Subgroups — #27203783

Order 13184.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13110105	C₂	1	1	yes	maximal, characteristic
2	#20267851	C₂	1	2	yes	maximal, complemented
3	#20267849	C₂	1	2	yes	maximal, complemented
4	#20267844	C₂	1	1	yes	maximal, characteristic, complemented
5	#20267842	C₂	1	1	yes	maximal, characteristic, complemented
6	#20267842	C₂	1	1	yes	maximal, characteristic, complemented
7	#20267816	C₂	1	2	yes	maximal, complemented
8	#20267756	C₂	1	2	yes	maximal
9	C₄×#13110088	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13110105	C₂²	1	1	yes	characteristic
13	C₂×#13110088	C₂²	1	2	yes	—
14	C₂×C₈×D₁₀₃	C₂²	1	2	yes	—
15	C₄×Dic₂₀₆	C₄	1	1	yes	characteristic
16	C₄×Dic₂₀₆	C₄	1	1	yes	characteristic
17	C₄×D₄₁₂	C₄	1	1	yes	characteristic
18	C₄×D₄₁₂	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₀₃	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₀₃	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₀₃	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₀₃	C₄	1	1	yes	characteristic
23	#15692980	C₂²	1	1	yes	characteristic
24	#15692978	C₂²	1	1	yes	characteristic
25	#15692978	C₂²	1	1	yes	characteristic
26	C₄²×D₁₀₃	C₂²	1	1	yes	characteristic
27	C₄²×D₁₀₃	C₂²	1	1	yes	characteristic
28	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
29	Dic₁₀₃⋊C₈	C₂²	1	2	yes	—
30	Dic₁₀₃⋊C₈	C₂²	1	2	yes	—
31	C₂²⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
32	(C₂×D₁₀₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₀₃)⋊C₈	C₂²	1	2	yes	—
34	#15692931	C₂²	1	2	yes	—
35	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×Dic₁₀₃	C₂²	1	2	yes	—
38	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₃	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₀₃	C₂²	1	2	yes	—
41	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
43	#13110105	—	2	4	no	—
44	C₂×Dic₂₀₆	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄₁₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₀₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₀₃	C₂³	1	1	yes	characteristic
54	#13110088	—	2	4	no	—
55	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₁₀₃⋊C₈	C₂³	1	2	yes	—
57	C₈×D₁₀₃	—	2	4	no	—
58	C₄⋊Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
60	Dic₁₀₃⋊C₄	C₂×C₄	1	1	yes	characteristic
61	Dic₁₀₃⋊C₄	C₂×C₄	1	1	yes	characteristic
62	Dic₁₀₃⋊C₄	C₂×C₄	1	1	yes	characteristic
63	Dic₁₀₃⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₁₀₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₁₀₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₁₀₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₁₀₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₀₃	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₀₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
80	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
81	C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
84	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
