Subgroups — C₂×#20265722 #27184410

Order 12928.

1728 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	2	yes	maximal, complemented
2	#20265722	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692241	C₂	1	2	yes	maximal, complemented
6	C₂×#15692229	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692227	C₂	1	2	yes	maximal, complemented
8	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692217	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692215	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#121×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108073	—	2	16	no	—
14	C₂×#13108073	C₂²	1	16	yes	complemented
15	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
16	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
20	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
21	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
24	#15692254	C₂²	1	8	yes	complemented
25	#15692251	C₂²	1	8	yes	complemented
26	#15692241	C₂²	1	16	yes	complemented
27	#15692229	C₂²	1	8	yes	complemented
28	#15692227	C₂²	1	16	yes	complemented
29	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
30	#15692217	C₂²	1	8	yes	complemented
31	#15692215	C₂²	1	16	yes	complemented
32	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×#13108053	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
35	C₂×#13108051	C₂²	1	2	yes	—
36	C₂×#13108051	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₁₀₁	C₂²	1	8	yes	complemented
42	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
43	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	#121×C₁₀₁	C₂²	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
48	C₂³×D₁₀₁	C₂³	1	2	yes	—
49	#13108073	—	2	64	no	—
50	C₂×Dic₂₀₂	—	2	8	no	—
51	C₂×Dic₂₀₂	C₂³	1	8	yes	—
52	C₂×D₄₀₄	—	2	4	no	—
53	C₂×D₄₀₄	C₂³	1	4	yes	—
54	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
55	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
56	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
59	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
60	C₂²×Dic₁₀₁	—	2	8	no	—
61	C₂²×Dic₁₀₁	—	2	4	no	—
62	C₂²×Dic₁₀₁	D₄	1	8	yes	complemented
63	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
64	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
65	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
67	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
70	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₀₁	—	2	8	no	—
72	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
73	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
74	#13108053	C₂³	1	8	yes	—
75	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
76	#13108051	C₂³	1	8	yes	—
77	#13108051	C₂³	1	8	yes	—
78	C₄×Dic₁₀₁	C₂³	1	4	yes	—
79	C₄×Dic₁₀₁	C₂³	1	4	yes	—
80	C₄×Dic₁₀₁	C₂³	1	4	yes	—
81	C₄×Dic₁₀₁	C₂³	1	4	yes	—
82	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
83	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₁₀₁	—	2	8	no	—
85	C₂×D₄×C₁₀₁	—	2	4	no	—
86	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
87	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
88	C₂²×C₄×C₁₀₁	—	2	4	no	—
89	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
90	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
92	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
94	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
95	C₂³×C₁₀₁	—	2	2	no	—
96	C₂³×C₁₀₁	—	2	2	no	—
97	C₂³×C₁₀₁	—	2	4	no	—
98	C₂³×C₁₀₁	—	2	4	no	—
99	C₂³×C₁₀₁	—	2	4	no	—
100	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
101	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
102	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
104	C₂²×D₁₀₁	—	2	4	no	—
105	C₂²×D₁₀₁	—	2	4	no	—
106	C₂²×D₁₀₁	—	2	4	no	—
107	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
108	D₄×C₁₀₁	—	2	16	no	—
109	D₄×C₁₀₁	—	2	8	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	—	2	2	no	—
112	C₂×C₄×C₁₀₁	—	2	8	no	—
