Subgroups — C₄×#15692215 #27173476

Order 12928.

924 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692215	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265028	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₄×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
8	C₄×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265024	C₂	1	2	yes	maximal, complemented
10	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₀₁	C₄	1	8	yes	complemented
17	#15692215	C₄	1	16	yes	direct product factor
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
21	C₂×#13108051	C₂²	1	1	yes	characteristic
22	C₂×#13108051	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	—	2	4	no	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₄²×D₁₀₁	—	2	4	no	—
32	#15692162	C₂²	1	2	yes	—
33	#15692162	C₂²	1	2	yes	—
34	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	2	yes	—
39	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	—
41	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	4	yes	—
42	C₂²×Dic₁₀₁	—	2	4	no	—
43	C₂²×Dic₁₀₁	—	2	4	no	—
44	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂×C₄×D₁₀₁	—	2	2	no	—
51	C₂×C₄×D₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	—	2	4	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
55	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
56	#13108053	C₂×C₄	1	8	yes	complemented
57	#13108053	C₂×C₄	1	4	yes	complemented
58	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
59	#13108051	C₂×C₄	1	8	yes	complemented
60	#13108051	C₂×C₄	1	4	yes	complemented
61	C₄×Dic₁₀₁	—	2	2	no	—
62	C₄×Dic₁₀₁	—	2	4	no	—
63	C₄×Dic₁₀₁	—	2	4	no	—
64	C₄×Dic₁₀₁	—	2	4	no	—
65	C₄×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
66	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
67	C₄×Dic₁₀₁	D₄	1	4	yes	complemented
68	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	—	2	2	no	—
70	C₂²×C₄×C₁₀₁	—	2	2	no	—
71	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
72	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
77	C₄²×C₁₀₁	—	2	2	no	—
78	C₄²×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	—	2	1	no	—
80	C₂³×C₁₀₁	—	2	1	no	—
81	C₂³×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
83	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
85	C₂²×D₁₀₁	—	2	1	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	1	no	—
88	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
89	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	1	no	—
92	C₂×C₄×C₁₀₁	—	2	2	no	—
93	C₂×C₄×C₁₀₁	—	2	1	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	8	no	—
97	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
98	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
99	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₁	#41	1	2	yes	—
103	C₂²⋊D₁₀₁	C₄²	1	16	yes	complemented
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	2	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	1	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	16	no	—
110	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
111	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
112	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
113	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
114	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
115	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
116	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
117	C₂×Dic₁₀₁	#41	1	4	yes	—
118	C₄×D₁₀₁	—	2	8	no	—
119	C₄×D₁₀₁	—	2	16	no	—
120	C₂²×C₁₀₁	—	2	4	no	—
121	C₂²×C₁₀₁	—	2	2	no	—
122	C₂²×C₁₀₁	—	2	4	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	complemented
125	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
126	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
129	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
130	C₂×D₁₀₁	—	2	4	no	—
131	C₂×D₁₀₁	—	2	2	no	—
132	C₂×D₁₀₁	—	2	4	no	—
133	C₂×D₁₀₁	—	2	2	no	—
134	C₂×D₁₀₁	C₂×C₄²	1	4	yes	complemented
135	C₄×C₁₀₁	—	2	4	no	—
136	C₄×C₁₀₁	—	2	2	no	—
137	C₄×C₁₀₁	—	2	4	no	—
138	C₄×C₁₀₁	C₄×D₄	1	4	yes	complemented
139	Dic₁₀₁	—	2	2	no	—
140	Dic₁₀₁	—	2	4	no	—
141	Dic₁₀₁	—	2	2	no	—
142	Dic₁₀₁	C₂×C₄²	1	4	yes	complemented
143	Dic₁₀₁	C₄×D₄	1	8	yes	complemented
144	C₂×C₁₀₁	—	2	4	no	—
145	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
147	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₀₁	C₄×#41	1	2	yes	—
149	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
150	D₁₀₁	—	2	4	no	—
151	C₄²×D₄	—	101	1	no	maximal, Sylow
152	C₁₀₁	C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄×D₄	—	101	1	no	—
154	C₂×C₄×D₄	—	101	2	no	—
155	C₂²×C₄²	—	101	1	no	—
156	C₂²×C₄²	—	101	1	no	—
157	C₄×C₄⋊C₄	—	101	1	no	—
158	C₄×C₄⋊C₄	—	101	2	no	—
159	C₄×C₂²⋊C₄	—	101	2	no	—
160	C₄×C₂²⋊C₄	—	101	1	no	—
161	C₄×C₂²⋊C₄	—	101	2	no	—
162	C₄×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
163	C₄³	—	101	1	no	—
164	C₂²×D₄	—	101	1	no	—
165	C₂³×C₄	—	101	2	no	—
166	C₂³×C₄	—	101	2	no	—
167	C₂³×C₄	—	101	1	no	—
168	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
169	C₄×D₄	—	101	8	no	—
170	C₄×D₄	—	101	16	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	1	no	—
173	C₂×C₂²⋊C₄	—	101	1	no	—
174	C₂×C₂²⋊C₄	—	101	1	no	—
175	C₂×C₂²⋊C₄	—	101	2	no	—
