Subgroups — C₂×#22590416 #32151832

Order 18176.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986434	C₂	1	2	yes	maximal, complemented
2	C₂×#17986376	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590416	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985820	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17985811	C₂	1	2	yes	maximal, complemented
6	C₂×#443×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₆₈	—	2	1	no	—
8	C₂²×D₅₆₈	C₂²	1	2	yes	—
9	C₂×#14556125	—	2	1	no	—
10	#17986434	—	2	4	no	—
11	#17986434	C₂²	1	8	yes	complemented
12	C₂×#14556071	—	2	2	no	—
13	C₂×#14556069	—	2	2	no	—
14	C₂×#14556068	C₂²	1	2	yes	—
15	#17986376	—	2	2	no	—
16	#17986376	C₂²	1	4	yes	complemented
17	#17985820	C₂²	1	4	yes	complemented
18	#17985811	C₂²	1	8	yes	complemented
19	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#131×C₇₁	—	2	1	no	—
21	#443×C₇₁	—	2	2	no	—
22	#443×C₇₁	C₂²	1	4	yes	complemented
23	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#104×C₇₁	—	2	1	no	—
25	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
26	C₂²×D₂₈₄	—	2	2	no	—
27	C₂²×D₂₈₄	—	4	2	no	—
28	C₂×D₅₆₈	—	2	2	no	—
29	C₂×D₅₆₈	—	2	2	no	—
30	C₂×D₅₆₈	—	2	2	no	—
31	C₂×D₅₆₈	—	2	4	no	—
32	C₂×D₅₆₈	—	2	2	no	—
33	C₂×D₅₆₈	—	2	2	no	—
34	C₂×D₅₆₈	—	4	1	no	—
35	C₂×D₅₆₈	—	4	1	no	—
36	C₂×D₅₆₈	—	4	2	no	—
37	C₂×D₅₆₈	C₂³	1	4	yes	—
38	#14556125	—	2	4	no	—
39	#14556125	—	4	2	no	—
40	#14556071	—	2	8	no	—
41	#14556069	—	2	8	no	—
42	#14556068	—	2	2	no	—
43	#14556068	C₂³	1	4	yes	—
44	C₂×C₇₁⋊C₁₆	—	2	2	no	—
45	C₂×#41×C₇₁	—	2	1	no	—
46	#135×C₇₁	—	2	1	no	—
47	#135×C₇₁	—	2	1	no	—
48	#135×C₇₁	D₄	1	2	yes	complemented
49	#135×C₇₁	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₇₁	D₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₇₁	—	2	1	no	—
52	C₂×D₈×C₇₁	D₄	1	1	yes	characteristic, complemented
53	#131×C₇₁	—	2	4	no	—
54	#131×C₇₁	—	4	2	no	—
55	C₂×M₁₆×C₇₁	—	2	1	no	—
56	C₂×M₁₆×C₇₁	—	4	1	no	—
57	C₂²×C₈×C₇₁	—	4	1	no	—
58	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
61	#108×C₇₁	—	2	1	no	—
62	#108×C₇₁	C₂³	1	2	yes	—
63	#104×C₇₁	—	2	4	no	—
64	C₄×C₈×C₇₁	D₄	1	2	yes	—
65	C₄×C₈×C₇₁	C₂³	1	2	yes	—
66	C₂³×D₇₁	—	4	2	no	—
67	C₂×D₂₈₄	—	2	4	no	—
68	C₂×D₂₈₄	—	4	2	no	—
69	C₂×D₂₈₄	—	4	2	no	—
70	C₂×D₂₈₄	—	4	2	no	—
71	C₂×D₂₈₄	—	4	2	no	—
72	C₂×D₂₈₄	—	4	8	no	—
73	C₂×D₂₈₄	—	4	2	no	—
74	C₂×D₂₈₄	—	8	8	no	—
75	D₅₆₈	—	2	4	no	—
76	D₅₆₈	—	2	4	no	—
77	D₅₆₈	—	4	2	no	—
78	D₅₆₈	—	4	2	no	—
79	D₅₆₈	—	4	4	no	—
80	D₅₆₈	—	4	4	no	—
81	C₇₁⋊C₁₆	—	2	4	no	—
82	#41×C₇₁	—	2	2	no	—
83	#41×C₇₁	—	2	2	no	—
84	#41×C₇₁	—	4	1	no	—
85	#41×C₇₁	—	4	1	no	—
86	C₂×Q₈×C₇₁	—	2	1	no	—