113	C₂×C₄×C₁₀₁	#41	1	4	yes	complemented
114	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
115	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
117	C₂²⋊D₁₀₁	—	2	16	no	—
118	C₂²⋊D₁₀₁	—	2	16	no	—
119	C₂²⋊D₁₀₁	—	2	16	no	—
120	C₂×Dic₁₀₁	—	2	4	no	—
121	C₂×Dic₁₀₁	—	2	2	no	—
122	C₂×Dic₁₀₁	—	2	16	no	—
123	C₂×Dic₁₀₁	—	2	8	no	—
124	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
125	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
126	C₂×Dic₁₀₁	C₂×D₄	1	16	yes	complemented
127	C₂×Dic₁₀₁	#41	1	4	yes	complemented
128	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
129	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
130	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
131	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
132	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
133	D₄₀₄	—	2	8	no	—
134	C₄×D₁₀₁	—	2	16	no	—
135	Dic₂₀₂	—	2	16	no	—
136	C₂²×C₁₀₁	—	2	8	no	—
137	C₂²×C₁₀₁	—	2	2	no	—
138	C₂²×C₁₀₁	—	2	2	no	—
139	C₂²×C₁₀₁	—	2	4	no	—
140	C₂²×C₁₀₁	—	2	4	no	—
141	C₂²×C₁₀₁	—	2	4	no	—
142	C₂²×C₁₀₁	—	4	8	no	—
143	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
144	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
145	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
146	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
147	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₀₁	—	2	4	no	—
149	C₂×D₁₀₁	—	2	4	no	—
150	C₂×D₁₀₁	—	2	4	no	—
151	C₂×D₁₀₁	—	4	8	no	—
152	C₄×C₁₀₁	—	2	4	no	—
153	C₄×C₁₀₁	—	2	2	no	—
154	C₄×C₁₀₁	C₂×#41	1	4	yes	complemented
155	Dic₁₀₁	—	2	4	no	—
156	Dic₁₀₁	—	2	2	no	—
157	Dic₁₀₁	C₂²×D₄	1	4	yes	—
158	Dic₁₀₁	C₂²×D₄	1	4	yes	—
159	Dic₁₀₁	C₂×#41	1	4	yes	complemented
160	C₂×C₁₀₁	—	2	4	no	—
161	C₂×C₁₀₁	—	4	4	no	—
162	C₂×C₁₀₁	#532	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	C₂²×#41	1	1	yes	characteristic
166	D₁₀₁	—	4	4	no	—
167	C₂×#532	—	101	1	no	maximal, Sylow
168	C₁₀₁	C₂×#532	1	1	yes	characteristic, complemented, Sylow
169	C₂²×#41	—	101	2	no	—
170	#532	—	101	16	no	—
171	C₂×C₄⋊Q₈	—	101	1	no	—
172	C₂×#127	—	101	1	no	—
173	C₂×#124	—	101	2	no	—
174	C₂×#121	—	101	2	no	—
175	C₂×#121	—	101	1	no	—
176	C₂×#121	D₁₀₁	1	1	yes	characteristic, complemented
177	C₂×C₄×D₄	—	101	1	no	—
178	C₂×C₄×D₄	—	101	1	no	—
179	C₂×C₄×D₄	—	101	2	no	—
180	C₂²×C₄²	—	101	1	no	—
181	C₂×#41	—	101	16	no	—
182	C₂×#41	—	202	16	no	—
183	C₂²×Q₈	—	101	2	no	—
184	C₂²×D₄	—	101	2	no	—
185	C₂²×D₄	—	101	2	no	—
186	C₂²×D₄	—	101	2	no	—
187	C₂²×D₄	—	101	1	no	—
188	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
189	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
190	C₂³×C₄	—	101	1	no	—
191	C₂³×C₄	—	101	2	no	—
192	C₂³×C₄	—	101	1	no	—
193	C₂³×C₄	—	101	2	no	—
194	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
195	C₄⋊Q₈	—	101	8	no	—
196	#127	—	101	8	no	—
197	#124	—	101	16	no	—
198	#121	—	101	16	no	—
199	#121	—	101	8	no	—
200	#121	C₂×D₁₀₁	1	8	yes	complemented
201	C₄×D₄	—	101	8	no	—
202	C₄×D₄	—	101	8	no	—
203	C₄×D₄	—	101	16	no	—
204	C₂×C₄⋊C₄	—	101	2	no	—
205	C₂×C₄⋊C₄	—	101	1	no	—
206	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	101	2	no	—
208	C₂×C₂²⋊C₄	—	101	2	no	—
209	C₂×C₂²⋊C₄	—	101	2	no	—
210	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
211	C₂×C₄²	—	101	1	no	—
212	C₂×C₄²	—	101	1	no	—
213	C₂×C₄²	—	101	1	no	—
214	C₂×C₄²	—	101	1	no	—
215	C₂×C₄²	—	101	8	no	—
216	C₂⁴	—	101	2	no	—
217	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
218	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
219	#41	—	202	64	no	—
220	C₂×Q₈	—	101	8	no	—
221	C₂×Q₈	—	202	8	no	—
222	C₂×D₄	—	2	8	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	101	8	no	—
225	C₂×D₄	—	101	8	no	—
226	C₂×D₄	—	101	8	no	—
227	C₂×D₄	—	101	4	no	—
228	C₂×D₄	—	202	8	no	—
229	C₂×D₄	—	202	8	no	—
230	C₂×D₄	—	202	8	no	—
231	C₂×D₄	—	202	4	no	—
232	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
233	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	101	8	no	—
