Subgroups — C₂×#122924 #72231986

Order 1024.

757 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11996	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1367	C₂	1	1	yes	maximal, characteristic
3	C₂×#11411	C₂	1	1	yes	maximal, characteristic
4	#122924	C₂	1	8	yes	maximal, direct product factor
5	C₂×#7099	C₂	1	4	yes	maximal, complemented
6	C₂×#2162	C₄	1	1	yes	characteristic
7	C₂×#2162	C₄	1	1	yes	characteristic
8	C₂×#2150	—	2	1	no	—
9	C₂²×C₄⋊C₁₆	—	2	1	no	—
10	C₂²×C₈²	C₂²	1	1	yes	characteristic
11	#11996	—	2	2	no	—
12	#11996	C₂²	1	4	yes	—
13	C₂×#1367	—	2	1	no	—
14	C₂×#1367	—	2	2	no	—
15	C₂×#1367	—	2	1	no	—
16	C₂×#1367	C₂²	1	2	yes	—
17	C₂×#1367	C₂²	1	2	yes	—
18	C₂×#1367	C₂²	1	4	yes	—
19	#11411	C₄	1	4	yes	—
20	#11411	C₂²	1	4	yes	—
21	C₂×#1312	C₄	1	2	yes	—
22	C₂×#1311	C₂²	1	2	yes	—
23	#7099	C₂²	1	16	yes	complemented
24	C₂²×M₃₂	—	4	1	no	—
25	C₂³×C₁₆	—	4	1	no	—
26	C₂²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈	Q₈	1	1	yes	characteristic
29	#2162	—	2	2	no	—
30	#2162	C₈	1	4	yes	—
31	#2162	C₂×C₄	1	4	yes	—
32	#2162	C₂×C₄	1	4	yes	—
33	#2150	—	2	4	no	—
34	#2150	—	4	2	no	—
35	C₂×C₄⋊C₁₆	—	2	1	no	—
36	C₂×C₄⋊C₁₆	—	2	2	no	—
37	C₂×C₄⋊C₁₆	—	2	4	no	—
38	C₂×C₄⋊C₁₆	—	2	2	no	—
39	C₂×C₄⋊C₁₆	—	4	2	no	—
40	C₂×C₄⋊C₁₆	—	4	1	no	—
41	C₂×C₄⋊C₁₆	—	4	1	no	—
42	C₂×C₄⋊C₁₆	C₈	1	2	yes	—
43	C₂×C₂²⋊C₁₆	—	2	1	no	—
44	C₂×C₂²⋊C₁₆	C₈	1	2	yes	—
45	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₈²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈²	C₂×C₄	1	2	yes	—
49	C₂×C₈²	D₄	1	1	yes	characteristic
50	C₂×C₈²	D₄	1	2	yes	—
51	C₂×C₈²	Q₈	1	2	yes	—
52	C₂×C₈²	Q₈	1	1	yes	characteristic
53	C₂×C₈²	C₂³	1	1	yes	characteristic
54	C₂×C₈²	C₂³	1	2	yes	—
55	#1367	—	2	2	no	—
56	#1367	—	2	4	no	—
57	#1367	—	2	4	no	—
58	#1367	—	2	2	no	—
59	#1367	C₂³	1	4	yes	—
60	#1367	C₂³	1	4	yes	—
61	#1312	C₂×C₄	1	8	yes	—
62	#1311	C₂×C₄	1	4	yes	—
63	#1311	C₂³	1	4	yes	—
64	C₂³×C₈	—	2	1	no	—
65	C₂³×C₈	C₂×C₈	1	1	yes	characteristic
66	C₂³×C₈	M₁₆	1	1	yes	characteristic
67	C₂³×C₈	SD₁₆	1	1	yes	characteristic
68	C₂³×C₈	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄²	C₄⋊C₄	1	1	yes	characteristic
70	C₂×M₃₂	—	4	4	no	—
71	C₂×M₃₂	—	8	4	no	—
72	C₂²×C₁₆	—	2	2	no	—
73	C₂²×C₁₆	—	4	4	no	—
74	C₂²×C₁₆	—	4	2	no	—
75	C₂²×C₁₆	—	4	2	no	—
76	C₂²×C₁₆	—	8	4	no	—
77	C₂²×C₁₆	C₂×C₈	1	4	yes	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	2	2	no	—
80	C₂×C₄×C₈	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
88	C₂×C₄×C₈	M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈	M₁₆	1	2	yes	—
90	C₂×C₄×C₈	SD₁₆	1	2	yes	—
91	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₈	SD₁₆	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₄⋊C₁₆	—	2	4	no	—
101	C₄⋊C₁₆	—	2	4	no	—
102	C₄⋊C₁₆	—	2	4	no	—
103	C₄⋊C₁₆	—	4	2	no	—
104	C₄⋊C₁₆	—	4	4	no	—
105	C₄⋊C₁₆	—	4	2	no	—
106	C₄⋊C₁₆	—	4	4	no	—
107	C₄⋊C₁₆	C₂×C₈	1	8	yes	—
108	C₂²⋊C₁₆	—	2	4	no	—
109	C₂²⋊C₁₆	C₂×C₈	1	8	yes	—
110	C₄×C₁₆	—	2	2	no	—
111	C₄×C₁₆	C₂×C₈	1	4	yes	—
112	C₄×C₁₆	C₂²×C₄	1	4	yes	—
113	C₄×C₁₆	C₂²×C₄	1	4	yes	—
