Subgroups — #27174400

Order 12928.

528 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265116	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265113	C₂	1	1	yes	maximal, characteristic
4	#20265110	C₂	1	2	yes	maximal, complemented
5	#20265100	C₂	1	2	yes	maximal
6	#20265097	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265089	C₂	1	1	yes	maximal, characteristic
8	C₄×#13108053	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₁₀₁	C₂	1	2	yes	maximal
11	C₂×#117×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₀₂	—	2	4	no	—
14	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×#13108053	C₂²	1	2	yes	—
17	C₂×#13108053	C₂²	1	2	yes	—
18	C₂×#13108053	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	#15692162	C₂²	1	2	yes	—
24	#15692162	C₂²	1	2	yes	—
25	#15692162	C₂²	1	1	yes	characteristic
26	#15692162	C₂²	1	1	yes	characteristic
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	#117×C₁₀₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×Dic₂₀₂	—	2	2	no	—
38	C₂×Dic₂₀₂	—	2	4	no	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
43	C₄⋊Dic₁₀₁	—	2	2	no	—
44	C₄⋊Dic₁₀₁	—	2	4	no	—
45	#13108053	—	2	4	no	—
46	#13108053	—	2	4	no	—
47	#13108053	—	2	4	no	—
48	C₂²⋊Dic₁₀₁	—	2	4	no	—
49	C₂²⋊Dic₁₀₁	—	2	4	no	—
50	C₄×Dic₁₀₁	—	2	4	no	—
51	C₄×Dic₁₀₁	—	2	4	no	—
52	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₁	—	2	2	no	—
54	C₂²×C₄×C₁₀₁	—	2	2	no	—
55	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	complemented
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₀₁	—	2	2	no	—
63	C₄⋊C₄×C₁₀₁	—	2	2	no	—
64	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
65	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
66	C₄²×C₁₀₁	—	2	2	no	—
67	C₄²×C₁₀₁	D₄	1	4	yes	—
68	C₂³×C₁₀₁	—	2	1	no	—
69	C₂³×C₁₀₁	—	2	1	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	2	no	—
72	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
73	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₄×C₁₀₁	—	2	1	no	—
75	C₂×C₄×C₁₀₁	—	2	1	no	—
76	C₂×C₄×C₁₀₁	—	2	2	no	—
77	C₂×C₄×C₁₀₁	—	2	2	no	—
78	C₂×C₄×C₁₀₁	—	2	8	no	—
79	C₂×C₄×C₁₀₁	—	2	1	no	—
80	C₂×C₄×C₁₀₁	—	2	1	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₁₀₁	#41	1	2	yes	—
89	C₂×Dic₁₀₁	—	2	2	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	2	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	#41	1	4	yes	—
100	C₂×Dic₁₀₁	#41	1	4	yes	—
101	Dic₂₀₂	—	2	8	no	—
102	C₂²×C₁₀₁	—	2	2	no	—
103	C₂²×C₁₀₁	—	2	4	no	—
104	C₂²×C₁₀₁	—	4	1	no	—
105	C₂²×C₁₀₁	—	4	2	no	—
106	C₂²×C₁₀₁	—	4	1	no	—
107	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
110	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
111	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	—	2	2	no	—
114	C₄×C₁₀₁	—	2	2	no	—
115	C₄×C₁₀₁	—	4	1	no	—
116	C₄×C₁₀₁	—	4	1	no	—
117	C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
118	Dic₁₀₁	—	2	4	no	—
119	Dic₁₀₁	—	2	4	no	—
120	Dic₁₀₁	—	4	2	no	—
121	Dic₁₀₁	—	4	2	no	—
122	C₂×C₁₀₁	—	4	2	no	—
123	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₁₀₁	#525	1	1	yes	characteristic
125	C₂×C₁₀₁	#529	1	2	yes	—
126	C₂×C₁₀₁	#532	1	1	yes	characteristic
127	C₂×C₁₀₁	#533	1	2	yes	—
128	#2396	—	101	1	no	maximal, Sylow
129	C₁₀₁	#2396	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₄×Q₈	—	101	1	no	—
131	C₂×#117	D₁₀₁	1	1	yes	characteristic
132	#400	—	101	1	no	—
133	#399	—	101	1	no	—
134	#398	—	101	2	no	—
135	#396	—	101	2	no	—
136	#395	—	101	1	no	—
