Subgroups — #28374775

Order 13952.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812721	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295685	C₂	1	2	yes	maximal, complemented
4	#20295410	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295267	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295262	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295249	C₂	1	2	yes	maximal, complemented
8	#20295190	C₂	1	2	yes	maximal, complemented
9	#20295167	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	2	yes	—
13	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
14	#16812721	—	2	4	no	—
15	C₂×Dic₄₃₆	—	2	4	no	—
16	C₂×Dic₄₃₆	C₂²	1	2	yes	—
17	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
19	C₂×#13182018	C₂²	1	2	yes	—
20	C₂×#13182018	C₂²	1	2	yes	—
21	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
22	#16812687	C₂²	1	2	yes	—
23	#16812685	—	2	2	no	—
24	#16812678	C₂²	1	2	yes	—
25	#16812677	—	2	2	no	—
26	#16812673	C₂²	1	2	yes	—
27	#16812671	—	2	2	no	—
28	C₄×Dic₂₁₈	C₂²	1	2	yes	—
29	#16812659	—	2	2	no	—
30	#16812633	C₂²	1	1	yes	characteristic
31	#16812624	C₂²	1	2	yes	—
32	#16812624	C₂²	1	2	yes	—
33	#16812621	C₂²	1	2	yes	—
34	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	#13182029	—	2	4	no	—
42	#13182029	—	4	2	no	—
43	C₂×Dic₂₁₈	—	2	4	no	—
44	C₂×Dic₂₁₈	—	2	2	no	—
45	C₂×Dic₂₁₈	—	4	2	no	—
46	C₂×Dic₂₁₈	C₂³	1	2	yes	—
47	C₂×Dic₂₁₈	C₂³	1	2	yes	—
48	C₂×D₄₃₆	C₂³	1	2	yes	—
49	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
50	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
51	C₂²×Dic₁₀₉	—	2	2	no	—
52	C₂×C₄×D₁₀₉	—	2	2	no	—
53	Dic₄₃₆	—	2	4	no	—
54	Dic₄₃₆	—	2	2	no	—
55	Dic₄₃₆	D₄	1	4	yes	complemented
56	#13182018	—	2	4	no	—
57	#13182018	—	4	2	no	—
58	D₈₇₂	—	2	2	no	—
59	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
60	#13182008	—	2	2	no	—
61	#13182008	—	2	2	no	—
62	C₂²⋊Dic₁₀₉	—	2	2	no	—
63	#13182006	—	2	2	no	—
64	#13182006	—	2	2	no	—
65	#13182006	—	2	2	no	—
66	C₄×Dic₁₀₉	—	2	2	no	—
67	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₀₉	—	2	2	no	—
71	C₂×C₈×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₉	—	2	2	no	—
83	D₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₁₀₉	—	4	4	no	—
94	C₂²⋊D₁₀₉	—	4	4	no	—
95	C₂×Dic₁₀₉	—	2	2	no	—
96	C₂×Dic₁₀₉	—	2	2	no	—
97	C₂×Dic₁₀₉	—	2	2	no	—
98	C₂×Dic₁₀₉	—	4	4	no	—
99	D₄₃₆	—	2	2	no	—
100	D₄₃₆	—	2	2	no	—
101	C₄×D₁₀₉	—	4	4	no	—
102	Dic₂₁₈	—	2	2	no	—
103	Dic₂₁₈	—	2	2	no	—
104	Dic₂₁₈	—	2	2	no	—
105	Dic₂₁₈	—	4	4	no	—
106	Dic₂₁₈	C₂×D₄	1	4	yes	—
107	C₈×C₁₀₉	—	2	1	no	—
108	C₈×C₁₀₉	—	2	1	no	—
109	C₈×C₁₀₉	—	2	1	no	—
110	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
111	C₂²×C₁₀₉	—	2	1	no	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	2	1	no	—
114	C₂²×C₁₀₉	—	2	1	no	—
115	C₂²×C₁₀₉	—	2	1	no	—
116	C₂²×C₁₀₉	#135	1	2	yes	complemented
117	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₉	—	4	2	no	—
119	C₂×D₁₀₉	—	4	2	no	—
120	C₂×D₁₀₉	—	4	2	no	—
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	—	2	1	no	—
123	C₄×C₁₀₉	—	2	1	no	—
124	C₄×C₁₀₉	—	2	1	no	—
125	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₀₉	—	2	4	no	—
128	Dic₁₀₉	—	4	2	no	—
129	Dic₁₀₉	—	4	2	no	—
130	C₂×C₁₀₉	—	2	2	no	—
131	C₂×C₁₀₉	—	4	1	no	—
132	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
133	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
134	C₂×C₁₀₉	#578	1	1	yes	characteristic
135	D₁₀₉	—	8	2	no	—
136	#3285	—	109	1	no	maximal, Sylow
137	C₁₀₉	#3285	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	109	1	no	—
139	C₂²×Q₁₆	—	109	1	no	—
140	#548	—	109	2	no	—
141	#495	—	109	1	no	—
142	#467	—	109	1	no	—
143	#466	—	109	1	no	—
144	#464	—	109	2	no	—
145	#452	—	109	2	no	—
146	#448	—	109	2	no	—
147	C₄×Q₁₆	—	109	1	no	—
