Subgroups — C₂×#20265890 #27184670

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265890	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692243	C₂	1	2	yes	maximal, complemented
4	C₂×#15692241	C₂	1	2	yes	maximal, complemented
5	C₂×#15692230	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692229	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692228	C₂	1	2	yes	maximal, complemented
8	C₂×#15692217	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692213	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692212	C₂	1	2	yes	maximal, complemented
12	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₄	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	#15692254	C₂²	1	8	yes	complemented
21	#15692243	C₂²	1	16	yes	complemented
22	#15692241	C₂²	1	16	yes	complemented
23	#15692230	C₂²	1	8	yes	complemented
24	#15692229	C₂²	1	8	yes	complemented
25	#15692228	C₂²	1	16	yes	complemented
26	#15692217	C₂²	1	8	yes	complemented
27	C₄×C₂²⋊D₁₀₁	C₂²	1	8	yes	complemented
28	#15692213	C₂²	1	8	yes	complemented
29	#15692212	C₂²	1	16	yes	complemented
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×#13108053	C₂²	1	1	yes	characteristic
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×#13108051	C₂²	1	2	yes	—
35	C₂×#13108051	C₂²	1	2	yes	—
36	C₂×#13108051	C₂²	1	2	yes	—
37	C₂×#13108051	C₂²	1	1	yes	characteristic
38	C₂×#13108051	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
42	#117×C₁₀₁	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
47	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
48	C₂³×D₁₀₁	C₂³	1	2	yes	—
49	C₂×Dic₂₀₂	—	4	4	no	—
50	C₂×Dic₂₀₂	C₂³	1	8	yes	—
51	C₂×D₄₀₄	—	2	4	no	—
52	C₂×D₄₀₄	—	4	4	no	—
53	C₂×D₄₀₄	C₂³	1	4	yes	—
54	C₂×D₄₀₄	C₂³	1	8	yes	—
55	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
56	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
59	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
61	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	—	2	4	no	—
64	C₂×C₄×D₁₀₁	—	2	8	no	—
65	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
68	#13108053	C₂³	1	4	yes	—
69	#13108053	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
71	#13108051	C₂³	1	8	yes	—
72	#13108051	C₂³	1	8	yes	—
73	#13108051	C₂³	1	8	yes	—
74	#13108051	C₂³	1	4	yes	—
75	#13108051	C₂³	1	8	yes	—
76	C₄×Dic₁₀₁	C₂³	1	8	yes	—
77	C₄×Dic₁₀₁	C₂³	1	4	yes	—
78	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	—	2	4	no	—
80	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
87	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
88	C₄²×C₁₀₁	C₂³	1	8	yes	—
89	C₂³×C₁₀₁	—	2	2	no	—
90	C₂³×C₁₀₁	—	2	2	no	—
91	C₂³×C₁₀₁	—	2	2	no	—
92	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
93	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
94	C₂²×D₁₀₁	—	2	4	no	—
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	2	2	no	—
97	C₂²×D₁₀₁	—	2	4	no	—
98	C₂²×D₁₀₁	—	2	4	no	—
99	C₂²×D₁₀₁	—	2	2	no	—
100	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
101	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	4	no	—
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	2	8	no	—
106	C₂×C₄×C₁₀₁	#41	1	4	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
110	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
111	C₂²⋊D₁₀₁	—	2	8	no	—
112	C₂²⋊D₁₀₁	—	4	8	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	4	no	—
115	C₂×Dic₁₀₁	—	2	2	no	—
116	C₂×Dic₁₀₁	#41	1	4	yes	—
117	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