137	#393	—	101	1	no	—
138	C₄×C₄⋊C₄	—	101	1	no	—
139	C₄×C₄⋊C₄	—	101	2	no	—
140	C₄×C₂²⋊C₄	—	101	2	no	—
141	C₂²×Q₈	—	101	1	no	—
142	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
143	C₄×Q₈	—	202	4	no	—
144	#117	—	2	4	no	—
145	C₂×C₄⋊C₄	—	101	2	no	—
146	C₂×C₄⋊C₄	—	101	1	no	—
147	C₂×C₄⋊C₄	—	101	2	no	—
148	C₂×C₄⋊C₄	—	101	2	no	—
149	C₂×C₄⋊C₄	—	101	2	no	—
150	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	—	101	2	no	—
153	C₂×C₂²⋊C₄	—	101	2	no	—
154	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
156	C₂×C₄²	—	101	2	no	—
157	C₂×C₄²	—	101	2	no	—
158	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
159	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
160	#95	—	101	2	no	—
161	#95	—	101	2	no	—
162	#95	—	101	2	no	—
163	#95	—	101	1	no	—
164	#95	—	101	1	no	—
165	#95	—	101	2	no	—
166	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
167	C₂×Q₈	—	202	4	no	—
168	C₂×Q₈	—	202	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	101	2	no	—
172	C₂²×C₄	—	101	2	no	—
173	C₂²×C₄	—	101	2	no	—
174	C₂²×C₄	—	101	2	no	—
175	C₂²×C₄	Dic₂₀₂	1	4	yes	complemented
176	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	202	4	no	—
185	C₄⋊C₄	—	202	4	no	—
186	C₄⋊C₄	—	202	4	no	—
187	C₄⋊C₄	—	202	4	no	—
188	C₄⋊C₄	—	202	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	202	4	no	—
192	C₂²⋊C₄	—	202	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	202	4	no	—
195	C₄²	—	202	4	no	—
196	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₂×Dic₂₀₂	1	2	yes	—
202	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
203	Q₈	—	202	8	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	101	4	no	—
214	C₂×C₄	—	101	4	no	—
215	C₂×C₄	—	202	2	no	—
216	C₂×C₄	—	202	2	no	—
217	C₂×C₄	—	202	2	no	—
218	C₂×C₄	—	202	2	no	—
219	C₂×C₄	—	202	2	no	—
220	C₂×C₄	—	202	2	no	—
221	C₂×C₄	—	202	2	no	—
222	C₂×C₄	—	202	2	no	—
223	C₂×C₄	—	202	2	no	—
224	C₂×C₄	—	202	2	no	—
225	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
227	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
228	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
229	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
230	C₂×C₄	#13108074	1	2	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
237	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
238	C₂²	#41×D₁₀₁	1	2	yes	—
239	C₂²	#41×D₁₀₁	1	2	yes	—
240	C₂²	C₂×#13108074	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	404	2	no	—
247	C₄	—	404	2	no	—
248	C₄	—	202	4	no	—
249	C₄	—	202	4	no	—
250	C₄	#15692234	1	4	yes	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#15692234	1	1	yes	characteristic
253	C₂	#20265684	1	1	yes	characteristic
254	C₂	#20265705	1	2	yes	—
255	C₂	#20265723	1	1	yes	characteristic
256	C₂	#20265730	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×#117×C₁₀₁	C₂
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂×#117	D₁₀₁
hypocenter	C₁₀₁	#2396
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Dic₂₀₂	C₂	semidirect
2. #20265116	C₂	semidirect
4. #20265110	C₂	semidirect
6. #20265097	C₂	semidirect
8. C₄×#13108053	C₂	semidirect
9. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
55. C₂²×C₄×C₁₀₁	Q₈	semidirect
129. C₁₀₁	#2396	semidirect
175. C₂²×C₄	Dic₂₀₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₀₂	2
#20265116	2
#20265113	2
#20265110	2
#20265100	2
#20265097	2
#20265089	2
C₄×#13108053	2
C₄×C₄⋊Dic₁₀₁	2
C₄×C₂²⋊Dic₁₀₁	2
C₂×#117×C₁₀₁	2
#2396	101