236	C₂²×C₄	—	101	4	no	—
237	C₂²×C₄	—	101	1	no	—
238	C₂²×C₄	—	101	1	no	—
239	C₂²×C₄	—	101	1	no	—
240	C₂²×C₄	—	101	4	no	—
241	C₂²×C₄	—	101	8	no	—
242	C₂²×C₄	—	101	1	no	—
243	C₂²×C₄	—	101	2	no	—
244	C₂²×C₄	—	101	8	no	—
245	C₂²×C₄	—	202	8	no	—
246	C₂²×C₄	—	202	4	no	—
247	C₂²×C₄	—	202	8	no	—
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	8	no	—
253	C₄⋊C₄	—	101	4	no	—
254	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
255	C₂²⋊C₄	—	101	8	no	—
256	C₂²⋊C₄	—	101	8	no	—
257	C₂²⋊C₄	—	101	8	no	—
258	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
259	C₄²	—	101	4	no	—
260	C₄²	—	101	4	no	—
261	C₄²	—	101	4	no	—
262	C₄²	—	101	4	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	101	4	no	—
269	C₂³	—	202	4	no	—
270	C₂³	—	202	4	no	—
271	C₂³	—	202	4	no	—
272	C₂³	D₄×D₁₀₁	1	4	yes	—
273	C₂³	C₂³×D₁₀₁	1	4	yes	—
274	C₂³	C₂³×D₁₀₁	1	2	yes	—
275	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
276	Q₈	—	202	16	no	—
277	D₄	—	2	16	no	—
278	D₄	—	2	8	no	—
279	D₄	—	202	16	no	—
280	D₄	—	202	16	no	—
281	D₄	—	202	16	no	—
282	D₄	—	202	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	101	4	no	—
287	C₂×C₄	—	101	4	no	—
288	C₂×C₄	—	101	16	no	—
289	C₂×C₄	—	101	4	no	—
290	C₂×C₄	—	101	2	no	—
291	C₂×C₄	—	101	2	no	—
292	C₂×C₄	—	101	2	no	—
293	C₂×C₄	—	101	2	no	—
294	C₂×C₄	—	101	4	no	—
295	C₂×C₄	—	202	16	no	—
296	C₂×C₄	—	202	16	no	—
297	C₂×C₄	—	202	8	no	—
298	C₂×C₄	—	202	4	no	—
299	C₂×C₄	—	202	2	no	—
300	C₂×C₄	#13108073	1	4	yes	complemented
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	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	404	8	no	—
312	C₂²	—	202	4	no	—
313	C₂²	—	202	4	no	—
314	C₂²	—	202	4	no	—
315	C₂²	C₂×D₄×D₁₀₁	1	4	yes	—
316	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
317	C₂²	#41×D₁₀₁	1	2	yes	—
318	C₂²	C₂×#13108073	1	2	yes	—
319	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	2	no	—
322	C₄	—	101	4	no	—
323	C₄	—	101	4	no	—
324	C₄	—	101	4	no	—
325	C₄	—	202	4	no	—
326	C₄	—	202	2	no	—
327	C₄	C₂×#13108073	1	4	yes	complemented
328	C₂	—	2	4	no	—
329	C₂	—	4	4	no	—
330	C₂	—	404	4	no	—
331	C₂	#20265722	1	4	yes	direct product factor
332	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
333	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
334	C₂	C₂²×#13108073	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#121×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#121	D₁₀₁
hypocenter	C₁₀₁	C₂×#532
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂²×#13108073	C₂	semidirect
2. #20265722	C₂	direct (N × H)
3. C₂×#15692254	C₂	semidirect
4. C₂×#15692251	C₂	semidirect
5. C₂×#15692241	C₂	semidirect
6. C₂×#15692229	C₂	semidirect
7. C₂×#15692227	C₂	semidirect
8. C₂×D₄×Dic₁₀₁	C₂	semidirect
9. C₂×#15692217	C₂	semidirect
10. C₂×#15692215	C₂	semidirect
11. C₂²×C₄×Dic₁₀₁	C₂	semidirect
12. C₂×#121×C₁₀₁	C₂	semidirect
14. C₂×#13108073	C₂²	semidirect
24. #15692254	C₂²	semidirect
25. #15692251	C₂²	semidirect
26. #15692241	C₂²	semidirect
27. #15692229	C₂²	semidirect
28. #15692227	C₂²	semidirect
29. D₄×Dic₁₀₁	C₂²	semidirect
30. #15692217	C₂²	semidirect
31. #15692215	C₂²	semidirect
41. C₂×C₄×Dic₁₀₁	C₂²	semidirect
45. #121×C₁₀₁	C₂²	semidirect
62. C₂²×Dic₁₀₁	D₄	semidirect
113. C₂×C₄×C₁₀₁	#41	semidirect
126. C₂×Dic₁₀₁	C₂×D₄	semidirect
127. C₂×Dic₁₀₁	#41	semidirect
154. C₄×C₁₀₁	C₂×#41	semidirect
159. Dic₁₀₁	C₂×#41	semidirect
162. C₂×C₁₀₁	#532	semidirect
168. C₁₀₁	C₂×#532	semidirect
176. C₂×#121	D₁₀₁	semidirect
200. #121	C₂×D₁₀₁	semidirect
300. C₂×C₄	#13108073	semidirect
327. C₄	C₂×#13108073	semidirect
331. C₂	#20265722	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
#20265722	2
C₂×#15692254	2
C₂×#15692251	2
C₂×#15692241	2
C₂×#15692229	2
C₂×#15692227	2
C₂×D₄×Dic₁₀₁	2
C₂×#15692217	2
C₂×#15692215	2
C₂²×C₄×Dic₁₀₁	2
C₂×#121×C₁₀₁	2
C₂×#532	101

Subgroups — C2×#20265722 #27184410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265722 #27184410