114	C₈²	C₄⋊C₄	1	2	yes	—
115	C₈²	C₄⋊C₄	1	2	yes	—
116	C₈²	C₄⋊C₄	1	2	yes	—
117	C₈²	C₂²×C₄	1	1	yes	characteristic
118	C₈²	C₂²×C₄	1	2	yes	—
119	C₈²	C₂×D₄	1	2	yes	—
120	C₈²	C₂×D₄	1	1	yes	characteristic
121	C₈²	C₂×Q₈	1	1	yes	characteristic
122	C₈²	C₂×Q₈	1	2	yes	—
123	C₈²	C₂⁴	1	1	yes	characteristic
124	C₂³×C₄	C₄⋊C₈	1	1	yes	characteristic
125	C₂³×C₄	#106	1	1	yes	characteristic
126	C₂³×C₄	#108	1	1	yes	characteristic
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	—	2	4	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	4	2	no	—
135	C₂²×C₈	—	4	4	no	—
136	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
137	C₂²×C₈	C₄⋊C₈	1	2	yes	—
138	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
139	C₂²×C₈	C₄⋊C₈	1	2	yes	—
140	C₂²×C₈	#106	1	2	yes	—
141	C₂²×C₈	#106	1	2	yes	—
142	C₂²×C₈	#106	1	1	yes	characteristic
143	C₂²×C₈	#106	1	1	yes	characteristic
144	C₂²×C₈	C₂²×C₈	1	2	yes	—
145	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
146	C₂²×C₈	C₂×M₁₆	1	1	yes	characteristic
147	C₂²×C₈	C₂×M₁₆	1	2	yes	—
148	C₂²×C₈	C₂×SD₁₆	1	2	yes	—
149	C₂²×C₈	C₂×SD₁₆	1	2	yes	—
150	C₂²×C₈	C₂×SD₁₆	1	1	yes	characteristic
151	C₂²×C₈	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄²	C₄⋊C₈	1	1	yes	characteristic
153	C₂×C₄²	C₄⋊C₈	1	2	yes	—
154	C₂×C₄²	#106	1	2	yes	—
155	C₂×C₄²	#106	1	1	yes	characteristic
156	C₂×C₄²	#108	1	1	yes	characteristic
157	C₂×C₄²	#108	1	2	yes	—
158	C₂×C₄²	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
160	M₃₂	—	8	8	no	—
161	C₂×C₁₆	—	2	8	no	—
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	—	4	4	no	—
164	C₂×C₁₆	—	4	4	no	—
165	C₂×C₁₆	—	4	4	no	—
166	C₂×C₁₆	—	8	8	no	—
167	C₂×C₁₆	—	8	4	no	—
168	C₂×C₁₆	—	8	4	no	—
169	C₂×C₁₆	C₂²×C₈	1	8	yes	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	C₄⋊C₈	1	2	yes	—
175	C₄×C₈	C₄⋊C₈	1	2	yes	—
176	C₄×C₈	C₄⋊C₈	1	2	yes	—
177	C₄×C₈	#106	1	2	yes	—
178	C₄×C₈	#106	1	2	yes	—
179	C₄×C₈	#106	1	2	yes	—
180	C₄×C₈	C₂×C₄⋊C₄	1	2	yes	—
181	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₈	C₂×C₄⋊C₄	1	2	yes	—
183	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄⋊C₄	1	2	yes	—
186	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
187	C₄×C₈	C₂²×C₈	1	2	yes	—
188	C₄×C₈	C₂×M₁₆	1	1	yes	characteristic
189	C₄×C₈	C₂×M₁₆	1	2	yes	—
190	C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
191	C₄×C₈	C₂×SD₁₆	1	2	yes	—
192	C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
193	C₄×C₈	C₂×SD₁₆	1	2	yes	—
194	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₄×C₈	C₂²×Q₈	1	1	yes	characteristic
197	C₂⁴	#334	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	#334	1	1	yes	characteristic
202	C₂²×C₄	#334	1	1	yes	characteristic
203	C₂²×C₄	#334	1	2	yes	—
204	C₂²×C₄	#334	1	2	yes	—
205	C₂²×C₄	#334	1	1	yes	characteristic
206	C₂²×C₄	#334	1	2	yes	—
207	C₂²×C₄	#365	1	4	yes	—
208	C₂²×C₄	#365	1	4	yes	—
209	C₂²×C₄	C₂×C₄⋊C₈	1	2	yes	—
210	C₂²×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×#106	1	2	yes	—
212	C₂²×C₄	C₂×#106	1	1	yes	characteristic