87	C₂×D₄×C₇₁	—	2	1	no	—
88	C₂×D₄×C₇₁	—	4	1	no	—
89	C₂²×C₄×C₇₁	—	2	1	no	—
90	C₂²×C₄×C₇₁	—	4	1	no	—
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₇₁	—	2	1	no	—
93	Q₁₆×C₇₁	C₂×D₄	1	2	yes	complemented
94	SD₁₆×C₇₁	—	2	2	no	—
95	SD₁₆×C₇₁	—	2	2	no	—
96	D₈×C₇₁	—	2	1	no	—
97	D₈×C₇₁	C₂×D₄	1	2	yes	complemented
98	M₁₆×C₇₁	—	2	2	no	—
99	M₁₆×C₇₁	—	4	1	no	—
100	M₁₆×C₇₁	—	4	4	no	—
101	C₂×C₈×C₇₁	—	4	1	no	—
102	C₂×C₈×C₇₁	—	4	4	no	—
103	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
104	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
105	C₂×C₈×C₇₁	D₈	1	2	yes	—
106	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
111	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
113	C₄²×C₇₁	—	2	1	no	—
114	C₄²×C₇₁	C₂×D₄	1	2	yes	—
115	C₂³×C₇₁	—	4	1	no	—
116	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂²×D₇₁	—	4	4	no	—
118	C₂²×D₇₁	—	8	8	no	—
119	C₂²×D₇₁	—	8	2	no	—
120	C₂²×D₇₁	—	8	2	no	—
121	Q₈×C₇₁	—	2	2	no	—
122	Q₈×C₇₁	—	4	1	no	—
123	D₄×C₇₁	—	2	2	no	—
124	D₄×C₇₁	—	4	2	no	—
125	D₄×C₇₁	—	4	2	no	—
126	D₄×C₇₁	—	4	1	no	—
127	C₂×C₄×C₇₁	—	2	4	no	—
128	C₂×C₄×C₇₁	—	2	2	no	—
129	C₂×C₄×C₇₁	—	4	1	no	—
130	C₂×C₄×C₇₁	—	4	2	no	—
131	C₂×C₄×C₇₁	—	4	2	no	—
132	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
138	D₂₈₄	—	4	4	no	—
139	D₂₈₄	—	4	4	no	—
140	D₂₈₄	—	8	16	no	—
141	D₂₈₄	—	8	2	no	—
142	D₂₈₄	—	8	2	no	—
143	C₈×C₇₁	—	4	2	no	—
144	C₈×C₇₁	#121	1	1	yes	characteristic
145	C₈×C₇₁	#121	1	1	yes	characteristic
146	C₈×C₇₁	C₂×D₈	1	2	yes	—
147	C₈×C₇₁	C₂×D₈	1	2	yes	—
148	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
150	C₂²×C₇₁	—	2	2	no	—
151	C₂²×C₇₁	—	4	2	no	—
152	C₂²×C₇₁	—	8	2	no	—
153	C₂²×C₇₁	#459	1	1	yes	characteristic
154	C₂²×C₇₁	#459	1	1	yes	characteristic
155	C₂²×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×D₇₁	—	8	8	no	—
157	C₂×D₇₁	—	8	4	no	—
158	C₂×D₇₁	—	16	4	no	—
159	C₂×D₇₁	—	16	4	no	—
160	C₄×C₇₁	—	2	4	no	—
161	C₄×C₇₁	—	4	2	no	—
162	C₄×C₇₁	#459	1	1	yes	characteristic
163	C₄×C₇₁	#459	1	1	yes	characteristic
164	C₄×C₇₁	C₂×#121	1	1	yes	characteristic
165	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
166	C₂×#2213	—	71	1	no	maximal, Sylow
167	C₂×C₇₁	—	2	1	no	—
168	C₂×C₇₁	—	2	1	no	—
169	C₂×C₇₁	—	8	2	no	—
170	C₂×C₇₁	#2213	1	2	yes	complemented
171	C₂×C₇₁	C₂×#459	1	1	yes	characteristic
172	D₇₁	—	16	4	no	—
173	C₂×#509	—	71	2	no	—
174	C₂×#493	—	71	1	no	—
175	C₂×#443	D₇₁	1	1	yes	characteristic, complemented
176	#2213	—	71	8	no	—
177	C₂×#364	—	71	1	no	—
178	C₂×#361	—	71	2	no	—
179	C₇₁	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
181	C₂²×D₈	—	71	2	no	—
182	C₂²×D₈	—	142	1	no	—
