Subgroups — #28371876

Order 13952.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295509	C₂	1	1	yes	maximal, characteristic
2	#20295504	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295133	C₂	1	2	yes	maximal, complemented
4	#20295127	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
7	#20295068	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295022	C₂	1	1	yes	maximal, characteristic
9	C₂×#16812623	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11724784	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₀₉	—	2	1	no	—
14	Q₈×Dic₁₀₉	—	2	1	no	—
15	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
16	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
17	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
18	D₄×Dic₁₀₉	—	2	2	no	—
19	D₄×Dic₁₀₉	C₂²	1	2	yes	—
20	#16812658	—	2	1	no	—
21	#16812658	—	2	2	no	—
22	#16812658	C₂²	1	1	yes	characteristic
23	#16812658	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
25	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
26	#16812633	C₂²	1	2	yes	—
27	#16812631	C₂²	1	2	yes	—
28	#16812623	C₂²	1	2	yes	—
29	#16812623	C₂²	1	2	yes	—
30	#16812623	C₂²	1	1	yes	characteristic
31	#16812623	C₂²	1	1	yes	characteristic
32	#16812620	C₂²	1	1	yes	characteristic
33	#16812620	C₂²	1	1	yes	characteristic
34	#16812609	C₂²	1	2	yes	—
35	C₈×Dic₁₀₉	C₂²	1	2	yes	—
36	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	#137×C₁₀₉	C₄	1	8	yes	complemented
39	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
40	C₂×SD₁₆×C₁₀₉	C₂²	1	2	yes	—
41	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂²×Dic₁₀₉	—	2	1	no	—
43	C₂²×Dic₁₀₉	—	2	2	no	—
44	C₂×#11724784	—	2	2	no	—
45	C₂×#11724784	C₂³	1	1	yes	characteristic
46	C₂×#11724784	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₀₉	—	2	1	no	—
48	C₄⋊Dic₁₀₉	—	2	2	no	—
49	C₄⋊Dic₁₀₉	—	2	2	no	—
50	C₄⋊Dic₁₀₉	—	2	2	no	—
51	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
52	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
56	C₂²⋊Dic₁₀₉	—	2	1	no	—
57	C₂²⋊Dic₁₀₉	—	2	2	no	—
58	C₂²⋊Dic₁₀₉	D₄	1	2	yes	—
59	C₄×Dic₁₀₉	—	2	1	no	—
60	C₄×Dic₁₀₉	—	2	2	no	—
61	C₄×Dic₁₀₉	—	2	2	no	—
62	C₄×Dic₁₀₉	—	2	2	no	—
63	C₄×Dic₁₀₉	C₂³	1	2	yes	—
64	#41×C₁₀₉	—	2	2	no	—
65	#41×C₁₀₉	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₁₀₉	—	2	1	no	—
67	C₂×Q₈×C₁₀₉	—	2	2	no	—
68	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
69	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
70	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₀₉	—	2	1	no	—
72	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	—	2	1	no	—
74	C₂²×C₄×C₁₀₉	—	2	1	no	—
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	Q₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
77	SD₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
78	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₂³×C₁₀₉	—	2	1	no	—
81	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₁₀₉	—	2	4	no	—
83	Q₈×C₁₀₉	—	2	2	no	—
84	Q₈×C₁₀₉	—	2	1	no	—
85	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
86	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₉	—	2	4	no	—
88	D₄×C₁₀₉	—	2	1	no	—
89	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	—	2	4	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	4	no	—
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#41	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₀₉	—	2	1	no	—
99	C₂×Dic₁₀₉	—	2	1	no	—
100	C₂×Dic₁₀₉	—	2	2	no	—
101	C₂×Dic₁₀₉	—	8	2	no	—
102	C₂×Dic₁₀₉	—	8	1	no	—
103	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
104	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
105	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
106	#11724784	—	2	1	no	—
107	#11724784	—	2	1	no	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	—	2	1	no	—
110	C₂²×C₁₀₉	—	2	2	no	—
111	C₂²×C₁₀₉	—	4	1	no	—
112	C₂²×C₁₀₉	—	4	1	no	—
113	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
114	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	—	2	4	no	—
117	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
118	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
120	Dic₁₀₉	—	2	2	no	—
121	Dic₁₀₉	—	2	2	no	—
122	Dic₁₀₉	—	4	1	no	—
123	Dic₁₀₉	—	4	2	no	—
124	Dic₁₀₉	—	4	1	no	—
125	C₂×C₁₀₉	—	2	2	no	—
126	C₂×C₁₀₉	—	4	2	no	—
127	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₁₀₉	#577	1	1	yes	characteristic
