Subgroups — C₂×#20265828 #27184589

Order 12928.

1632 conjugacy classes of subgroups.

The subgroup lattice has 328 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	2	yes	maximal, complemented
2	#20265828	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692240	C₂	1	2	yes	maximal, complemented
5	C₂×#15692238	C₂	1	2	yes	maximal, complemented
6	C₂×#15692233	C₂	1	2	yes	maximal, complemented
7	C₂×#15692230	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692229	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692217	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108074	—	2	16	no	—
14	C₂×#13108074	C₂²	1	16	yes	complemented
15	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
16	C₂²×D₄₀₄	C₂²	1	2	yes	—
17	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
18	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
21	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
22	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
23	#15692254	C₂²	1	8	yes	complemented
24	#15692240	C₂²	1	16	yes	complemented
25	#15692238	C₂²	1	16	yes	complemented
26	#15692233	C₂²	1	16	yes	complemented
27	#15692230	C₂²	1	8	yes	complemented
28	#15692229	C₂²	1	8	yes	complemented
29	#15692217	C₂²	1	8	yes	complemented
30	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	complemented
31	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×#13108053	C₂²	1	1	yes	characteristic
34	C₂×#13108053	C₂²	1	2	yes	—
35	C₄⋊C₄×D₁₀₁	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×#13108051	C₂²	1	1	yes	characteristic
38	C₂×#13108051	C₂²	1	2	yes	—
39	C₂×#13108051	C₂²	1	2	yes	—
40	C₂×#13108051	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
43	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	8	yes	complemented
48	C₂³×D₁₀₁	C₂³	1	2	yes	—
49	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
50	#13108074	—	2	64	no	—
51	C₂×Dic₂₀₂	—	2	8	no	—
52	C₂×Dic₂₀₂	C₂³	1	8	yes	—
53	C₂×D₄₀₄	—	2	8	no	—
54	C₂×D₄₀₄	—	2	4	no	—
55	C₂×D₄₀₄	C₂³	1	8	yes	—
56	C₂×D₄₀₄	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
58	C₂×C₂²⋊D₁₀₁	D₄	1	8	yes	complemented
59	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
61	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
62	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₁	—	2	8	no	—
66	C₂×C₄×D₁₀₁	—	2	4	no	—
67	C₂×C₄×D₁₀₁	—	2	8	no	—
68	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
69	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
70	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
71	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
72	#13108053	C₂³	1	8	yes	—
73	#13108053	C₂³	1	4	yes	—
74	#13108053	C₂³	1	8	yes	—
75	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
76	#13108051	C₂³	1	4	yes	—
77	#13108051	C₂³	1	8	yes	—
78	#13108051	C₂³	1	8	yes	—
79	#13108051	C₂³	1	8	yes	—
80	C₄×Dic₁₀₁	C₂³	1	4	yes	—
81	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	—	2	8	no	—
83	C₂²×C₄×C₁₀₁	—	2	4	no	—
84	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
86	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
87	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
89	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
91	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
92	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
93	C₂³×C₁₀₁	—	2	2	no	—
94	C₂³×C₁₀₁	—	2	2	no	—
95	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
96	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
97	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
98	C₂²×D₁₀₁	—	2	2	no	—
99	C₂²×D₁₀₁	—	2	2	no	—
100	C₂²×D₁₀₁	—	2	4	no	—
101	C₂²×D₁₀₁	—	2	4	no	—
102	C₂²×D₁₀₁	—	2	4	no	—
103	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
104	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
105	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	4	no	—
108	C₂×C₄×C₁₀₁	—	2	16	no	—
109	C₂×C₄×C₁₀₁	—	2	2	no	—
110	C₂×C₄×C₁₀₁	—	2	8	no	—
111	C₂×C₄×C₁₀₁	#41	1	4	yes	complemented