85	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₀₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₀₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₃	—	2	2	no	—
89	C₂×C₄×C₁₀₃	—	2	2	no	—
90	C₂×C₄×C₁₀₃	—	2	2	no	—
91	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₃	—	2	2	no	—
98	C₂²⋊D₁₀₃	—	2	2	no	—
99	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₃	C₂²×C₄	1	1	yes	characteristic
105	D₄₁₂	—	2	2	no	—
106	C₄×D₁₀₃	—	2	2	no	—
107	C₄×D₁₀₃	—	2	2	no	—
108	C₄×D₁₀₃	—	2	2	no	—
109	C₄×D₁₀₃	—	2	2	no	—
110	C₄×D₁₀₃	—	2	2	no	—
111	C₄×D₁₀₃	#41	1	4	yes	—
112	Dic₂₀₆	—	2	2	no	—
113	C₈×C₁₀₃	—	2	2	no	—
114	C₈×C₁₀₃	—	2	2	no	—
115	C₁₀₃⋊C₈	—	2	2	no	—
116	C₁₀₃⋊C₈	—	2	2	no	—
117	C₂²×C₁₀₃	—	2	1	no	—
118	C₂²×C₁₀₃	—	2	1	no	—
119	C₂²×C₁₀₃	—	2	1	no	—
120	C₂²×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₀₃	—	2	1	no	—
122	C₂×D₁₀₃	—	2	1	no	—
123	C₂×D₁₀₃	—	2	1	no	—
124	C₂×D₁₀₃	—	2	1	no	—
125	C₂×D₁₀₃	—	2	1	no	—
126	C₂×D₁₀₃	#117	1	2	yes	—
127	C₄×C₁₀₃	—	2	1	no	—
128	C₄×C₁₀₃	—	2	1	no	—
129	C₄×C₁₀₃	—	2	1	no	—
130	C₄×C₁₀₃	#131	1	2	yes	—
131	C₄×C₁₀₃	#131	1	2	yes	—
132	C₄×C₁₀₃	C₂×#41	1	2	yes	—
133	Dic₁₀₃	—	2	1	no	—
134	Dic₁₀₃	—	2	1	no	—
135	Dic₁₀₃	—	2	1	no	—
136	Dic₁₀₃	—	2	1	no	—
137	Dic₁₀₃	—	2	1	no	—
138	Dic₁₀₃	#117	1	2	yes	—
139	C₂×C₁₀₃	—	4	1	no	—
140	C₂×C₁₀₃	C₂×#117	1	1	yes	characteristic
141	C₂×C₁₀₃	C₂×#131	1	1	yes	characteristic
142	C₂×C₁₀₃	C₂×#131	1	1	yes	characteristic
143	D₁₀₃	—	2	2	no	—
144	D₁₀₃	—	4	1	no	—
145	#2922	—	103	1	no	maximal, Sylow
146	C₁₀₃	#2922	1	1	yes	characteristic, complemented, Sylow
147	C₄×#41	—	103	1	no	—
148	#433	—	103	2	no	—
149	#433	—	103	2	no	—
150	#431	—	103	1	no	—
151	#431	—	103	1	no	—
152	#431	—	103	1	no	—
153	#431	D₁₀₃	1	1	yes	characteristic, complemented
154	#424	—	103	2	no	—
155	#408	—	103	2	no	—
156	C₄×M₁₆	—	103	1	no	—
157	C₂×C₄×C₈	—	103	1	no	—
158	C₂×#41	—	103	1	no	—
159	C₂×M₁₆	—	103	2	no	—
160	C₂²×C₈	—	103	2	no	—
161	C₄×Q₈	—	103	1	no	—
162	C₄×Q₈	—	103	1	no	—
163	C₄×D₄	—	103	1	no	—
164	C₄×D₄	—	103	1	no	—
165	C₄×D₄	—	103	1	no	—
166	C₄×D₄	—	103	1	no	—
167	C₄×D₄	—	103	1	no	—
168	C₄×D₄	—	103	1	no	—
169	#117	—	103	1	no	—
170	#117	—	103	1	no	—
171	#117	—	103	1	no	—
172	C₂×C₄²	—	103	1	no	—
173	C₂×C₄²	—	103	1	no	—
174	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
175	C₄⋊C₈	—	103	2	no	—
176	C₄⋊C₈	—	103	2	no	—
177	C₄⋊C₈	—	103	2	no	—
178	C₄⋊C₈	C₂×D₁₀₃	1	2	yes	—
179	C₂²⋊C₈	—	103	2	no	—
180	C₂²⋊C₈	—	103	2	no	—
181	C₂²⋊C₈	—	103	2	no	—
182	C₂²⋊C₈	C₂×D₁₀₃	1	2	yes	—
183	#97	—	103	2	no	—
184	C₄×C₈	—	103	2	no	—
185	C₄×C₈	—	103	1	no	—
186	C₄×C₈	—	103	1	no	—
187	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
188	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
189	#41	—	206	4	no	—
190	C₂×Q₈	—	103	1	no	—
191	C₂×D₄	—	103	1	no	—
192	C₂×D₄	—	103	1	no	—
193	C₂×D₄	—	103	1	no	—
194	C₂²×C₄	—	103	1	no	—
195	C₂²×C₄	—	103	1	no	—
196	C₂²×C₄	—	103	1	no	—
197	C₂²×C₄	—	103	1	no	—
