Subgroups — #28376424

Order 13952.

1040 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	#20295650	C₂	1	4	yes	maximal, complemented
2	#20295639	C₂	1	2	yes	maximal, complemented
3	#20295632	C₂	1	2	yes	maximal, complemented
4	#20295625	C₂	1	4	yes	maximal, complemented
5	#20295616	C₂	1	4	yes	maximal, complemented
6	#20295576	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295563	C₂	1	8	yes	maximal, complemented
8	#20295538	C₂	1	2	yes	maximal, complemented
9	#20295494	C₂	1	2	yes	maximal, complemented
10	C₂×#16812656	C₂	1	1	yes	maximal, characteristic, complemented
11	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812699	C₂²	1	1	yes	characteristic, complemented
15	#16812696	C₂²	1	1	yes	characteristic, complemented
16	#16812694	C₂²	1	8	yes	—
17	#16812694	C₂²	1	8	yes	complemented
18	#16812690	C₂²	1	8	yes	complemented
19	#16812688	C₂²	1	8	yes	—
20	#16812687	C₂²	1	2	yes	complemented
21	#16812684	C₂²	1	8	yes	—
22	#16812684	C₂²	1	8	yes	complemented
23	#16812683	C₂²	1	4	yes	—
24	#16812679	C₂²	1	4	yes	—
25	#16812675	C₂²	1	8	yes	complemented
26	#16812674	C₂²	1	4	yes	—
27	#16812674	C₂²	1	8	yes	—
28	#16812673	C₂²	1	4	yes	—
29	#16812669	C₂²	1	4	yes	—
30	#16812664	C₂²	1	4	yes	complemented
31	C₄×D₄₃₆	C₂²	1	8	yes	—
32	C₄×D₄₃₆	C₂²	1	4	yes	complemented
33	#16812661	C₂²	1	4	yes	complemented
34	#16812659	C₂²	1	8	yes	—
35	#16812656	C₂²	1	4	yes	—
36	#16812656	C₂²	1	2	yes	—
37	#16812656	C₂²	1	2	yes	—
38	#16812656	C₂²	1	4	yes	complemented
39	C₂×#13182008	C₂²	1	2	yes	—
40	C₂×#13182006	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
42	C₄²×D₁₀₉	C₂²	1	4	yes	complemented
43	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
44	#125×C₁₀₉	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
46	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	complemented
47	#117×C₁₀₉	C₂²	1	2	yes	—
48	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
49	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
50	#13182029	—	2	8	no	—
51	C₂×Dic₂₁₈	C₂³	1	2	yes	—
52	C₂×D₄₃₆	C₂³	1	2	yes	—
53	C₂×D₄₃₆	C₂³	1	8	yes	—
54	C₂×D₄₃₆	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
56	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
58	C₂×C₄×D₁₀₉	—	2	4	no	—
59	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
60	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
61	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
62	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
63	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
64	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
65	#13182008	C₂³	1	4	yes	—
66	#13182008	C₂³	1	4	yes	—
67	#13182008	C₂³	1	8	yes	—
68	#13182006	C₂³	1	4	yes	—
69	#13182006	C₂³	1	4	yes	—
70	#13182006	C₂³	1	8	yes	—
71	#13182006	C₂³	1	8	yes	—
72	#13182006	C₂³	1	8	yes	—
73	#13182006	C₂³	1	8	yes	—
74	C₄×Dic₁₀₉	C₂³	1	2	yes	—
75	C₄×Dic₁₀₉	C₂³	1	8	yes	—
76	C₄×Dic₁₀₉	C₂³	1	2	yes	—
77	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
81	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
82	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
84	C₄²×C₁₀₉	C₂³	1	2	yes	—
85	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₀₉	C₂³	1	4	yes	—
