C₁₃	#508	1	2	yes	—
177	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×D₁₃	—	4	4	no	—
179	C₂×D₁₃	—	8	2	no	—
180	C₂×D₁₃	—	8	2	no	—
181	C₂×D₁₃	—	8	8	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	C₄×D₈	1	2	yes	—
185	C₄×C₁₃	C₂×D₁₆	1	4	yes	complemented
186	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
188	Dic₁₃	—	8	2	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂²×D₄	—	26	1	no	—
191	C₂³×C₄	—	4	1	no	—
192	C₂³×C₄	—	52	1	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	13	4	no	—
196	C₂×D₈	—	13	2	no	—
197	C₂×D₈	—	26	2	no	—
198	C₂×D₈	—	26	1	no	—
199	C₂×D₈	C₄×D₁₃	1	4	yes	complemented
200	C₂×D₈	C₂²×D₁₃	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	4	2	no	—
205	C₄×D₄	—	26	4	no	—
206	C₄×D₄	—	52	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	26	1	no	—
209	C₂×C₂²⋊C₄	—	4	1	no	—
210	C₂×C₂²⋊C₄	—	52	1	no	—
211	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
212	D₁₆	—	13	16	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	2	no	—
216	C₂×C₁₆	—	26	2	no	—
217	#107	—	2	1	no	—
218	#107	—	13	2	no	—
219	#107	—	26	1	no	—
220	#107	C₂²×D₁₃	1	2	yes	—
221	#102	—	2	4	no	—
222	#102	—	26	4	no	—
223	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
224	C₄×C₈	C₂²×D₁₃	1	2	yes	—
225	C₂×C₁₃	—	8	4	no	—
226	C₂×C₁₃	C₄×D₁₆	1	4	yes	complemented
227	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
228	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
229	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
230	D₁₃	—	8	4	no	—
231	C₂⁴	—	4	1	no	—
232	C₂⁴	—	52	1	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	26	4	no	—
238	C₂×D₄	—	26	2	no	—
239	C₂×D₄	—	52	1	no	—
240	C₂×D₄	—	52	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	8	4	no	—
244	C₂²×C₄	—	52	1	no	—
245	C₂²×C₄	—	52	4	no	—
246	C₂²×C₄	—	104	4	no	—
247	C₂²×C₄	#6104	1	1	yes	characteristic
248	C₂²×C₄	#6104	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	D₈	—	2	4	no	—
251	D₈	—	2	2	no	—
252	D₈	—	13	4	no	—
253	D₈	—	26	4	no	—
254	D₈	—	26	2	no	—
255	D₈	C₂×C₄×D₁₃	1	4	yes	complemented
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
258	C₂×C₈	#6127	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	52	1	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	52	4	no	—
268	C₄²	#6104	1	2	yes	—
269	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
270	C₁₆	—	13	4	no	—
271	C₁₆	—	26	2	no	—
272	C₁₃	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	4	4	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	52	4	no	—
278	C₂³	—	52	2	no	—
279	C₂³	—	104	2	no	—
280	C₂³	—	104	2	no	—
281	C₂³	C₂×#6104	1	1	yes	characteristic
282	D₄	—	2	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	2	no	—
285	D₄	—	26	4	no	—
286	D₄	—	52	4	no	—
287	D₄	—	52	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	8	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	104	2	no	—
294	C₂×C₄	—	104	8	no	—
295	C₂×C₄	#88066	1	4	yes	direct product factor
296	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
297	C₂×C₄	C₂×#6104	1	2	yes	—
298	C₂×C₄	C₂×#6104	1	1	yes	characteristic
299	C₂×C₄	C₂×#6104	1	2	yes	—
300	C₂×C₄	#88204	1	2	yes	—
301	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
304	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
305	C₈	C₂×#6127	1	1	yes	characteristic
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	8	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	52	4	no	—
311	C₂²	—	104	2	no	—
312	C₂²	—	104	8	no	—
313	C₂²	—	104	2	no	—
314	C₂²	C₄×#6104	1	2	yes	—
315	C₂²	C₂×#88066	1	2	yes	—
316	C₂²	#11718855	1	2	yes	—
317	C₂²	C₂²×#6104	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	104	2	no	—
321	C₄	C₄×#6104	1	2	yes	—
322	C₄	C₂×#88066	1	4	yes	direct product factor
323	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
324	C₄	C₂×#88204	1	1	yes	characteristic
325	C₂	—	8	4	no	—
326	C₂	—	104	4	no	—
327	C₂	C₄×#88066	1	4	yes	direct product factor
328	C₂	C₂×C₄×#6104	1	1	yes	characteristic
329	C₂	C₂²×#88066	1	1	yes	characteristic
330	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×D₈	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×#88066	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₁₀₄	C₂	semidirect
3. C₄×#88066	C₂	direct (N × H)
4. C₂×#11718081	C₂	semidirect
5. C₂×#11718154	C₂	semidirect
6. C₂×#11718083	C₂	semidirect
7. C₂×C₄×#6090	C₂	semidirect
8. C₂×C₄×C₁₃×D₈	C₂	semidirect
14. C₄×C₁₃×D₈	C₂²	semidirect
18. C₂×#88066	C₄	direct (N × H)
25. C₄×D₁₀₄	C₂²	semidirect
26. #11718154	C₂²	semidirect
27. #11718083	C₂²	semidirect
28. #11718081	C₂²	semidirect
29. C₄×#6090	C₂²	semidirect
36. C₂×C₁₃×D₈	C₂×C₄	semidirect
49. C₂×D₁₀₄	C₂×C₄	semidirect
58. #88066	C₂×C₄	direct (N × H)
63. C₂×#6090	C₂×C₄	semidirect
88. C₁₃×D₈	C₂²×C₄	semidirect
105. D₁₀₄	C₂²×C₄	semidirect
107. #6090	C₂²×C₄	semidirect
110. C₂×C₄×D₈	D₁₃	semidirect
132. C₂×C₄×C₁₃	D₁₆	semidirect
160. C₄×D₈	C₂×D₁₃	semidirect
185. C₄×C₁₃	C₂×D₁₆	semidirect
199. C₂×D₈	C₄×D₁₃	semidirect
226. C₂×C₁₃	C₄×D₁₆	semidirect
255. D₈	C₂×C₄×D₁₃	semidirect
272. C₁₃	C₂×C₄×D₁₆	semidirect
295. C₂×C₄	#88066	direct (N × H)
322. C₄	C₂×#88066	direct (N × H)
327. C₂	C₄×#88066	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×#88066	2
C₂×C₄×D₁₀₄	2
C₄×#88066	2
C₂×#11718081	2
C₂×#11718154	2
C₂×#11718083	2
C₂×C₄×#6090	2
C₂×C₄×C₁₃×D₈	2
C₂×C₄×D₁₆	13

Subgroups — C2×C4×#88066 #16016178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88066 #16016178