Subgroups — C₂×#7277 #127713

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	#7277	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₁₆	—	2	1	no	—
6	C₂×#431	—	2	1	no	—
7	C₂²×#108	—	2	1	no	—
8	C₂²×#108	C₄	1	2	yes	—
9	C₂×#423	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	—	2	1	no	—
13	C₂×C₄⋊C₁₆	—	2	1	no	—
14	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
15	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	#1848	—	2	1	no	—
18	#1848	—	2	1	no	—
19	#1848	C₄	1	2	yes	—
20	#1848	C₂²	1	2	yes	—
21	#1564	—	2	1	no	—
22	#1564	—	2	1	no	—
23	#1564	C₄	1	2	yes	—
24	#1564	C₂²	1	2	yes	—
25	C₂×#335	C₄	1	2	yes	—
26	C₂×#334	—	2	1	no	—
27	C₂²×M₁₆	—	2	2	no	—
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	—	2	2	no	—
33	C₂²×C₁₆	—	2	2	no	—
34	C₂²×C₁₆	—	4	2	no	—
35	C₂²×C₁₆	—	4	2	no	—
36	#431	—	2	2	no	—
37	#431	—	2	2	no	—
38	#431	—	4	1	no	—
39	#431	—	4	1	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	4	4	no	—
44	C₂×#108	—	4	2	no	—
45	C₂×#108	—	4	2	no	—
46	C₂×#108	C₂×C₄	1	4	yes	—
47	#423	—	2	2	no	—
48	#423	—	2	2	no	—
49	#423	—	4	1	no	—
50	#423	—	4	1	no	—
51	C₂×C₄⋊C₈	—	2	2	no	—
52	C₂×C₄⋊C₈	—	4	1	no	—
53	C₂×C₄⋊C₈	—	4	1	no	—
54	C₂×C₂²⋊C₈	—	4	1	no	—
55	C₂×C₄×C₈	—	4	1	no	—
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	D₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	1	yes	characteristic
65	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₄⋊C₁₆	—	2	2	no	—
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
72	#335	—	2	2	no	—
73	#335	C₂×C₄	1	4	yes	—
74	#334	—	2	2	no	—
75	#334	—	2	2	no	—
76	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₄	SD₁₆	1	1	yes	characteristic
78	C₂³×C₄	Q₁₆	1	1	yes	characteristic
79	C₂×M₁₆	—	2	4	no	—
80	C₂×M₁₆	—	4	2	no	—
81	C₂×M₁₆	—	4	4	no	—
82	C₂×M₁₆	—	4	4	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	4	2	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₄²	1	1	yes	characteristic
88	C₂²×C₈	C₄²	1	1	yes	characteristic
89	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	D₈	1	2	yes	—
98	C₂²×C₈	SD₁₆	1	2	yes	—
99	C₂²×C₈	SD₁₆	1	2	yes	—
100	C₂²×C₈	Q₁₆	1	2	yes	—
101	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	D₈	1	1	yes	characteristic
109	C₂×C₄²	D₈	1	1	yes	characteristic
110	C₂×C₄²	SD₁₆	1	1	yes	characteristic
111	C₂×C₄²	SD₁₆	1	1	yes	characteristic
112	C₂×C₄²	SD₁₆	1	1	yes	characteristic
113	C₂×C₄²	Q₁₆	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	4	4	no	—
117	C₂×C₁₆	—	4	4	no	—
118	C₂×C₁₆	—	4	2	no	—
119	C₂×C₁₆	—	4	2	no	—
120	#108	—	4	4	no	—
121	#108	—	4	4	no	—
122	#108	—	4	4	no	—
123	#108	—	4	2	no	—
124	#108	—	4	2	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	4	2	no	—
129	C₄⋊C₈	—	4	2	no	—
130	C₄⋊C₈	—	4	2	no	—
131	C₂²⋊C₈	—	4	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₄×C₈	—	4	2	no	—
134	C₄×C₈	—	4	2	no	—
135	C₄×C₈	C₄²	1	1	yes	characteristic
136	C₄×C₈	C₄²	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
150	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
151	C₂⁴	#103	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	#95	1	1	yes	characteristic
155	C₂²×C₄	#95	1	1	yes	characteristic
156	C₂²×C₄	#100	1	2	yes	—
157	C₂²×C₄	#101	1	2	yes	—
158	C₂²×C₄	#102	1	1	yes	characteristic
159	C₂²×C₄	#102	1	1	yes	characteristic
160	C₂²×C₄	#102	1	2	yes	—
161	C₂²×C₄	#102	1	1	yes	characteristic
162	C₂²×C₄	#102	1	1	yes	characteristic
163	C₂²×C₄	#103	1	1	yes	characteristic
164	C₂²×C₄	#103	1	1	yes	characteristic
165	C₂²×C₄	#103	1	1	yes	characteristic
166	C₂²×C₄	#103	1	1	yes	characteristic
167	C₂²×C₄	#103	1	1	yes	characteristic
168	C₂²×C₄	#103	1	2	yes	—
169	C₂²×C₄	#106	1	1	yes	characteristic
170	C₂²×C₄	#106	1	1	yes	characteristic
171	C₂²×C₄	#107	1	1	yes	characteristic
172	C₂²×C₄	#107	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