88	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
89	C₂²×D₁₀₉	—	2	2	no	—
90	C₂²×D₁₀₉	—	4	2	no	—
91	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
92	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
93	Q₈×C₁₀₉	—	4	4	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	#41	1	4	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₉	—	2	8	no	—
102	C₂×Dic₁₀₉	—	2	2	no	—
103	C₂×Dic₁₀₉	—	4	2	no	—
104	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
105	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
106	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
107	D₄₃₆	—	2	4	no	—
108	D₄₃₆	—	4	8	no	—
109	D₄₃₆	—	4	4	no	—
110	C₄×D₁₀₉	—	2	4	no	—
111	C₄×D₁₀₉	—	2	4	no	—
112	C₄×D₁₀₉	—	2	8	no	—
113	C₄×D₁₀₉	—	4	8	no	—
114	Dic₂₁₈	—	2	4	no	—
115	C₂²×C₁₀₉	—	2	2	no	—
116	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
117	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₀₉	—	2	2	no	—
119	C₂×D₁₀₉	—	2	4	no	—
120	C₂×D₁₀₉	—	2	8	no	—
121	C₂×D₁₀₉	—	2	4	no	—
122	C₂×D₁₀₉	—	4	4	no	—
123	C₄×C₁₀₉	—	2	8	no	—
124	C₄×C₁₀₉	—	2	2	no	—
125	C₄×C₁₀₉	—	2	2	no	—
126	C₄×C₁₀₉	C₂×#41	1	2	yes	—
127	C₄×C₁₀₉	C₂×#41	1	2	yes	—
128	Dic₁₀₉	—	2	4	no	—
129	Dic₁₀₉	—	2	2	no	—
130	Dic₁₀₉	—	2	4	no	—
131	C₂×C₁₀₉	—	2	2	no	—
132	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
133	C₂×C₁₀₉	#585	1	2	yes	—
134	D₁₀₉	—	4	4	no	—
135	D₁₀₉	—	4	2	no	—
136	#3455	—	109	1	no	maximal, Sylow
137	C₁₀₉	#3455	1	1	yes	characteristic, complemented, Sylow
138	#542	—	109	4	no	—
139	#541	—	109	2	no	—
140	#540	—	109	2	no	—
141	#539	—	109	4	no	—
142	#538	—	109	4	no	—
143	#533	D₁₀₉	1	1	yes	characteristic, complemented
144	#532	—	109	1	no	—
145	#529	—	109	8	no	—
146	#525	—	109	2	no	—
147	C₄×#41	—	109	2	no	—
148	C₂×#117	—	109	1	no	—
149	C₂×#41	—	109	2	no	—
150	C₂³×C₄	—	109	1	no	—
151	C₄⋊Q₈	—	109	1	no	—
152	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
153	#127	—	109	1	no	—
154	#126	—	109	8	no	—
155	#126	—	109	8	no	—
156	#125	—	109	8	no	—
157	#125	C₂×D₁₀₉	1	2	yes	complemented
158	#124	—	109	8	no	—
159	#124	—	109	2	no	—
160	#123	—	109	8	no	—
161	#123	—	109	8	no	—
162	#123	—	109	4	no	—
163	C₂²⋊Q₈	—	109	8	no	—
164	C₂²⋊Q₈	—	109	4	no	—
165	C₂²⋊Q₈	C₂×D₁₀₉	1	4	yes	—
166	#121	—	109	4	no	—
167	#121	—	109	4	no	—
168	#121	—	109	8	no	—
169	#120	—	109	4	no	—
170	C₄×Q₈	—	109	4	no	—
171	C₄×Q₈	C₂×D₁₀₉	1	4	yes	complemented
172	C₄×D₄	—	109	4	no	—
173	C₄×D₄	—	109	8	no	—
174	C₄×D₄	—	109	8	no	—
175	C₄×D₄	—	109	4	no	—
176	#117	—	109	4	no	—
177	#117	—	109	2	no	—
178	#117	—	109	2	no	—
179	#117	—	109	4	no	—
180	#117	C₂×D₁₀₉	1	2	yes	—
181	C₂×C₄⋊C₄	—	109	2	no	—
182	C₂×C₂²⋊C₄	—	109	2	no	—
183	C₂×C₄²	—	109	1	no	—
184	C₂×C₄²	—	109	4	no	—
185	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂⁴	—	109	1	no	—
187	#41	—	218	8	no	—
188	C₂×Q₈	—	109	2	no	—
189	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
190	C₂×D₄	—	109	4	no	—
191	C₂×D₄	—	109	8	no	—
192	C₂×D₄	—	109	4	no	—
193	C₂×D₄	—	109	2	no	—
194	C₂²×C₄	—	109	2	no	—
195	C₂²×C₄	—	109	1	no	—
196	C₂²×C₄	—	109	8	no	—
197	C₂²×C₄	—	109	2	no	—
198	C₂²×C₄	—	109	4	no	—
199	C₂²×C₄	—	109	2	no	—
200	C₂²×C₄	—	218	4	no	—