112	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
114	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
116	C₂²⋊D₁₀₁	—	2	16	no	—
117	C₂²⋊D₁₀₁	C₂×D₄	1	16	yes	complemented
118	C₂×Dic₁₀₁	—	2	4	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	—	2	2	no	—
121	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
122	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
123	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
124	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
125	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
126	D₄₀₄	—	2	16	no	—
127	D₄₀₄	—	2	8	no	—
128	C₄×D₁₀₁	—	2	16	no	—
129	C₄×D₁₀₁	—	2	8	no	—
130	C₄×D₁₀₁	—	2	16	no	—
131	Dic₂₀₂	—	2	16	no	—
132	C₂²×C₁₀₁	—	2	8	no	—
133	C₂²×C₁₀₁	—	2	2	no	—
134	C₂²×C₁₀₁	—	2	2	no	—
135	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
136	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
137	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
138	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
139	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
140	C₂×D₁₀₁	—	2	8	no	—
141	C₂×D₁₀₁	—	2	2	no	—
142	C₂×D₁₀₁	—	2	2	no	—
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	2	4	no	—
145	C₂×D₁₀₁	—	2	4	no	—
146	C₂×D₁₀₁	—	4	8	no	—
147	C₂×D₁₀₁	C₂²×D₄	1	4	yes	—
148	C₄×C₁₀₁	—	2	4	no	—
149	C₄×C₁₀₁	—	2	4	no	—
150	C₄×C₁₀₁	—	2	2	no	—
151	C₄×C₁₀₁	C₂×#41	1	4	yes	complemented
152	Dic₁₀₁	—	2	4	no	—
153	Dic₁₀₁	—	2	2	no	—
154	Dic₁₀₁	—	2	2	no	—
155	Dic₁₀₁	C₂²×D₄	1	4	yes	—
156	C₂×C₁₀₁	—	2	4	no	—
157	C₂×C₁₀₁	#547	1	4	yes	complemented
158	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
159	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
160	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
161	D₁₀₁	—	2	4	no	—
162	D₁₀₁	—	4	4	no	—
163	C₂×#547	—	101	1	no	maximal, Sylow
164	C₁₀₁	C₂×#547	1	1	yes	characteristic, complemented, Sylow
165	C₂²×#41	—	101	2	no	—
166	#547	—	101	16	no	—
167	C₂×C₄⋊Q₈	—	101	1	no	—
168	C₂×#123	—	101	2	no	—
169	C₂×#123	—	101	2	no	—
170	C₂×C₂²⋊Q₈	—	101	2	no	—
171	C₂×#121	—	101	1	no	—
172	C₂×#121	—	101	1	no	—
173	C₂×C₄×D₄	—	101	1	no	—
174	C₂×C₄×D₄	—	101	1	no	—
175	C₂²×C₄⋊C₄	—	101	1	no	—
176	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
177	C₂×#41	—	101	16	no	—
178	C₂×#41	—	202	16	no	—
179	C₂²×Q₈	—	101	2	no	—
180	C₂²×D₄	—	101	1	no	—
181	C₂²×D₄	—	101	2	no	—
182	C₂²×D₄	—	101	1	no	—
183	C₂²×D₄	—	101	2	no	—
184	C₂³×C₄	—	101	1	no	—
185	C₂³×C₄	—	101	2	no	—
186	C₂³×C₄	—	101	2	no	—
187	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
188	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
189	C₄⋊Q₈	—	101	8	no	—
190	#123	—	101	16	no	—
191	#123	—	101	16	no	—
192	C₂²⋊Q₈	—	101	16	no	—
193	#121	—	101	8	no	—
194	#121	—	101	8	no	—
195	C₄×D₄	—	101	8	no	—
196	C₄×D₄	—	101	8	no	—
197	C₂×C₄⋊C₄	—	101	1	no	—
198	C₂×C₄⋊C₄	—	101	2	no	—
199	C₂×C₄⋊C₄	—	101	1	no	—
200	C₂×C₄⋊C₄	—	101	2	no	—
201	C₂×C₄⋊C₄	—	101	8	no	—
202	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×D₁₀₁	1	8	yes	complemented
206	C₂×C₂²⋊C₄	—	101	1	no	—
207	C₂×C₂²⋊C₄	—	101	1	no	—
208	C₂×C₂²⋊C₄	—	101	2	no	—
209	C₂×C₂²⋊C₄	—	101	2	no	—
210	C₂×C₂²⋊C₄	—	101	2	no	—
211	C₂×C₄²	—	101	1	no	—
212	C₂⁴	—	101	1	no	—
213	C₂⁴	—	101	2	no	—
214	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
215	#41	—	202	64	no	—
216	C₂×Q₈	—	101	8	no	—
217	C₂×Q₈	—	202	8	no	—
218	C₂×D₄	—	101	8	no	—
219	C₂×D₄	—	101	4	no	—
220	C₂×D₄	—	101	8	no	—
221	C₂×D₄	—	101	4	no	—
222	C₂×D₄	—	101	8	no	—
223	C₂×D₄	—	202	8	no	—
224	C₂×D₄	—	202	8	no	—
225	C₂×D₄	—	202	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	101	1	no	—
229	C₂²×C₄	—	101	1	no	—
230	C₂²×C₄	—	101	1	no	—
231	C₂²×C₄	—	101	2	no	—
232	C₂²×C₄	—	101	4	no	—
233	C₂²×C₄	—	101	8	no	—
234	C₂²×C₄	—	101	8	no	—
235	C₂²×C₄	—	202	8	no	—
236	C₂²×C₄	—	202	8	no	—
237	C₂²×C₄	—	202	4	no	—
238	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
239	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