176	M₁₆	—	4	4	no	—
177	M₁₆	—	4	4	no	—
178	M₁₆	—	8	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	8	4	no	—
190	C₂×C₈	—	8	4	no	—
191	C₂×C₈	#95	1	1	yes	characteristic
192	C₂×C₈	#95	1	1	yes	characteristic
193	C₂×C₈	#95	1	1	yes	characteristic
194	C₂×C₈	#95	1	1	yes	characteristic
195	C₂×C₈	#102	1	2	yes	—
196	C₂×C₈	#102	1	2	yes	—
197	C₂×C₈	#102	1	2	yes	—
198	C₂×C₈	#103	1	2	yes	—
199	C₂×C₈	#103	1	2	yes	—
200	C₂×C₈	#103	1	2	yes	—
201	C₂×C₈	#106	1	2	yes	—
202	C₂×C₈	#106	1	2	yes	—
203	C₂×C₈	#107	1	2	yes	—
204	C₂×C₈	#107	1	2	yes	—
205	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₈	C₂×D₈	1	1	yes	characteristic
210	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
211	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
212	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
213	C₄²	#95	1	1	yes	characteristic
214	C₄²	#95	1	1	yes	characteristic
215	C₄²	#102	1	1	yes	characteristic
216	C₄²	#102	1	1	yes	characteristic
217	C₄²	#102	1	1	yes	characteristic
218	C₄²	#102	1	1	yes	characteristic
219	C₄²	#103	1	1	yes	characteristic
220	C₄²	#103	1	1	yes	characteristic
221	C₄²	#106	1	1	yes	characteristic
222	C₄²	#106	1	1	yes	characteristic
223	C₄²	#107	1	1	yes	characteristic
224	C₄²	#107	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
227	C₄²	C₂×D₈	1	1	yes	characteristic
228	C₄²	C₂×SD₁₆	1	1	yes	characteristic
229	C₁₆	—	4	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	#332	1	2	yes	—
233	C₂³	#333	1	2	yes	—
234	C₂³	#340	1	1	yes	characteristic
235	C₂³	#340	1	1	yes	characteristic
236	C₂³	C₂×#103	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	#331	1	2	yes	—
249	C₂×C₄	#331	1	2	yes	—
250	C₂×C₄	#332	1	2	yes	—
251	C₂×C₄	#332	1	2	yes	—
252	C₂×C₄	#332	1	2	yes	—
253	C₂×C₄	#333	1	2	yes	—
254	C₂×C₄	#340	1	1	yes	characteristic
255	C₂×C₄	#340	1	1	yes	characteristic
256	C₂×C₄	#340	1	1	yes	characteristic
257	C₂×C₄	#340	1	1	yes	characteristic
258	C₂×C₄	#340	1	1	yes	characteristic
259	C₂×C₄	#340	1	2	yes	—
260	C₂×C₄	#340	1	2	yes	—
261	C₂×C₄	#340	1	1	yes	characteristic
262	C₂×C₄	#343	1	2	yes	—
263	C₂×C₄	#343	1	2	yes	—
264	C₂×C₄	#359	1	4	yes	—
265	C₂×C₄	C₂×#95	1	1	yes	characteristic
266	C₂×C₄	C₂×#100	1	1	yes	characteristic
267	C₂×C₄	C₂×#101	1	1	yes	characteristic
268	C₂×C₄	C₂×#102	1	1	yes	characteristic
269	C₂×C₄	C₂×#102	1	1	yes	characteristic
270	C₂×C₄	C₂×#102	1	1	yes	characteristic
271	C₂×C₄	C₂×#103	1	1	yes	characteristic
272	C₂×C₄	C₂×#103	1	1	yes	characteristic
273	C₂×C₄	C₂×#106	1	1	yes	characteristic
274	C₂×C₄	C₂×#107	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	8	4	no	—
280	C₈	—	8	4	no	—
281	C₈	#340	1	2	yes	—
282	C₈	#340	1	2	yes	—
283	C₈	C₂×#102	1	1	yes	characteristic
284	C₈	C₂×#103	1	1	yes	characteristic
285	C₈	C₂×#106	1	1	yes	characteristic
286	C₈	C₂×#107	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	#1296	1	2	yes	—
294	C₂²	#1296	1	2	yes	—
295	C₂²	#1382	1	2	yes	—
296	C₂²	#1388	1	2	yes	—
297	C₂²	C₂×#332	1	1	yes	characteristic
298	C₂²	C₂×#333	1	1	yes	characteristic
299	C₂²	C₂×#340	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	#1296	1	2	yes	—
307	C₄	#1296	1	2	yes	—
308	C₄	#1382	1	2	yes	—
309	C₄	C₂×#331	1	1	yes	characteristic
310	C₄	C₂×#332	1	1	yes	characteristic
311	C₄	C₂×#340	1	1	yes	characteristic
312	C₄	C₂×#343	1	1	yes	characteristic
313	C₄	C₂×#359	1	2	yes	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	#7277	1	4	yes	direct product factor
318	C₂	C₂×#1296	1	1	yes	characteristic
319	C₂	C₂×#1382	1	1	yes	characteristic
320	C₂	C₂×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #7277	C₂	direct (N × H)
317. C₂	#7277	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⋊C₁₆	2
C₂×#1848	2
C₂×#1564	2
#7277	2

Subgroups — C2×#7277 #127713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7277 #127713