Subgroups — C₂×#20265826 #27184587

Order 12928.

1632 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108073	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265826	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692238	C₂	1	2	yes	maximal, complemented
5	C₂×#15692236	C₂	1	2	yes	maximal, complemented
6	C₂×#15692234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692231	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692227	C₂	1	2	yes	maximal, complemented
9	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#13108053	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108073	—	2	8	no	—
14	C₂×#13108073	C₂²	1	8	yes	complemented
15	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
18	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
19	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
21	#15692253	C₂²	1	8	yes	complemented
22	#15692238	C₂²	1	16	yes	complemented
23	#15692236	C₂²	1	16	yes	complemented
24	#15692234	C₂²	1	8	yes	complemented
25	#15692231	C₂²	1	8	yes	complemented
26	#15692227	C₂²	1	16	yes	complemented
27	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×#13108053	C₂²	1	1	yes	characteristic
31	C₂×#13108053	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×#13108053	C₂²	1	2	yes	—
34	C₂×#13108053	C₂²	1	2	yes	—
35	C₂×#13108053	C₂²	1	16	yes	complemented
36	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
38	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
39	C₂×#13108051	C₂²	1	1	yes	characteristic
40	C₂×#13108051	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
42	C₂×#41×C₁₀₁	—	2	8	no	—
43	C₂×#41×C₁₀₁	C₂²	1	8	yes	complemented
44	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
47	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
48	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
49	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
50	#13108073	—	2	32	no	—
51	C₂×Dic₂₀₂	—	2	4	no	—
52	C₂×Dic₂₀₂	C₂³	1	4	yes	—
53	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
54	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
55	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
57	C₂²×Dic₁₀₁	—	2	8	no	—
58	C₂²×Dic₁₀₁	—	2	8	no	—
59	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
61	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
62	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
63	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
64	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₀₁	—	2	4	no	—
67	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
68	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
69	#13108053	C₂³	1	4	yes	—
70	#13108053	C₂³	1	8	yes	—
71	#13108053	C₂³	1	8	yes	—
72	#13108053	C₂³	1	8	yes	—
73	#13108053	C₂³	1	8	yes	—
74	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
75	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
77	#13108051	C₂³	1	4	yes	—
78	#13108051	C₂³	1	8	yes	—
79	C₄×Dic₁₀₁	C₂³	1	4	yes	—
80	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
81	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
82	#41×C₁₀₁	—	2	32	no	—
83	C₂×Q₈×C₁₀₁	—	2	4	no	—
84	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
85	C₂×D₄×C₁₀₁	—	2	8	no	—
86	C₂×D₄×C₁₀₁	D₄	1	8	yes	complemented
87	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
88	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
89	C₂²×C₄×C₁₀₁	—	2	8	no	—
90	C₂²×C₄×C₁₀₁	—	2	4	no	—
91	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
94	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
96	C₂³×C₁₀₁	—	2	4	no	—
97	C₂³×C₁₀₁	—	2	4	no	—
98	C₂³×C₁₀₁	—	2	2	no	—
99	C₂³×C₁₀₁	—	2	2	no	—
100	C₂³×C₁₀₁	—	2	2	no	—
101	C₂³×C₁₀₁	C₂×D₄	1	8	yes	—
102	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
104	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
105	C₂²×D₁₀₁	—	2	2	no	—
106	C₂²×D₁₀₁	—	2	2	no	—
107	C₂²×D₁₀₁	—	2	2	no	—
108	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
109	Q₈×C₁₀₁	—	2	8	no	—
110	D₄×C₁₀₁	—	2	16	no	—
111	D₄×C₁₀₁	C₂×D₄	1	16	yes	complemented
112	C₂×C₄×C₁₀₁	—	2	2	no	—
113	C₂×C₄×C₁₀₁	—	2	4	no	—
