Subgroups — C₂×#20265150 #27179588

Order 12928.

788 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265150	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15692169	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108074	—	2	1	no	—
7	C₂×#13108074	—	2	1	no	—
8	C₂×#13108074	—	2	2	no	—
9	C₂×#13108074	—	2	1	no	—
10	C₂×#13108074	—	2	1	no	—
11	C₂×#13108074	—	2	2	no	—
12	C₂×#13108074	C₄	1	4	yes	complemented
13	C₂×#13108074	C₂²	1	4	yes	—
14	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₄₀₄	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₀₁	C₄	1	2	yes	complemented
18	#15692214	—	2	2	no	—
19	#15692214	C₂²	1	4	yes	complemented
20	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	—	2	1	no	—
23	#15692169	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
25	#117×C₁₀₁	—	2	2	no	—
26	#117×C₁₀₁	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×C₁₀₁	—	2	1	no	—
28	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₀₁	—	2	1	no	—
30	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
31	#13108074	—	2	4	no	—
32	#13108074	—	2	4	no	—
33	#13108074	—	2	4	no	—
34	#13108074	—	2	4	no	—
35	#13108074	—	2	4	no	—
36	#13108074	—	2	4	no	—
37	#13108074	—	2	4	no	—
38	#13108074	—	2	4	no	—
39	C₂×Dic₂₀₂	—	2	2	no	—
40	C₂×Dic₂₀₂	—	2	1	no	—
41	C₂×Dic₂₀₂	—	2	1	no	—
42	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
43	C₂×D₄₀₄	—	2	2	no	—
44	C₂×D₄₀₄	—	2	1	no	—
45	C₂×D₄₀₄	—	2	1	no	—
46	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
47	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
48	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
49	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
50	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
51	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
53	C₂²×Dic₁₀₁	—	2	2	no	—
54	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	—	2	2	no	—
57	C₂×C₄×D₁₀₁	—	2	2	no	—
58	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₁₀₁	—	2	4	no	—
60	C₂²⋊Dic₁₀₁	—	2	2	no	—
61	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
62	C₄×Dic₁₀₁	—	2	4	no	—
63	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	—	2	2	no	—
65	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
71	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₀₁	—	2	4	no	—
76	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
77	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
78	C₄²×C₁₀₁	—	2	4	no	—
79	C₂³×C₁₀₁	—	4	2	no	—
80	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
81	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
82	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	—	4	4	no	—
90	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	4	4	no	—
93	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂²⋊D₁₀₁	—	2	4	no	—
110	C₂²⋊D₁₀₁	—	2	4	no	—
111	C₂²⋊D₁₀₁	—	2	4	no	—
112	C₂²⋊D₁₀₁	—	2	4	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	2	no	—
115	C₂×Dic₁₀₁	—	2	4	no	—
116	C₂×Dic₁₀₁	—	4	4	no	—
117	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
118	D₄₀₄	—	2	4	no	—
119	D₄₀₄	—	2	4	no	—
120	C₄×D₁₀₁	—	2	8	no	—
121	C₄×D₁₀₁	—	2	8	no	—
122	Dic₂₀₂	—	2	4	no	—
123	Dic₂₀₂	—	2	4	no	—
124	C₂²×C₁₀₁	—	2	2	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	4	1	no	—
128	C₂²×C₁₀₁	—	4	1	no	—
129	C₂²×C₁₀₁	—	4	1	no	—
130	C₂²×C₁₀₁	—	4	1	no	—
131	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
136	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₀₁	—	2	4	no	—
139	C₂×D₁₀₁	—	4	4	no	—
140	C₂×D₁₀₁	—	4	4	no	—
141	C₂×D₁₀₁	—	4	4	no	—
142	C₄×C₁₀₁	—	4	4	no	—
143	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
144	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
145	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
147	Dic₁₀₁	—	2	4	no	—
148	Dic₁₀₁	—	4	4	no	—
149	C₂×C₁₀₁	—	2	1	no	—
150	C₂×C₁₀₁	—	2	1	no	—
151	C₂×C₁₀₁	—	2	1	no	—
152	C₂×C₁₀₁	—	2	1	no	—
153	C₂×C₁₀₁	—	2	1	no	—
154	C₂×C₁₀₁	—	2	1	no	—
155	C₂×C₁₀₁	#410	1	2	yes	complemented
156	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	D₁₀₁	—	4	4	no	—
158	C₂×#410	—	101	1	no	maximal, Sylow
159	C₁₀₁	C₂×#410	1	1	yes	characteristic, complemented, Sylow
160	C₂²×#41	—	101	1	no	—
161	C₂×#117	—	101	1	no	—
162	C₂×#117	D₁₀₁	1	1	yes	characteristic, complemented
163	#410	—	101	8	no	—
164	C₂×#99	—	101	4	no	—
165	C₂×#41	—	101	4	no	—
166	C₂×#41	—	101	4	no	—
167	C₂×#41	—	202	1	no	—
168	C₂×#41	—	202	2	no	—
169	C₂×#41	—	202	1	no	—
170	C₂×#41	—	202	1	no	—
171	C₂×#41	—	202	2	no	—
172	C₂×#41	—	202	1	no	—
173	C₂²×Q₈	—	101	1	no	—
174	C₂²×D₄	—	101	2	no	—
175	C₂²×D₄	—	101	1	no	—
176	C₂³×C₄	—	101	2	no	—
177	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
178	#117	—	2	2	no	—
179	#117	—	101	4	no	—
180	#117	—	202	2	no	—
181	#117	C₂×D₁₀₁	1	4	yes	complemented
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	202	1	no	—