176	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
177	C₂×C₄²	—	101	2	no	—
178	C₂×C₄²	—	101	1	no	—
179	C₂×C₄²	—	101	1	no	—
180	C₂×C₄²	—	101	2	no	—
181	C₂×C₄²	—	101	2	no	—
182	C₂×C₄²	—	101	2	no	—
183	C₂×C₄²	—	101	1	no	—
184	C₂×C₄²	—	202	4	no	—
185	C₂×C₄²	—	202	4	no	—
186	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
187	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
188	#95	—	101	2	no	—
189	#95	—	101	2	no	—
190	#95	C₂×D₁₀₁	1	2	yes	—
191	C₂⁴	—	101	1	no	—
192	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
193	C₂×D₄	—	101	4	no	—
194	C₂×D₄	—	101	8	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	101	2	no	—
198	C₂²×C₄	—	101	1	no	—
199	C₂²×C₄	—	101	8	no	—
200	C₂²×C₄	—	101	1	no	—
201	C₂²×C₄	—	101	2	no	—
202	C₂²×C₄	—	101	1	no	—
203	C₂²×C₄	—	101	8	no	—
204	C₂²×C₄	—	101	4	no	—
205	C₂²×C₄	—	202	4	no	—
206	C₂²×C₄	—	202	4	no	—
207	C₂²×C₄	—	202	4	no	—
208	C₂²×C₄	—	202	4	no	—
209	C₂²×C₄	—	202	2	no	—
210	C₂²×C₄	—	202	2	no	—
211	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
212	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
213	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
214	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
216	C₄⋊C₄	—	101	4	no	—
217	C₄⋊C₄	—	101	8	no	—
218	C₂²⋊C₄	—	101	4	no	—
219	C₂²⋊C₄	—	101	4	no	—
220	C₂²⋊C₄	—	101	8	no	—
221	C₂²⋊C₄	C₄×D₁₀₁	1	8	yes	complemented
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	101	8	no	—
225	C₄²	—	101	4	no	—
226	C₄²	—	101	4	no	—
227	C₄²	—	202	4	no	—
228	C₄²	—	202	4	no	—
229	C₄²	—	202	4	no	—
230	C₄²	—	202	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	101	4	no	—
235	C₂³	—	101	2	no	—
236	C₂³	—	202	1	no	—
237	C₂³	—	202	2	no	—
238	C₂³	—	202	1	no	—
239	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
240	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
241	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
242	D₄	—	101	16	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	101	4	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	101	4	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	4	no	—
255	C₂×C₄	—	101	2	no	—
256	C₂×C₄	—	101	4	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	202	16	no	—
259	C₂×C₄	—	202	16	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	1	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	—	202	1	no	—
265	C₂×C₄	—	202	8	no	—
266	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
268	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
269	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
270	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
271	C₂×C₄	#13108073	1	2	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	101	4	no	—
277	C₂²	—	202	4	no	—
278	C₂²	—	202	2	no	—
279	C₂²	—	202	4	no	—
280	C₂²	—	202	2	no	—
281	C₂²	C₄²×D₁₀₁	1	4	yes	complemented
282	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
284	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
285	C₂²	#41×D₁₀₁	1	2	yes	—
286	C₂²	C₂×#13108073	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	101	4	no	—
291	C₄	—	101	8	no	—
292	C₄	—	202	4	no	—
293	C₄	—	202	2	no	—
294	C₄	—	202	2	no	—
295	C₄	#15692215	1	4	yes	direct product factor
296	C₂	—	2	4	no	—
297	C₂	—	202	4	no	—
298	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
299	C₂	C₄×D₄×D₁₀₁	1	2	yes	—
300	C₂	C₂×#15692215	1	1	yes	characteristic
301	C₂	C₄×#13108073	1	2	yes	—
302	C₂	#20265645	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₄×C₂²⋊C₄×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₁	C₂²
hypercenter	C₄×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₄²×D₄
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×Dic₁₀₁	C₂	semidirect
4. C₂×C₄²×D₁₀₁	C₂	semidirect
5. C₄×#13108053	C₂	semidirect
6. #20265028	C₂	semidirect
7. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
8. C₄×#13108051	C₂	semidirect
9. #20265024	C₂	semidirect
10. C₄²×Dic₁₀₁	C₂	semidirect
11. C₄×C₂²⋊C₄×C₁₀₁	C₂	semidirect
16. C₄×C₂²⋊D₁₀₁	C₄	semidirect
17. #15692215	C₄	direct (N × H)
44. C₂²×Dic₁₀₁	C₂×C₄	semidirect
54. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
55. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
56. #13108053	C₂×C₄	semidirect
57. #13108053	C₂×C₄	semidirect
58. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
59. #13108051	C₂×C₄	semidirect
60. #13108051	C₂×C₄	semidirect
65. C₄×Dic₁₀₁	C₂×C₄	semidirect
66. C₄×Dic₁₀₁	C₂×C₄	semidirect
67. C₄×Dic₁₀₁	D₄	semidirect
71. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
76. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
103. C₂²⋊D₁₀₁	C₄²	semidirect
124. C₂²×C₁₀₁	C₂×C₄²	semidirect
134. C₂×D₁₀₁	C₂×C₄²	semidirect
138. C₄×C₁₀₁	C₄×D₄	semidirect
142. Dic₁₀₁	C₂×C₄²	semidirect
143. Dic₁₀₁	C₄×D₄	semidirect
152. C₁₀₁	C₄²×D₄	semidirect
162. C₄×C₂²⋊C₄	D₁₀₁	semidirect
211. C₂²×C₄	C₄×D₁₀₁	semidirect
221. C₂²⋊C₄	C₄×D₁₀₁	semidirect
281. C₂²	C₄²×D₁₀₁	semidirect
295. C₄	#15692215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692215	2
C₂²×C₄×Dic₁₀₁	2
C₂×C₄²×D₁₀₁	2
C₄×#13108053	2
#20265028	2
C₂²⋊C₄×Dic₁₀₁	2
C₄×#13108051	2
#20265024	2
C₄²×Dic₁₀₁	2
C₄×C₂²⋊C₄×C₁₀₁	2
C₄²×D₄	101

Subgroups — C4×#15692215 #27173476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15692215 #27173476