114	C₂×C₄×C₁₀₁	—	2	16	no	—
115	C₂×C₄×C₁₀₁	—	2	8	no	—
116	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
117	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
120	C₂²⋊D₁₀₁	—	2	16	no	—
121	C₂²⋊D₁₀₁	—	2	8	no	—
122	C₂×Dic₁₀₁	—	2	2	no	—
123	C₂×Dic₁₀₁	—	2	4	no	—
124	C₂×Dic₁₀₁	—	2	16	no	—
125	C₂×Dic₁₀₁	—	2	4	no	—
126	C₂×Dic₁₀₁	—	2	2	no	—
127	C₂×Dic₁₀₁	—	2	16	no	—
128	C₂×Dic₁₀₁	#41	1	4	yes	complemented
129	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
130	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
131	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
132	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
133	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
134	C₄×D₁₀₁	—	2	8	no	—
135	Dic₂₀₂	—	2	8	no	—
136	C₂²×C₁₀₁	—	2	16	no	—
137	C₂²×C₁₀₁	—	2	4	no	—
138	C₂²×C₁₀₁	—	2	4	no	—
139	C₂²×C₁₀₁	—	2	2	no	—
140	C₂²×C₁₀₁	—	2	2	no	—
141	C₂²×C₁₀₁	—	2	2	no	—
142	C₂²×C₁₀₁	—	4	4	no	—
143	C₂²×C₁₀₁	C₂²×D₄	1	8	yes	—
144	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
145	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
146	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
147	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₀₁	—	2	2	no	—
149	C₂×D₁₀₁	—	2	2	no	—
150	C₂×D₁₀₁	—	2	2	no	—
151	C₂×D₁₀₁	—	4	4	no	—
152	C₄×C₁₀₁	—	2	2	no	—
153	C₄×C₁₀₁	—	2	4	no	—
154	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
155	Dic₁₀₁	—	2	4	no	—
156	Dic₁₀₁	—	2	2	no	—
157	Dic₁₀₁	—	2	4	no	—
158	Dic₁₀₁	—	2	2	no	—
159	Dic₁₀₁	C₂×#41	1	4	yes	complemented
160	C₂×C₁₀₁	—	2	8	no	—
161	C₂×C₁₀₁	—	4	2	no	—
162	C₂×C₁₀₁	#547	1	4	yes	complemented
163	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
164	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
165	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
166	D₁₀₁	—	4	2	no	—
167	C₂×#547	—	101	1	no	maximal, Sylow
168	C₁₀₁	C₂×#547	1	1	yes	characteristic, complemented, Sylow
169	C₂²×#41	—	101	1	no	—
170	C₂²×#41	D₁₀₁	1	1	yes	characteristic, complemented
171	#547	—	101	16	no	—
172	C₂×C₄⋊Q₈	—	101	1	no	—
173	C₂×#123	—	101	2	no	—
174	C₂×#123	—	101	2	no	—
175	C₂×C₂²⋊Q₈	—	101	1	no	—
176	C₂×C₂²⋊Q₈	—	101	1	no	—
177	C₂×#121	—	101	2	no	—
178	C₂×C₄×D₄	—	101	1	no	—
179	C₂×C₄×D₄	—	101	1	no	—
180	C₂²×C₄⋊C₄	—	101	2	no	—
181	C₂×#41	—	2	8	no	—
182	C₂×#41	—	101	8	no	—
183	C₂×#41	—	202	8	no	—
184	C₂×#41	C₂×D₁₀₁	1	8	yes	complemented
185	C₂²×Q₈	—	101	1	no	—
186	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
187	C₂²×D₄	—	101	1	no	—
188	C₂²×D₄	—	101	2	no	—
189	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
190	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
191	C₂³×C₄	—	101	2	no	—
192	C₂³×C₄	—	101	2	no	—
193	C₂³×C₄	—	101	1	no	—
194	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
195	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
196	C₄⋊Q₈	—	101	8	no	—
197	#123	—	101	16	no	—
198	#123	—	101	16	no	—
199	C₂²⋊Q₈	—	101	8	no	—
200	C₂²⋊Q₈	—	101	8	no	—
201	#121	—	101	16	no	—
202	C₄×D₄	—	101	8	no	—
203	C₄×D₄	—	101	8	no	—
204	C₂×C₄⋊C₄	—	101	1	no	—
205	C₂×C₄⋊C₄	—	101	2	no	—
206	C₂×C₄⋊C₄	—	101	2	no	—
207	C₂×C₄⋊C₄	—	101	2	no	—
208	C₂×C₄⋊C₄	—	101	16	no	—
209	C₂×C₄⋊C₄	—	101	1	no	—
210	C₂×C₄⋊C₄	—	101	2	no	—
211	C₂×C₂²⋊C₄	—	101	2	no	—
212	C₂×C₂²⋊C₄	—	101	1	no	—
213	C₂×C₂²⋊C₄	—	101	2	no	—
214	C₂×C₂²⋊C₄	—	101	1	no	—
215	C₂×C₂²⋊C₄	—	101	2	no	—
216	C₂×C₄²	—	101	1	no	—
217	C₂⁴	—	101	1	no	—
218	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
219	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
220	#41	—	2	32	no	—
221	#41	—	202	32	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	101	4	no	—
224	C₂×Q₈	—	202	4	no	—
225	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
226	C₂×D₄	—	2	8	no	—
227	C₂×D₄	—	101	4	no	—
228	C₂×D₄	—	101	8	no	—
229	C₂×D₄	—	202	4	no	—
230	C₂×D₄	—	202	8	no	—
231	C₂×D₄	C₂²⋊D₁₀₁	1	8	yes	complemented
232	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
233	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	101	1	no	—
237	C₂²×C₄	—	101	8	no	—
238	C₂²×C₄	—	101	8	no	—
239	C₂²×C₄	—	101	1	no	—
240	C₂²×C₄	—	101	2	no	—
241	C₂²×C₄	—	101	1	no	—