184	C₂×C₂²⋊C₄	—	101	2	no	—
185	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	202	1	no	—
188	#99	—	101	16	no	—
189	C₂⁴	—	101	2	no	—
190	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
191	#41	—	202	4	no	—
192	#41	—	202	4	no	—
193	#41	—	202	4	no	—
194	#41	—	202	4	no	—
195	#41	—	202	4	no	—
196	#41	—	202	4	no	—
197	#41	—	202	4	no	—
198	#41	—	202	4	no	—
199	C₂×Q₈	—	101	4	no	—
200	C₂×Q₈	—	202	2	no	—
201	C₂×Q₈	—	202	1	no	—
202	C₂×Q₈	—	202	1	no	—
203	C₂×D₄	—	101	4	no	—
204	C₂×D₄	—	101	4	no	—
205	C₂×D₄	—	101	4	no	—
206	C₂×D₄	—	202	2	no	—
207	C₂×D₄	—	202	2	no	—
208	C₂×D₄	—	202	2	no	—
209	C₂×D₄	—	202	2	no	—
210	C₂×D₄	—	202	2	no	—
211	C₂×D₄	—	202	1	no	—
212	C₂×D₄	—	202	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	101	2	no	—
215	C₂²×C₄	—	101	8	no	—
216	C₂²×C₄	—	202	2	no	—
217	C₂²×C₄	—	202	4	no	—
218	C₂²×C₄	—	202	2	no	—
219	C₂²×C₄	—	202	2	no	—
220	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
221	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
222	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
223	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
224	C₂²×C₄	D₄₀₄	1	2	yes	—
225	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
228	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	202	4	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	101	4	no	—
234	C₂²⋊C₄	—	202	2	no	—
235	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
236	C₄²	—	2	4	no	—
237	C₄²	—	202	4	no	—
238	C₂³	—	4	2	no	—
239	C₂³	—	101	4	no	—
240	C₂³	—	404	4	no	—
241	C₂³	—	202	2	no	—
242	C₂³	—	202	2	no	—
243	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
244	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
245	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
246	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
247	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
248	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
249	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
250	Q₈	—	202	4	no	—
251	Q₈	—	202	4	no	—
252	D₄	—	202	4	no	—
253	D₄	—	202	4	no	—
254	D₄	—	202	4	no	—
255	D₄	—	202	4	no	—
256	D₄	—	202	4	no	—
257	D₄	—	202	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	101	4	no	—
261	C₂×C₄	—	404	4	no	—
262	C₂×C₄	—	202	4	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	—	202	2	no	—
265	C₂×C₄	—	202	8	no	—
266	C₂×C₄	—	202	8	no	—
267	C₂×C₄	#13108051	1	1	yes	characteristic
268	C₂×C₄	#13108051	1	2	yes	—
269	C₂×C₄	#13108051	1	2	yes	—
270	C₂×C₄	#13108051	1	1	yes	characteristic
271	C₂×C₄	#13108051	1	1	yes	characteristic
272	C₂×C₄	#13108051	1	2	yes	—
273	C₂×C₄	#13108051	1	2	yes	—
274	C₂×C₄	#13108051	1	2	yes	—
275	C₂×C₄	#13108051	1	2	yes	—
276	C₂×C₄	#13108051	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
278	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
281	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
282	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	404	4	no	—
291	C₂²	—	404	4	no	—
292	C₂²	—	404	4	no	—
293	C₂²	—	202	4	no	—
294	C₂²	C₂×#13108051	1	1	yes	characteristic
295	C₂²	C₂×#13108051	1	1	yes	characteristic
296	C₂²	C₂×#13108051	1	1	yes	characteristic
297	C₂²	C₂×#13108051	1	1	yes	characteristic
298	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
299	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
300	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
301	C₄	—	4	4	no	—
302	C₄	—	404	4	no	—
303	C₄	—	202	4	no	—
304	C₄	C₂×#13108051	1	2	yes	—
305	C₄	C₂×#13108051	1	2	yes	—
306	C₄	C₂×#13108051	1	2	yes	—
307	C₄	C₂×#13108051	1	2	yes	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	404	4	no	—
315	C₂	#20265150	1	2	yes	direct product factor
316	C₂	C₂²×#13108051	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13108051
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×#117×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×#13108074	C₂
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	C₂×#410
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#15692214	C₂	semidirect
3. #20265150	C₂	direct (N × H)
4. C₂×#15692169	C₂	semidirect
5. C₂×#117×C₁₀₁	C₂	semidirect
12. C₂×#13108074	C₄	semidirect
14. C₂²×Dic₂₀₂	C₄	semidirect
15. C₂²×D₄₀₄	C₄	semidirect
17. C₂²×C₄×D₁₀₁	C₄	semidirect
19. #15692214	C₂²	semidirect
23. #15692169	C₂²	semidirect
26. #117×C₁₀₁	C₂²	semidirect
42. C₂×Dic₂₀₂	C₂×C₄	semidirect
46. C₂×D₄₀₄	C₂×C₄	semidirect
58. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
155. C₂×C₁₀₁	#410	semidirect
159. C₁₀₁	C₂×#410	semidirect
162. C₂×#117	D₁₀₁	semidirect
181. #117	C₂×D₁₀₁	semidirect
315. C₂	#20265150	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×#13108074	2
C₂×#15692214	2
#20265150	2
C₂×#15692169	2
C₂×#117×C₁₀₁	2
C₂×#410	101

Subgroups — C2×#20265150 #27179588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265150 #27179588