Subgroups — #28379810

Order 13952.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296106	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296104	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296071	C₂	1	4	yes	maximal, complemented
4	#20296069	C₂	1	4	yes	maximal, complemented
5	D₁₀₉⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296025	C₂	1	1	yes	maximal, characteristic, complemented
7	#20296024	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295989	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812720	C₂	1	1	yes	maximal, characteristic
10	#16812787	—	2	2	no	—
11	#16812787	C₂²	1	4	yes	complemented
12	#16812786	—	2	2	no	—
13	#16812786	C₂²	1	4	yes	complemented
14	D₁₀₉⋊M₁₆	—	2	2	no	—
15	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
16	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
17	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#13182038	—	2	2	no	—
19	#16812778	C₂²	1	1	yes	characteristic
20	#16812770	C₂²	1	2	yes	complemented
21	#16812769	C₂²	1	2	yes	complemented
22	#16812764	C₂²	1	4	yes	complemented
23	#16812763	C₂²	1	4	yes	complemented
24	#16812753	C₂²	1	2	yes	complemented
25	C₈×#89137	C₂²	1	2	yes	complemented
26	C₂×#13182030	C₂²	1	1	yes	characteristic
27	C₂×#13182028	C₂²	1	1	yes	characteristic
28	#16812720	C₄	1	2	yes	complemented
29	#16812720	C₄	1	2	yes	complemented
30	#16812720	C₂²	1	2	yes	—
31	#16812720	C₂²	1	2	yes	—
32	C₂×#13182019	C₄	1	1	yes	characteristic, complemented
33	C₂×#13182018	C₄	1	1	yes	characteristic, complemented
34	C₂×#13182014	C₄	1	1	yes	characteristic, complemented
35	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
37	#13182039	—	2	4	no	—
38	#13182039	D₄	1	2	yes	—
39	#13182039	C₂³	1	2	yes	—
40	#13182039	C₂³	1	2	yes	—
41	#13182038	—	2	1	no	—
42	#13182038	—	2	1	no	—
43	#13182038	—	2	4	no	—
44	#13182038	D₄	1	2	yes	—
45	D₁₀₉⋊C₈	—	2	1	no	—
46	D₁₀₉⋊C₈	—	2	1	no	—
47	D₁₀₉⋊C₈	—	2	4	no	—
48	D₁₀₉⋊C₈	C₂³	1	2	yes	—
49	C₂×#11724786	—	2	4	no	—
50	C₄⋊#89137	—	2	1	no	—
51	C₂²⋊#89137	—	2	1	no	—
52	C₄×#89137	C₂³	1	2	yes	—
53	#13182030	—	2	1	no	—
54	#13182030	—	2	1	no	—
55	#13182030	C₂×C₄	1	2	yes	—
56	#13182030	C₂³	1	2	yes	—
57	#13182028	—	2	1	no	—
58	#13182028	—	2	1	no	—
59	#13182028	C₂×C₄	1	2	yes	—
60	#13182028	C₂³	1	2	yes	—
61	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
65	C₂²×Dic₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
68	#13182019	C₂×C₄	1	2	yes	—
69	#13182019	C₂×C₄	1	2	yes	—
70	#13182018	C₂×C₄	1	2	yes	—
71	#13182018	C₂×C₄	1	2	yes	—
72	#13182014	C₂×C₄	1	2	yes	—
73	#13182014	C₂×C₄	1	2	yes	—
74	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
78	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
82	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
83	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
84	C₂³×C₁₀₉	—	2	1	no	—
85	C₂²×D₁₀₉	—	2	1	no	—
86	C₂²×D₁₀₉	#41	1	1	yes	characteristic
87	C₂×#89137	—	2	2	no	—
88	Q₈×C₁₀₉	—	2	1	no	—
89	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
90	D₄×C₁₀₉	—	2	1	no	—