213	C₂²×C₄	C₂×#108	1	1	yes	characteristic
214	C₂²×C₄	C₂×#108	1	2	yes	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄⋊C₈	1	2	yes	—
239	C₂×C₈	C₂×C₄⋊C₈	1	2	yes	—
240	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
241	C₂×C₈	C₂×#106	1	1	yes	characteristic
242	C₂×C₈	C₂×#106	1	1	yes	characteristic
243	C₂×C₈	C₂×#106	1	2	yes	—
244	C₂×C₈	C₂×#106	1	2	yes	—
245	C₂×C₈	#504	1	4	yes	—
246	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
247	C₂×C₈	C₂²×M₁₆	1	1	yes	characteristic
248	C₂×C₈	C₂²×SD₁₆	1	1	yes	characteristic
249	C₂×C₈	C₂²×SD₁₆	1	1	yes	characteristic
250	C₄²	#334	1	2	yes	—
251	C₄²	#334	1	2	yes	—
252	C₄²	#334	1	2	yes	—
253	C₄²	C₂×C₄⋊C₈	1	2	yes	—
254	C₄²	C₂×C₄⋊C₈	1	1	yes	characteristic
255	C₄²	C₂×#106	1	2	yes	—
256	C₄²	C₂×#106	1	1	yes	characteristic
257	C₄²	C₂×#108	1	2	yes	—
258	C₄²	C₂×#108	1	1	yes	characteristic
259	C₄²	C₂²×C₄⋊C₄	1	1	yes	characteristic
260	C₁₆	—	2	8	no	—
261	C₁₆	—	4	4	no	—
262	C₁₆	—	8	4	no	—
263	C₁₆	—	8	4	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	#1368	1	4	yes	—
267	C₂³	C₂×#334	1	2	yes	—
268	C₂³	C₂×#334	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#1368	1	8	yes	—
278	C₂×C₄	#1368	1	4	yes	—
279	C₂×C₄	C₂×#334	1	2	yes	—
280	C₂×C₄	C₂×#334	1	1	yes	characteristic
281	C₂×C₄	C₂×#334	1	2	yes	—
282	C₂×C₄	C₂×#334	1	1	yes	characteristic
283	C₂×C₄	C₂×#334	1	2	yes	—
284	C₂×C₄	C₂×#334	1	1	yes	characteristic
285	C₂×C₄	#2151	1	2	yes	—
286	C₂×C₄	#2153	1	2	yes	—
287	C₂×C₄	C₂×#365	1	4	yes	—
288	C₂×C₄	C₂×#365	1	4	yes	—
289	C₂×C₄	C₂×#365	1	2	yes	—
290	C₂×C₄	C₂×#365	1	2	yes	—
291	C₂×C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×#106	1	1	yes	characteristic
293	C₂×C₄	C₂²×#108	1	1	yes	characteristic
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	2	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	2	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	#2151	1	2	yes	—
308	C₈	C₂×#504	1	2	yes	—
309	C₂²	—	2	8	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	#11997	1	2	yes	—
313	C₂²	C₂×#1368	1	4	yes	—
314	C₂²	C₂×#1368	1	2	yes	—
315	C₂²	#12002	1	2	yes	—
316	C₂²	C₂²×#334	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	#11997	1	2	yes	—
322	C₄	C₂×#1368	1	4	yes	—
323	C₄	C₂×#1368	1	2	yes	—
324	C₄	C₂×#2151	1	1	yes	characteristic
325	C₄	C₂×#2153	1	1	yes	characteristic
326	C₄	C₂²×#365	1	1	yes	characteristic
327	C₄	C₂²×#365	1	1	yes	characteristic
328	C₂	—	2	4	no	—
329	C₂	#122924	1	4	yes	direct product factor
330	C₂	C₂×#11997	1	1	yes	characteristic
331	C₂	C₂²×#1368	1	1	yes	characteristic
332	C₂	C₂×#12002	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂³	C₂×#334
prime radical	C₂⁴	#334
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #122924	C₂	direct (N × H)
5. C₂×#7099	C₂	semidirect
23. #7099	C₂²	semidirect
329. C₂	#122924	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×#11996	2
C₂²×#1367	2
C₂×#11411	2
#122924	2
C₂×#7099	2

Subgroups — C2×#122924 #72231986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#122924 #72231986