183	C₂×#131	—	2	1	no	—
184	C₂×#127	—	142	1	no	—
185	#509	—	71	8	no	—
186	#509	—	142	4	no	—
187	C₂×SD₃₂	—	142	2	no	—
188	C₂×D₁₆	—	142	2	no	—
189	C₂×M₃₂	—	71	2	no	—
190	#493	—	71	4	no	—
191	#493	—	142	2	no	—
192	#443	—	2	2	no	—
193	#443	C₂×D₇₁	1	4	yes	complemented
194	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
195	C₂×#104	—	2	1	no	—
196	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
197	#364	—	71	4	no	—
198	#361	—	71	8	no	—
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	284	2	no	—
201	C₂²×D₄	—	142	2	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	D₂₈₄	1	2	yes	complemented
205	#135	C₂²×D₇₁	1	2	yes	—
206	C₂×Q₁₆	D₂₈₄	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	284	1	no	—
209	C₂×D₈	—	284	1	no	—
210	C₂×D₈	—	284	2	no	—
211	C₂×D₈	—	71	4	no	—
212	C₂×D₈	—	142	2	no	—
213	C₂×D₈	—	142	4	no	—
214	C₂×D₈	—	142	2	no	—
215	C₂×D₈	—	142	2	no	—
216	C₂×D₈	—	142	2	no	—
217	C₂×D₈	—	142	2	no	—
218	C₂×D₈	D₂₈₄	1	1	yes	characteristic, complemented
219	#131	—	2	4	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	4	1	no	—
223	C₂²×C₈	—	4	1	no	—
224	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
226	#127	—	284	2	no	—
227	#127	—	142	4	no	—
228	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
229	SD₃₂	—	142	8	no	—
230	D₁₆	—	142	8	no	—
231	M₃₂	—	71	4	no	—
232	M₃₂	—	142	2	no	—
233	C₂×C₁₆	—	142	2	no	—
234	#108	—	2	1	no	—
235	#108	C₂²×D₇₁	1	2	yes	—
236	#104	—	2	4	no	—
237	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
238	C₄×C₈	C₂²×D₇₁	1	2	yes	—
239	C₂⁴	—	284	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	284	8	no	—
248	C₂×D₄	—	284	2	no	—
249	C₂×D₄	—	284	2	no	—
250	C₂×D₄	—	284	2	no	—
251	C₂×D₄	—	284	2	no	—
252	C₂×D₄	—	284	2	no	—
253	C₂×D₄	—	568	8	no	—
254	C₂×D₄	—	142	4	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	C₂×D₂₈₄	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	284	2	no	—
264	D₈	—	284	2	no	—
265	D₈	—	284	4	no	—
266	D₈	—	284	4	no	—
267	D₈	—	142	4	no	—
268	D₈	—	142	4	no	—
269	D₈	C₂×D₂₈₄	1	2	yes	complemented
270	M₁₆	—	2	2	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	#11765850	1	1	yes	characteristic
276	C₂×C₈	#11765850	1	1	yes	characteristic
277	C₂×C₈	#11765850	1	2	yes	—
278	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
280	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
281	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
282	C₂×C₈	#11765865	1	1	yes	characteristic
283	C₂×C₈	#11765865	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
287	C₁₆	—	142	4	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	284	4	no	—