91	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²⋊D₁₀₉	—	2	2	no	—
95	C₂²⋊D₁₀₉	—	2	2	no	—
96	C₂×Dic₁₀₉	—	2	1	no	—
97	C₂×Dic₁₀₉	—	2	2	no	—
98	C₂×Dic₁₀₉	—	2	2	no	—
99	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
100	D₄₃₆	—	2	2	no	—
101	D₄₃₆	—	2	2	no	—
102	D₄₃₆	—	2	1	no	—
103	D₄₃₆	C₂²×C₄	1	2	yes	—
104	C₄×D₁₀₉	—	2	2	no	—
105	C₄×D₁₀₉	—	2	2	no	—
106	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₄×D₁₀₉	#41	1	1	yes	characteristic
109	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
110	Dic₂₁₈	—	2	1	no	—
111	Dic₂₁₈	C₂²×C₄	1	2	yes	—
112	#11724786	—	2	2	no	—
113	#11724786	—	2	2	no	—
114	#11724786	C₂×D₄	1	4	yes	—
115	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
117	#11724784	C₂²×C₄	1	1	yes	characteristic
118	#11724784	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₁₀₉	—	2	2	no	—
120	C₂²×C₁₀₉	—	8	1	no	—
121	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
122	C₂×D₁₀₉	—	2	2	no	—
123	C₂×D₁₀₉	—	4	1	no	—
124	C₂×D₁₀₉	—	8	1	no	—
125	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
126	C₂×D₁₀₉	C₂×#41	1	1	yes	characteristic
127	#89137	—	4	2	no	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₀₉	—	2	2	no	—
132	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
133	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₁₀₉	—	2	1	no	—
135	C₂×C₁₀₉	—	4	2	no	—
136	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
137	D₁₀₉	—	2	1	no	—
138	D₁₀₉	—	2	1	no	—
139	D₁₀₉	—	4	2	no	—
140	#2957	—	109	1	no	maximal, Sylow
141	C₁₀₉	#2957	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	109	1	no	—
143	#568	—	109	1	no	—
144	#568	—	109	1	no	—
145	#444	—	109	4	no	—
146	#443	—	109	4	no	—
147	C₂×#108	—	109	1	no	—
148	#421	—	109	1	no	—
149	#421	—	109	1	no	—
150	#405	—	109	1	no	—
151	C₂×#41	—	109	1	no	—
152	C₂×#41	—	109	1	no	—
153	#135	—	109	2	no	—
154	#135	—	109	2	no	—
155	#135	—	109	2	no	—
156	#135	—	109	2	no	—
157	C₂×Q₁₆	—	109	1	no	—
158	C₂×SD₁₆	—	109	1	no	—
159	C₂×SD₁₆	#89137	1	1	yes	characteristic, complemented
160	C₂×D₈	—	109	1	no	—
161	#131	—	109	4	no	—
162	#131	—	109	4	no	—
163	#131	—	218	2	no	—
164	#131	—	218	2	no	—
165	C₂×M₁₆	—	109	1	no	—
166	C₂×M₁₆	—	109	1	no	—
167	C₂×M₁₆	—	109	1	no	—
168	C₂×M₁₆	—	218	2	no	—
169	C₂×M₁₆	—	218	2	no	—
170	C₂²×C₈	—	109	1	no	—
171	#117	—	109	1	no	—
172	#108	—	109	2	no	—
173	#108	—	109	2	no	—
174	#104	—	109	4	no	—
175	#104	—	109	4	no	—
176	#97	—	109	2	no	—
177	C₄×C₈	—	109	2	no	—
178	#41	—	109	2	no	—
179	#41	—	109	2	no	—
180	#41	—	109	2	no	—
181	#41	—	109	2	no	—
182	#41	—	218	1	no	—
183	#41	—	218	1	no	—
184	#41	—	218	1	no	—
185	#41	—	218	1	no	—
186	C₂×Q₈	—	109	1	no	—
187	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
188	C₂×D₄	—	109	1	no	—
189	C₂×D₄	—	218	1	no	—
190	C₂×D₄	—	218	1	no	—
191	C₂×D₄	C₂×#89137	1	1	yes	characteristic
192	C₂²×C₄	—	109	1	no	—
193	C₂²×C₄	—	218	1	no	—
194	C₂²×C₄	—	218	1	no	—
195	Q₁₆	—	109	2	no	—
196	Q₁₆	—	109	2	no	—
197	SD₁₆	—	109	2	no	—
198	SD₁₆	—	109	2	no	—
199	SD₁₆	C₂×#89137	1	2	yes	—
200	SD₁₆	C₂×#89137	1	2	yes	—
201	D₈	—	109	2	no	—