118	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
119	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
120	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
121	D₄₀₄	—	2	8	no	—
122	D₄₀₄	—	4	8	no	—
123	C₄×D₁₀₁	—	2	8	no	—
124	C₄×D₁₀₁	—	2	16	no	—
125	Dic₂₀₂	—	4	8	no	—
126	C₂²×C₁₀₁	—	2	2	no	—
127	C₂²×C₁₀₁	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	4	4	no	—
130	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
131	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
132	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
133	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₀₁	—	2	4	no	—
135	C₂×D₁₀₁	—	2	2	no	—
136	C₂×D₁₀₁	—	2	4	no	—
137	C₂×D₁₀₁	—	2	2	no	—
138	C₂×D₁₀₁	—	2	2	no	—
139	C₂×D₁₀₁	—	2	4	no	—
140	C₂×D₁₀₁	—	4	4	no	—
141	C₂×D₁₀₁	—	4	8	no	—
142	C₄×C₁₀₁	—	2	4	no	—
143	C₄×C₁₀₁	—	2	2	no	—
144	C₄×C₁₀₁	—	2	4	no	—
145	C₄×C₁₀₁	C₂×#41	1	4	yes	—
146	Dic₁₀₁	—	2	2	no	—
147	Dic₁₀₁	—	2	2	no	—
148	Dic₁₀₁	—	2	4	no	—
149	Dic₁₀₁	C₂×#41	1	4	yes	—
150	C₂×C₁₀₁	—	4	2	no	—
151	C₂×C₁₀₁	#555	1	4	yes	complemented
152	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
153	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
154	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
155	D₁₀₁	—	4	4	no	—
156	D₁₀₁	—	4	2	no	—
157	C₂×#555	—	101	1	no	maximal, Sylow
158	C₁₀₁	C₂×#555	1	1	yes	characteristic, complemented, Sylow
159	#555	—	101	16	no	—
160	C₂×C₄⋊Q₈	—	101	1	no	—
161	C₂×#124	—	101	2	no	—
162	C₂×#124	—	101	2	no	—
163	C₂×#121	—	101	1	no	—
164	C₂×#121	—	101	1	no	—
165	C₂×#121	—	101	2	no	—
166	C₂×C₄×D₄	—	101	1	no	—
167	C₂×C₄×D₄	—	101	1	no	—
168	C₂×#117	—	101	1	no	—
169	C₂×#117	—	101	2	no	—
170	C₂×#117	D₁₀₁	1	1	yes	characteristic, complemented
171	C₂²×Q₈	—	101	2	no	—
172	C₂²×D₄	—	101	2	no	—
173	C₂²×D₄	—	101	1	no	—
174	C₂²×D₄	—	101	2	no	—
175	C₂²×D₄	—	101	1	no	—
176	C₂³×C₄	—	101	2	no	—
177	C₂³×C₄	—	101	1	no	—
178	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
179	C₄⋊Q₈	—	101	8	no	—
180	#124	—	101	16	no	—
181	#124	—	101	16	no	—
182	#121	—	101	8	no	—
183	#121	—	101	8	no	—
184	#121	—	101	16	no	—
185	C₄×D₄	—	101	8	no	—
186	C₄×D₄	—	101	8	no	—
187	#117	—	101	8	no	—
188	#117	—	101	16	no	—
189	#117	C₂×D₁₀₁	1	8	yes	complemented
190	C₂×C₄⋊C₄	—	101	1	no	—
191	C₂×C₄⋊C₄	—	101	2	no	—
192	C₂×C₄⋊C₄	—	101	1	no	—
193	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	101	1	no	—
196	C₂×C₂²⋊C₄	—	101	2	no	—
197	C₂×C₂²⋊C₄	—	101	2	no	—
198	C₂×C₂²⋊C₄	—	101	2	no	—
199	C₂×C₂²⋊C₄	—	101	2	no	—
200	C₂×C₂²⋊C₄	—	101	1	no	—
201	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
202	C₂×C₄²	—	101	2	no	—
203	C₂×C₄²	—	101	1	no	—
204	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
205	C₂⁴	—	101	2	no	—
206	C₂⁴	—	101	1	no	—
207	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂×Q₈	—	101	8	no	—
209	C₂×Q₈	—	404	4	no	—
210	C₂×D₄	—	101	8	no	—
211	C₂×D₄	—	101	4	no	—
212	C₂×D₄	—	101	8	no	—
213	C₂×D₄	—	101	4	no	—
214	C₂×D₄	—	404	4	no	—
215	C₂×D₄	—	404	4	no	—
216	C₂×D₄	—	202	4	no	—
217	C₂×D₄	—	202	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	101	1	no	—
220	C₂²×C₄	—	101	1	no	—
221	C₂²×C₄	—	101	1	no	—
222	C₂²×C₄	—	101	2	no	—
223	C₂²×C₄	—	101	8	no	—
224	C₂²×C₄	—	101	4	no	—
225	C₂²×C₄	—	202	8	no	—
226	C₂²×C₄	—	202	4	no	—
227	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
228	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
231	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
232	C₄⋊C₄	—	101	4	no	—