240	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
242	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
243	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₄⋊C₄	—	101	8	no	—
245	C₄⋊C₄	—	101	4	no	—
246	C₄⋊C₄	—	101	8	no	—
247	C₄⋊C₄	—	101	4	no	—
248	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
249	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
250	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
251	C₂²⋊C₄	—	101	4	no	—
252	C₂²⋊C₄	—	101	4	no	—
253	C₂²⋊C₄	—	101	8	no	—
254	C₂²⋊C₄	—	101	8	no	—
255	C₂²⋊C₄	—	101	8	no	—
256	C₄²	—	101	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	101	4	no	—
260	C₂³	—	101	2	no	—
261	C₂³	—	101	4	no	—
262	C₂³	—	202	4	no	—
263	C₂³	—	202	4	no	—
264	C₂³	—	202	4	no	—
265	C₂³	—	202	2	no	—
266	C₂³	—	202	2	no	—
267	C₂³	D₄×D₁₀₁	1	4	yes	—
268	C₂³	C₂³×D₁₀₁	1	2	yes	—
269	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
270	Q₈	—	202	16	no	—
271	D₄	—	101	16	no	—
272	D₄	—	202	16	no	—
273	D₄	—	202	16	no	—
274	D₄	—	202	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	16	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	101	4	no	—
281	C₂×C₄	—	101	2	no	—
282	C₂×C₄	—	101	2	no	—
283	C₂×C₄	—	101	2	no	—
284	C₂×C₄	—	101	4	no	—
285	C₂×C₄	—	202	16	no	—
286	C₂×C₄	—	202	16	no	—
287	C₂×C₄	—	202	4	no	—
288	C₂×C₄	—	202	2	no	—
289	C₂×C₄	—	202	2	no	—
290	C₂×C₄	—	202	8	no	—
291	C₂×C₄	#13108074	1	4	yes	complemented
292	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
293	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
294	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
295	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	101	4	no	—
300	C₂²	—	404	8	no	—
301	C₂²	—	202	4	no	—
302	C₂²	—	202	4	no	—
303	C₂²	—	202	4	no	—
304	C₂²	—	202	2	no	—
305	C₂²	—	202	2	no	—
306	C₂²	—	202	8	no	—
307	C₂²	C₂×D₄×D₁₀₁	1	4	yes	—
308	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
309	C₂²	C₂×#13108074	1	2	yes	—
310	C₂²	#15692303	1	2	yes	—
311	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	101	4	no	—
316	C₄	—	202	4	no	—
317	C₄	—	202	2	no	—
318	C₄	—	202	2	no	—
319	C₄	C₂×#13108074	1	4	yes	complemented
320	C₂	—	2	4	no	—
321	C₂	—	404	4	no	—
322	C₂	—	202	4	no	—
323	C₂	#20265828	1	4	yes	direct product factor
324	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
325	C₂	C₂²×#13108074	1	1	yes	characteristic
326	C₂	C₂×#15692303	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#547
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108074	C₂	semidirect
2. #20265828	C₂	direct (N × H)
3. C₂×#15692254	C₂	semidirect
4. C₂×#15692240	C₂	semidirect
5. C₂×#15692238	C₂	semidirect
6. C₂×#15692233	C₂	semidirect
7. C₂×#15692230	C₂	semidirect
8. C₂×#15692229	C₂	semidirect
9. C₂×#15692217	C₂	semidirect
10. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
12. C₂²×C₄⋊C₄×C₁₀₁	C₂	semidirect
14. C₂×#13108074	C₂²	semidirect
23. #15692254	C₂²	semidirect
24. #15692240	C₂²	semidirect
25. #15692238	C₂²	semidirect
26. #15692233	C₂²	semidirect
27. #15692230	C₂²	semidirect
28. #15692229	C₂²	semidirect
29. #15692217	C₂²	semidirect
30. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
35. C₄⋊C₄×D₁₀₁	C₂²	semidirect
47. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
58. C₂×C₂²⋊D₁₀₁	D₄	semidirect
111. C₂×C₄×C₁₀₁	#41	semidirect
117. C₂²⋊D₁₀₁	C₂×D₄	semidirect
151. C₄×C₁₀₁	C₂×#41	semidirect
157. C₂×C₁₀₁	#547	semidirect
164. C₁₀₁	C₂×#547	semidirect
176. C₂²×C₄⋊C₄	D₁₀₁	semidirect
205. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
291. C₂×C₄	#13108074	semidirect
319. C₄	C₂×#13108074	semidirect
323. C₂	#20265828	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
#20265828	2
C₂×#15692254	2
C₂×#15692240	2
C₂×#15692238	2
C₂×#15692233	2
C₂×#15692230	2
C₂×#15692229	2
C₂×#15692217	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂²×C₄⋊C₄×C₁₀₁	2
C₂×#547	101

Subgroups — C2×#20265828 #27184589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265828 #27184589