148	C₈×D₄	D₁₀₉	1	1	yes	characteristic, complemented
149	C₂×#41	—	109	2	no	—
150	C₂²×Q₈	—	109	2	no	—
151	#135	—	218	4	no	—
152	C₂×Q₁₆	—	109	1	no	—
153	C₂×Q₁₆	—	109	2	no	—
154	C₂×Q₁₆	—	109	1	no	—
155	C₂×Q₁₆	—	218	4	no	—
156	C₂×SD₁₆	—	109	2	no	—
157	C₂×SD₁₆	—	109	2	no	—
158	C₂×D₈	—	109	1	no	—
159	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
161	#124	—	109	2	no	—
162	#124	—	218	2	no	—
163	C₂²⋊Q₈	—	109	2	no	—
164	C₂²⋊Q₈	—	218	2	no	—
165	#121	—	109	2	no	—
166	#121	—	218	2	no	—
167	C₄×Q₈	—	109	2	no	—
168	C₄×D₄	—	218	2	no	—
169	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
170	#107	—	109	1	no	—
171	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
172	#103	—	109	2	no	—
173	#103	—	109	2	no	—
174	#102	—	109	2	no	—
175	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
177	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
178	#41	—	436	2	no	—
179	#41	—	218	4	no	—
180	C₂×Q₈	—	109	2	no	—
181	C₂×Q₈	—	109	2	no	—
182	C₂×Q₈	—	436	2	no	—
183	C₂×Q₈	—	218	4	no	—
184	C₂×Q₈	—	218	2	no	—
185	C₂×D₄	—	109	2	no	—
186	C₂×D₄	—	218	2	no	—
187	C₂×D₄	—	218	2	no	—
188	C₂×D₄	D₄₃₆	1	1	yes	characteristic
189	C₂²×C₄	—	218	2	no	—
190	C₂²×C₄	—	218	2	no	—
191	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
193	Q₁₆	—	109	4	no	—
194	Q₁₆	—	218	4	no	—
195	Q₁₆	—	218	2	no	—
196	SD₁₆	—	436	2	no	—
197	SD₁₆	—	218	4	no	—
198	D₈	—	218	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
205	C₄⋊C₄	—	109	2	no	—
206	C₄⋊C₄	—	218	2	no	—
207	C₄⋊C₄	—	218	2	no	—
208	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
209	C₂²⋊C₄	—	218	2	no	—
210	C₂²⋊C₄	—	218	2	no	—
211	C₂²⋊C₄	—	218	2	no	—
212	C₂²⋊C₄	—	218	2	no	—
213	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
214	C₂²⋊C₄	D₄₃₆	1	1	yes	characteristic
215	C₄²	—	218	2	no	—
216	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
217	C₂³	—	218	2	no	—
218	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
219	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
220	Q₈	—	109	4	no	—
221	Q₈	—	436	4	no	—
222	Q₈	—	218	2	no	—
223	Q₈	—	218	2	no	—
224	Q₈	—	218	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	436	4	no	—
227	D₄	—	436	4	no	—
228	D₄	—	218	2	no	—
229	D₄	—	218	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	436	4	no	—
235	C₂×C₄	—	436	4	no	—
236	C₂×C₄	—	218	2	no	—
237	C₂×C₄	—	218	2	no	—
238	C₂×C₄	—	218	2	no	—
239	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₁₀₉	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	436	2	no	—
254	C₂²	—	436	2	no	—
255	C₂²	—	436	2	no	—
256	C₂²	#16812721	1	2	yes	complemented
257	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	436	2	no	—
263	C₄	—	436	2	no	—
264	C₄	—	218	4	no	—
265	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
266	C₄	#16812745	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	872	2	no	—
270	C₂	D₄×D₄₃₆	1	1	yes	characteristic
271	C₂	C₂×#16812721	1	1	yes	characteristic
272	C₂	#20295843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₈×D₄×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	C₈×D₄	D₁₀₉
hypocenter	C₁₀₉	#3285
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16812721	C₂	semidirect
2. C₂²×Dic₄₃₆	C₂	semidirect
3. #20295685	C₂	semidirect
4. #20295410	C₂	semidirect
5. #20295267	C₂	semidirect
6. #20295262	C₂	semidirect
7. #20295249	C₂	semidirect
8. #20295190	C₂	semidirect
9. #20295167	C₂	semidirect
10. C₄×Dic₄₃₆	C₂	semidirect
11. C₈×D₄×C₁₀₉	C₂	semidirect
55. Dic₄₃₆	D₄	semidirect
116. C₂²×C₁₀₉	#135	semidirect
137. C₁₀₉	#3285	semidirect
148. C₈×D₄	D₁₀₉	semidirect
256. C₂²	#16812721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812721	2
C₂²×Dic₄₃₆	2
#20295685	2
#20295410	2
#20295267	2
#20295262	2
#20295249	2
#20295190	2
#20295167	2
C₄×Dic₄₃₆	2
C₈×D₄×C₁₀₉	2
#3285	109