233	C₄⋊C₄	—	101	8	no	—
234	C₄⋊C₄	—	101	4	no	—
235	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
236	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
237	C₂²⋊C₄	—	101	4	no	—
238	C₂²⋊C₄	—	101	8	no	—
239	C₂²⋊C₄	—	101	8	no	—
240	C₂²⋊C₄	—	101	8	no	—
241	C₂²⋊C₄	—	101	8	no	—
242	C₂²⋊C₄	—	101	4	no	—
243	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
244	C₄²	—	101	8	no	—
245	C₄²	—	101	4	no	—
246	C₄²	C₂²×D₁₀₁	1	8	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	101	4	no	—
251	C₂³	—	101	2	no	—
252	C₂³	—	202	4	no	—
253	C₂³	—	202	2	no	—
254	C₂³	—	202	2	no	—
255	C₂³	—	202	4	no	—
256	C₂³	—	202	2	no	—
257	C₂³	—	202	4	no	—
258	C₂³	C₂³×D₁₀₁	1	2	yes	—
259	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
260	Q₈	—	404	8	no	—
261	D₄	—	404	8	no	—
262	D₄	—	404	8	no	—
263	D₄	—	202	8	no	—
264	D₄	—	202	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	101	4	no	—
270	C₂×C₄	—	101	2	no	—
271	C₂×C₄	—	101	2	no	—
272	C₂×C₄	—	101	2	no	—
273	C₂×C₄	—	101	4	no	—
274	C₂×C₄	—	202	16	no	—
275	C₂×C₄	—	202	2	no	—
276	C₂×C₄	—	202	4	no	—
277	C₂×C₄	—	202	2	no	—
278	C₂×C₄	—	202	8	no	—
279	C₂×C₄	#13108074	1	4	yes	—
280	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
281	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
282	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
283	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	404	8	no	—
289	C₂²	—	404	4	no	—
290	C₂²	—	202	4	no	—
291	C₂²	—	202	2	no	—
292	C₂²	—	202	2	no	—
293	C₂²	—	202	4	no	—
294	C₂²	—	202	2	no	—
295	C₂²	—	202	4	no	—
296	C₂²	#41×D₁₀₁	1	2	yes	—
297	C₂²	C₂×#13108074	1	2	yes	—
298	C₂²	#15692301	1	2	yes	—
299	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	101	4	no	—
304	C₄	—	202	2	no	—
305	C₄	—	202	4	no	—
306	C₄	—	202	2	no	—
307	C₄	C₂×#13108074	1	4	yes	—
308	C₂	—	4	2	no	—
309	C₂	—	404	4	no	—
310	C₂	—	404	2	no	—
311	C₂	#20265890	1	4	yes	direct product factor
312	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
313	C₂	C₂²×#13108074	1	1	yes	characteristic
314	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#117×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	C₂×#555
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #20265890	C₂	direct (N × H)
2. C₂×#15692254	C₂	semidirect
3. C₂×#15692243	C₂	semidirect
4. C₂×#15692241	C₂	semidirect
5. C₂×#15692230	C₂	semidirect
6. C₂×#15692229	C₂	semidirect
7. C₂×#15692228	C₂	semidirect
8. C₂×#15692217	C₂	semidirect
9. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
10. C₂×#15692213	C₂	semidirect
11. C₂×#15692212	C₂	semidirect
12. C₂×#117×C₁₀₁	C₂	semidirect
20. #15692254	C₂²	semidirect
21. #15692243	C₂²	semidirect
22. #15692241	C₂²	semidirect
23. #15692230	C₂²	semidirect
24. #15692229	C₂²	semidirect
25. #15692228	C₂²	semidirect
26. #15692217	C₂²	semidirect
27. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
28. #15692213	C₂²	semidirect
29. #15692212	C₂²	semidirect
42. #117×C₁₀₁	C₂²	semidirect
151. C₂×C₁₀₁	#555	semidirect
158. C₁₀₁	C₂×#555	semidirect
170. C₂×#117	D₁₀₁	semidirect
189. #117	C₂×D₁₀₁	semidirect
311. C₂	#20265890	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20265890	2
C₂×#15692254	2
C₂×#15692243	2
C₂×#15692241	2
C₂×#15692230	2
C₂×#15692229	2
C₂×#15692228	2
C₂×#15692217	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692213	2
C₂×#15692212	2
C₂×#117×C₁₀₁	2
C₂×#555	101

Subgroups — C2×#20265890 #27184670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265890 #27184670