201	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
203	C₄⋊C₄	—	109	4	no	—
204	C₄⋊C₄	—	109	4	no	—
205	C₄⋊C₄	—	109	8	no	—
206	C₄⋊C₄	—	109	4	no	—
207	C₄⋊C₄	—	109	8	no	—
208	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
209	C₄⋊C₄	C₂²×D₁₀₉	1	8	yes	—
210	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
211	C₂²⋊C₄	—	109	4	no	—
212	C₂²⋊C₄	—	109	4	no	—
213	C₂²⋊C₄	—	109	8	no	—
214	C₂²⋊C₄	—	109	8	no	—
215	C₂²⋊C₄	—	109	8	no	—
216	C₂²⋊C₄	—	109	8	no	—
217	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
218	C₄²	—	109	2	no	—
219	C₄²	—	109	8	no	—
220	C₄²	—	109	2	no	—
221	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
222	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
223	C₄²	C₂²×D₁₀₉	1	4	yes	—
224	C₄²	C₂²×D₁₀₉	1	2	yes	—
225	C₂³	—	109	2	no	—
226	C₂³	—	109	4	no	—
227	C₂³	—	436	2	no	—
228	C₂³	—	218	2	no	—
229	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	218	4	no	—
232	D₄	—	436	4	no	—
233	D₄	—	436	8	no	—
234	D₄	—	218	8	no	—
235	D₄	—	218	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	109	4	no	—
238	C₂×C₄	—	109	4	no	—
239	C₂×C₄	—	109	2	no	—
240	C₂×C₄	—	436	8	no	—
241	C₂×C₄	—	436	2	no	—
242	C₂×C₄	—	218	2	no	—
243	C₂×C₄	—	218	4	no	—
244	C₂×C₄	—	218	4	no	—
245	C₂×C₄	—	218	8	no	—
246	C₂×C₄	#13182030	1	4	yes	—
247	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
248	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
250	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	436	4	no	—
254	C₂²	—	218	2	no	—
255	C₂²	—	218	4	no	—
256	C₂²	—	218	8	no	—
257	C₂²	—	218	4	no	—
258	C₂²	C₂×#13182030	1	2	yes	—
259	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	218	2	no	—
264	C₄	—	218	4	no	—
265	C₄	—	218	4	no	—
266	C₄	C₂×#13182030	1	2	yes	—
267	C₄	C₂×#13182030	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	436	2	no	—
270	C₂	—	436	4	no	—
271	C₂	C₂²×#13182030	1	1	yes	characteristic
272	C₂	#20295862	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	#533×C₁₀₉	C₂
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3455
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #20295650	C₂	semidirect
2. #20295639	C₂	semidirect
3. #20295632	C₂	semidirect
4. #20295625	C₂	semidirect
5. #20295616	C₂	semidirect
6. #20295576	C₂	semidirect
7. #20295563	C₂	semidirect
8. #20295538	C₂	semidirect
9. #20295494	C₂	semidirect
10. C₂×#16812656	C₂	semidirect
11. #533×C₁₀₉	C₂	semidirect
14. #16812699	C₂²	semidirect
15. #16812696	C₂²	semidirect
17. #16812694	C₂²	semidirect
18. #16812690	C₂²	semidirect
20. #16812687	C₂²	semidirect
22. #16812684	C₂²	semidirect
25. #16812675	C₂²	semidirect
30. #16812664	C₂²	semidirect
32. C₄×D₄₃₆	C₂²	semidirect
33. #16812661	C₂²	semidirect
38. #16812656	C₂²	semidirect
42. C₄²×D₁₀₉	C₂²	semidirect
43. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
44. #125×C₁₀₉	C₂²	semidirect
46. C₄×Q₈×C₁₀₉	C₂²	semidirect
137. C₁₀₉	#3455	semidirect
143. #533	D₁₀₉	semidirect
152. C₄⋊Q₈	C₂×D₁₀₉	semidirect
157. #125	C₂×D₁₀₉	semidirect
171. C₄×Q₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295650	2
#20295639	2
#20295632	2
#20295625	2
#20295616	2
#20295576	2
#20295563	2
#20295538	2
#20295494	2
C₂×#16812656	2
#533×C₁₀₉	2
#3455	109