129	C₂×C₁₀₉	#578	1	1	yes	characteristic
130	#2947	—	109	1	no	maximal, Sylow
131	C₁₀₉	#2947	1	1	yes	characteristic, complemented, Sylow
132	C₂×#137	D₁₀₉	1	1	yes	characteristic
133	#520	—	109	1	no	—
134	#519	—	109	1	no	—
135	#441	—	109	2	no	—
136	#440	—	109	2	no	—
137	C₄×Q₁₆	—	109	2	no	—
138	C₄×SD₁₆	—	109	2	no	—
139	#428	—	109	1	no	—
140	#416	—	109	1	no	—
141	C₂×#103	—	109	1	no	—
142	#405	—	109	1	no	—
143	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
144	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
145	#137	Dic₁₀₉	1	8	yes	complemented
146	C₂×Q₁₆	C₂×D₁₀₉	1	2	yes	—
147	C₂×SD₁₆	C₂×D₁₀₉	1	2	yes	—
148	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
149	C₂²×C₈	—	109	1	no	—
150	C₄×Q₈	—	109	2	no	—
151	C₄×Q₈	—	109	2	no	—
152	C₄×Q₈	—	109	2	no	—
153	C₄×Q₈	—	218	1	no	—
154	C₄×Q₈	—	218	1	no	—
155	C₄×D₄	—	109	2	no	—
156	C₄×D₄	—	218	2	no	—
157	#117	—	109	1	no	—
158	#117	—	109	1	no	—
159	#117	—	218	2	no	—
160	#117	—	218	1	no	—
161	C₂×C₄⋊C₄	—	109	1	no	—
162	C₂×C₄⋊C₄	—	218	1	no	—
163	#107	—	109	2	no	—
164	#106	—	109	2	no	—
165	#103	—	109	2	no	—
166	#103	—	109	2	no	—
167	#103	—	109	1	no	—
168	#103	—	109	1	no	—
169	#102	—	109	1	no	—
170	#102	—	109	1	no	—
171	#97	—	109	2	no	—
172	C₄×C₈	—	109	2	no	—
173	#41	—	2	2	no	—
174	#41	C₂×Dic₁₀₉	1	4	yes	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂×Dic₁₀₉	1	4	yes	complemented
178	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
179	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	218	2	no	—
185	C₂²×C₄	—	218	1	no	—
186	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
187	Q₁₆	C₂×Dic₁₀₉	1	8	yes	complemented
188	SD₁₆	C₂×Dic₁₀₉	1	8	yes	complemented
189	M₁₆	C₂×Dic₁₀₉	1	4	yes	complemented
190	C₂×C₈	—	109	1	no	—
191	C₂×C₈	—	109	1	no	—
192	C₂×C₈	—	218	2	no	—
193	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
194	C₄⋊C₄	—	109	1	no	—
195	C₄⋊C₄	—	109	1	no	—
196	C₄⋊C₄	—	109	1	no	—
197	C₄⋊C₄	—	109	1	no	—
198	C₄⋊C₄	—	109	2	no	—
199	C₄⋊C₄	—	218	2	no	—
200	C₄⋊C₄	—	218	2	no	—
201	C₄⋊C₄	—	218	2	no	—
202	C₄⋊C₄	—	218	1	no	—
203	C₂²⋊C₄	—	109	2	no	—
204	C₂²⋊C₄	—	218	2	no	—
205	C₂²⋊C₄	—	218	1	no	—
206	C₄²	—	109	2	no	—
207	C₄²	—	218	2	no	—
208	C₄²	—	218	2	no	—
209	C₄²	—	218	2	no	—
210	C₄²	—	218	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	C₂²×Dic₁₀₉	1	4	yes	—
217	Q₈	C₂²×Dic₁₀₉	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂²×Dic₁₀₉	1	2	yes	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	872	1	no	—
226	C₂×C₄	—	872	2	no	—
227	C₂×C₄	—	109	2	no	—
228	C₂×C₄	—	109	2	no	—
229	C₂×C₄	—	218	1	no	—
230	C₂×C₄	—	218	1	no	—
231	C₂×C₄	—	218	2	no	—
232	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
233	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
234	C₂×C₄	#13182028	1	2	yes	—
235	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
236	C₈	—	218	1	no	—
237	C₈	—	218	1	no	—
238	C₈	C₂²×Dic₁₀₉	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	D₄×Dic₁₀₉	1	2	yes	—
245	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	436	2	no	—
249	C₄	—	436	1	no	—
250	C₄	—	436	1	no	—
251	C₄	—	218	2	no	—
252	C₄	—	218	2	no	—
253	C₄	D₄×Dic₁₀₉	1	2	yes	—
254	C₄	C₂³×Dic₁₀₉	1	1	yes	characteristic
255	C₄	C₂×#13182028	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
259	C₂	#20295835	1	1	yes	characteristic
260	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#137×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#137	D₁₀₉
hypocenter	C₁₀₉	#2947
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. #20295504	C₂	semidirect
3. #20295133	C₂	semidirect
4. #20295127	C₂	semidirect
5. Q₁₆×Dic₁₀₉	C₂	semidirect
6. SD₁₆×Dic₁₀₉	C₂	semidirect
7. #20295068	C₂	semidirect
9. C₂×#16812623	C₂	semidirect
10. #20294984	C₂	semidirect
38. #137×C₁₀₉	C₄	semidirect
68. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
76. Q₁₆×C₁₀₉	C₂×C₄	semidirect
77. SD₁₆×C₁₀₉	C₂×C₄	semidirect
78. M₁₆×C₁₀₉	C₂×C₄	semidirect
131. C₁₀₉	#2947	semidirect
145. #137	Dic₁₀₉	semidirect
177. C₂×Q₈	C₂×Dic₁₀₉	semidirect
187. Q₁₆	C₂×Dic₁₀₉	semidirect
188. SD₁₆	C₂×Dic₁₀₉	semidirect
189. M₁₆	C₂×Dic₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295509	2
#20295504	2
#20295133	2
#20295127	2
Q₁₆×Dic₁₀₉	2
SD₁₆×Dic₁₀₉	2
#20295068	2
#20295022	2
C₂×#16812623	2
#20294984	2
C₂×#137×C₁₀₉	2
#2947	109