198	C₂²×C₄	—	103	1	no	—
199	C₂²×C₄	—	103	1	no	—
200	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
203	M₁₆	—	206	4	no	—
204	C₂×C₈	—	103	2	no	—
205	C₂×C₈	—	103	2	no	—
206	C₂×C₈	—	206	4	no	—
207	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
208	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
209	C₄⋊C₄	—	103	1	no	—
210	C₄⋊C₄	—	103	1	no	—
211	C₄⋊C₄	—	103	1	no	—
212	C₄⋊C₄	—	103	1	no	—
213	C₄⋊C₄	—	103	1	no	—
214	C₄⋊C₄	—	103	1	no	—
215	C₂²⋊C₄	—	103	1	no	—
216	C₂²⋊C₄	—	103	1	no	—
217	C₂²⋊C₄	—	103	1	no	—
218	C₂²⋊C₄	—	103	1	no	—
219	C₂²⋊C₄	—	103	1	no	—
220	C₂²⋊C₄	—	103	1	no	—
221	C₄²	—	103	1	no	—
222	C₄²	—	103	1	no	—
223	C₄²	—	103	1	no	—
224	C₄²	—	103	1	no	—
225	C₄²	—	103	1	no	—
226	C₄²	—	103	1	no	—
227	C₄²	C₄×D₁₀₃	1	1	yes	characteristic
228	C₄²	C₄×D₁₀₃	1	1	yes	characteristic
229	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
230	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
231	C₂³	—	103	1	no	—
232	C₂³	—	103	1	no	—
233	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
234	Q₈	—	206	2	no	—
235	D₄	—	206	2	no	—
236	D₄	—	206	2	no	—
237	D₄	—	206	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	103	1	no	—
242	C₂×C₄	—	103	4	no	—
243	C₂×C₄	—	103	1	no	—
244	C₂×C₄	—	103	1	no	—
245	C₂×C₄	—	103	1	no	—
246	C₂×C₄	—	103	1	no	—
247	C₂×C₄	—	103	1	no	—
248	C₂×C₄	—	206	2	no	—
249	C₂×C₄	—	206	2	no	—
250	C₂×C₄	—	206	2	no	—
251	C₂×C₄	—	206	2	no	—
252	C₂×C₄	—	206	2	no	—
253	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
258	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	206	2	no	—
262	C₈	—	206	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	103	2	no	—
267	C₂²	—	206	1	no	—
268	C₂²	—	206	1	no	—
269	C₂²	—	206	1	no	—
270	C₂²	—	206	1	no	—
271	C₂²	—	206	1	no	—
272	C₂²	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	103	2	no	—
277	C₄	—	206	1	no	—
278	C₄	—	206	1	no	—
279	C₄	—	206	1	no	—
280	C₄	—	206	1	no	—
281	C₄	—	206	1	no	—
282	C₄	#15693027	1	2	yes	—
283	C₄	#15693029	1	2	yes	—
284	C₄	#41×D₁₀₃	1	2	yes	—
285	C₂	—	4	1	no	—
286	C₂	—	412	1	no	—
287	C₂	—	206	2	no	—
288	C₂	#117×D₁₀₃	1	1	yes	characteristic
289	C₂	C₂×#15693027	1	1	yes	characteristic
290	C₂	C₂×#15693029	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#431×C₁₀₃	C₂
prime radical	C₂×#13110105	C₂²
cosocle	C₂×C₄×C₁₀₃	C₂⁴
hypercenter	#431	D₁₀₃
hypocenter	C₁₀₃	#2922
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #20267851	C₂	semidirect
3. #20267849	C₂	semidirect
4. #20267844	C₂	semidirect
5. #20267842	C₂	semidirect
6. #20267842	C₂	semidirect
7. #20267816	C₂	semidirect
9. C₄×#13110088	C₂	semidirect
10. C₄×C₈×D₁₀₃	C₂	semidirect
11. #431×C₁₀₃	C₂	semidirect
146. C₁₀₃	#2922	semidirect
153. #431	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13110105	2
#20267851	2
#20267849	2
#20267844	2
#20267842	2
#20267816	2
#20267756	2
C₄×#13110088	2
C₄×C₈×D₁₀₃	2
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