290	C₂³	—	568	8	no	—
291	C₂³	—	568	2	no	—
292	C₂³	—	568	2	no	—
293	C₂³	#14556153	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	284	4	no	—
301	D₄	—	284	4	no	—
302	D₄	—	568	16	no	—
303	D₄	—	568	2	no	—
304	D₄	—	568	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	#14556102	1	1	yes	characteristic
311	C₂×C₄	#14556102	1	1	yes	characteristic
312	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
313	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
314	C₂×C₄	#14556153	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	#14556102	1	1	yes	characteristic
318	C₈	#14556102	1	1	yes	characteristic
319	C₈	C₂×#11765850	1	2	yes	—
320	C₈	C₂×#11765850	1	2	yes	—
321	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
322	C₈	C₂×#11765865	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	568	8	no	—
327	C₂²	—	568	4	no	—
328	C₂²	—	1136	4	no	—
329	C₂²	—	1136	4	no	—
330	C₂²	#17986194	1	1	yes	characteristic
331	C₂²	#17986194	1	1	yes	characteristic
332	C₂²	C₂×#14556153	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#17986194	1	1	yes	characteristic
336	C₄	#17986194	1	1	yes	characteristic
337	C₄	C₂×#14556102	1	1	yes	characteristic
338	C₄	C₂²×#11765850	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	1136	4	no	—
343	C₂	#22590416	1	2	yes	direct product factor
344	C₂	C₂×#17986194	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17986194
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#443×C₇₁	C₂
socle	C₂²×C₇₁	#459
hypercenter	C₂×#443	D₇₁
hypocenter	C₇₁	C₂×#2213
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#17986434	C₂	semidirect
2. C₂×#17986376	C₂	semidirect
3. #22590416	C₂	direct (N × H)
4. C₂×#17985820	C₂	semidirect
5. C₂×#17985811	C₂	semidirect
6. C₂×#443×C₇₁	C₂	semidirect
11. #17986434	C₂²	semidirect
16. #17986376	C₂²	semidirect
17. #17985820	C₂²	semidirect
18. #17985811	C₂²	semidirect
22. #443×C₇₁	C₂²	semidirect
48. #135×C₇₁	D₄	semidirect
50. C₂×Q₁₆×C₇₁	D₄	semidirect
52. C₂×D₈×C₇₁	D₄	semidirect
93. Q₁₆×C₇₁	C₂×D₄	semidirect
97. D₈×C₇₁	C₂×D₄	semidirect
170. C₂×C₇₁	#2213	semidirect
175. C₂×#443	D₇₁	semidirect
179. C₇₁	C₂×#2213	semidirect
193. #443	C₂×D₇₁	semidirect
204. #135	D₂₈₄	semidirect
206. C₂×Q₁₆	D₂₈₄	semidirect
218. C₂×D₈	D₂₈₄	semidirect
259. Q₁₆	C₂×D₂₈₄	semidirect
269. D₈	C₂×D₂₈₄	semidirect
343. C₂	#22590416	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×#17986434	2
C₂×#17986376	2
#22590416	2
C₂×#17985820	2
C₂×#17985811	2
C₂×#443×C₇₁	2
C₂×#2213	71

Subgroups — C2×#22590416 #32151832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590416 #32151832