242	C₂²×C₄	—	101	2	no	—
243	C₂²×C₄	—	101	4	no	—
244	C₂²×C₄	—	202	8	no	—
245	C₂²×C₄	—	202	8	no	—
246	C₂²×C₄	—	202	4	no	—
247	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
248	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
249	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
251	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
252	C₄⋊C₄	—	101	4	no	—
253	C₄⋊C₄	—	101	8	no	—
254	C₄⋊C₄	—	101	8	no	—
255	C₄⋊C₄	—	101	8	no	—
256	C₄⋊C₄	—	101	8	no	—
257	C₄⋊C₄	—	101	4	no	—
258	C₂²⋊C₄	—	101	8	no	—
259	C₂²⋊C₄	—	101	4	no	—
260	C₂²⋊C₄	—	101	8	no	—
261	C₂²⋊C₄	—	101	4	no	—
262	C₂²⋊C₄	—	101	8	no	—
263	C₄²	—	101	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	101	2	no	—
270	C₂³	—	202	2	no	—
271	C₂³	—	202	2	no	—
272	C₂³	—	202	2	no	—
273	C₂³	C₂×C₂²⋊D₁₀₁	1	8	yes	—
274	C₂³	C₂³×D₁₀₁	1	2	yes	—
275	C₂³	C₂³×D₁₀₁	1	4	yes	—
276	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	202	8	no	—
279	D₄	—	2	16	no	—
280	D₄	—	202	8	no	—
281	D₄	—	202	16	no	—
282	D₄	C₂×C₂²⋊D₁₀₁	1	16	yes	complemented
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	16	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	101	4	no	—
288	C₂×C₄	—	101	2	no	—
289	C₂×C₄	—	101	2	no	—
290	C₂×C₄	—	101	4	no	—
291	C₂×C₄	—	101	2	no	—
292	C₂×C₄	—	101	4	no	—
293	C₂×C₄	—	202	16	no	—
294	C₂×C₄	—	202	2	no	—
295	C₂×C₄	—	202	4	no	—
296	C₂×C₄	—	202	16	no	—
297	C₂×C₄	—	202	4	no	—
298	C₂×C₄	—	202	2	no	—
299	C₂×C₄	—	202	8	no	—
300	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
301	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
302	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
303	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	16	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	404	4	no	—
312	C₂²	—	202	2	no	—
313	C₂²	—	202	2	no	—
314	C₂²	—	202	2	no	—
315	C₂²	C₂²×C₂²⋊D₁₀₁	1	8	yes	—
316	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
317	C₂²	#41×D₁₀₁	1	2	yes	—
318	C₂²	#15692302	1	2	yes	—
319	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	101	4	no	—
323	C₄	—	202	2	no	—
324	C₄	—	202	4	no	—
325	C₄	—	202	4	no	—
326	C₄	—	202	2	no	—
327	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
328	C₂	—	2	8	no	—
329	C₂	—	4	2	no	—
330	C₂	—	404	2	no	—
331	C₂	#20265826	1	4	yes	direct product factor
332	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
333	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
334	C₂	C₂×#15692302	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×#41×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×#41	D₁₀₁
hypocenter	C₁₀₁	C₂×#547
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108073	C₂	semidirect
2. #20265826	C₂	direct (N × H)
3. C₂×#15692253	C₂	semidirect
4. C₂×#15692238	C₂	semidirect
5. C₂×#15692236	C₂	semidirect
6. C₂×#15692234	C₂	semidirect
7. C₂×#15692231	C₂	semidirect
8. C₂×#15692227	C₂	semidirect
9. C₂×D₄×Dic₁₀₁	C₂	semidirect
10. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
11. C₂²×#13108053	C₂	semidirect
12. C₂²×#41×C₁₀₁	C₂	semidirect
14. C₂×#13108073	C₂²	semidirect
21. #15692253	C₂²	semidirect
22. #15692238	C₂²	semidirect
23. #15692236	C₂²	semidirect
24. #15692234	C₂²	semidirect
25. #15692231	C₂²	semidirect
26. #15692227	C₂²	semidirect
27. D₄×Dic₁₀₁	C₂²	semidirect
28. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
35. C₂×#13108053	C₂²	semidirect
43. C₂×#41×C₁₀₁	C₂²	semidirect
86. C₂×D₄×C₁₀₁	D₄	semidirect
111. D₄×C₁₀₁	C₂×D₄	semidirect
128. C₂×Dic₁₀₁	#41	semidirect
159. Dic₁₀₁	C₂×#41	semidirect
162. C₂×C₁₀₁	#547	semidirect
168. C₁₀₁	C₂×#547	semidirect
170. C₂²×#41	D₁₀₁	semidirect
184. C₂×#41	C₂×D₁₀₁	semidirect
231. C₂×D₄	C₂²⋊D₁₀₁	semidirect
282. D₄	C₂×C₂²⋊D₁₀₁	semidirect
331. C₂	#20265826	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×#13108073	2
#20265826	2
C₂×#15692253	2
C₂×#15692238	2
C₂×#15692236	2
C₂×#15692234	2
C₂×#15692231	2
C₂×#15692227	2
C₂×D₄×Dic₁₀₁	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂²×#13108053	2
C₂²×#41×C₁₀₁	2
C₂×#547	101

Subgroups — C2×#20265826 #27184587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265826 #27184587