202	D₈	—	109	2	no	—
203	M₁₆	—	109	2	no	—
204	M₁₆	—	109	2	no	—
205	M₁₆	—	109	2	no	—
206	M₁₆	—	109	2	no	—
207	M₁₆	—	218	4	no	—
208	M₁₆	—	218	4	no	—
209	M₁₆	—	218	1	no	—
210	M₁₆	—	218	1	no	—
211	C₂×C₈	—	109	2	no	—
212	C₂×C₈	—	109	1	no	—
213	C₂×C₈	—	109	1	no	—
214	C₂×C₈	—	109	1	no	—
215	C₂×C₈	—	109	1	no	—
216	C₂×C₈	—	109	1	no	—
217	C₂×C₈	—	218	4	no	—
218	C₂×C₈	—	218	4	no	—
219	C₂×C₈	—	218	1	no	—
220	C₂×C₈	—	218	1	no	—
221	C₂×C₈	C₂×#89137	1	1	yes	characteristic
222	C₄⋊C₄	—	218	1	no	—
223	C₂²⋊C₄	—	218	1	no	—
224	C₄²	—	109	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	109	1	no	—
227	C₂³	—	218	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	109	2	no	—
230	Q₈	—	218	1	no	—
231	Q₈	C₂²×#89137	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	109	2	no	—
234	D₄	—	218	2	no	—
235	D₄	—	218	2	no	—
236	D₄	—	218	2	no	—
237	D₄	—	218	2	no	—
238	D₄	—	218	1	no	—
239	D₄	C₂²×#89137	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	109	1	no	—
242	C₂×C₄	—	109	1	no	—
243	C₂×C₄	—	109	1	no	—
244	C₂×C₄	—	109	1	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	218	2	no	—
247	C₂×C₄	—	218	2	no	—
248	C₂×C₄	—	218	2	no	—
249	C₂×C₄	—	218	2	no	—
250	C₂×C₄	—	218	2	no	—
251	C₂×C₄	—	218	1	no	—
252	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
253	C₈	—	109	4	no	—
254	C₈	—	109	1	no	—
255	C₈	—	109	1	no	—
256	C₈	—	218	2	no	—
257	C₈	—	218	2	no	—
258	C₈	C₂²×#89137	1	1	yes	characteristic
259	C₈	C₂²×#89137	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	872	1	no	—
263	C₂²	—	109	1	no	—
264	C₂²	—	109	1	no	—
265	C₂²	—	436	1	no	—
266	C₂²	—	218	2	no	—
267	C₂²	D₄×#89137	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	109	1	no	—
270	C₄	—	109	1	no	—
271	C₄	—	436	2	no	—
272	C₄	—	218	2	no	—
273	C₄	D₄×#89137	1	1	yes	characteristic
274	C₄	C₂³×#89137	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	436	2	no	—
278	C₂	—	218	1	no	—
279	C₂	—	218	1	no	—
280	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	C₂×SD₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	C₂×#16812720	C₂
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	C₂×SD₁₆	#89137
hypocenter	C₁₀₉	#2957
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20296106	C₂	semidirect
2. #20296104	C₂	semidirect
3. #20296071	C₂	semidirect
4. #20296069	C₂	semidirect
5. D₁₀₉⋊#108	C₂	semidirect
6. #20296025	C₂	semidirect
7. #20296024	C₂	semidirect
8. #20295989	C₂	semidirect
11. #16812787	C₂²	semidirect
13. #16812786	C₂²	semidirect
20. #16812770	C₂²	semidirect
21. #16812769	C₂²	semidirect
22. #16812764	C₂²	semidirect
23. #16812763	C₂²	semidirect
24. #16812753	C₂²	semidirect
25. C₈×#89137	C₂²	semidirect
28. #16812720	C₄	semidirect
29. #16812720	C₄	semidirect
32. C₂×#13182019	C₄	semidirect
33. C₂×#13182018	C₄	semidirect
34. C₂×#13182014	C₄	semidirect
36. C₂×SD₁₆×C₁₀₉	C₄	semidirect
141. C₁₀₉	#2957	semidirect
159. C₂×SD₁₆	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296106	2
#20296104	2
#20296071	2
#20296069	2
D₁₀₉⋊#108	2
#20296025	2
#20296024	2
#20295989	2
C₂×#16812720	2
#2957	109