Subgroups — #28371759

Order 13952.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812706	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295104	C₂	1	2	yes	maximal, complemented
4	#20295098	C₂	1	2	yes	maximal, complemented
5	#20295098	C₂	1	2	yes	maximal, complemented
6	D₄×#11724784	C₂	1	2	yes	maximal, complemented
7	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294993	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	#16812706	—	2	4	no	—
13	C₂×#13182013	—	2	4	no	—
14	C₂×#13182013	C₂²	1	1	yes	characteristic
15	C₂×#13182013	C₂²	1	2	yes	—
16	C₂×#13182013	C₂²	1	1	yes	characteristic
17	C₂×#13182013	C₂²	1	2	yes	complemented
18	C₂²×#11724784	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₂²×#11724784	C₂²	1	2	yes	—
21	C₂²×#11724784	C₂²	1	2	yes	complemented
22	C₄⋊#11724784	C₂²	1	2	yes	complemented
23	C₄⋊#11724784	C₂²	1	2	yes	complemented
24	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
25	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
26	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
27	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
28	C₂²⋊#11724784	C₂²	1	2	yes	complemented
29	C₂²⋊#11724784	C₂²	1	2	yes	complemented
30	#16812610	C₂²	1	2	yes	complemented
31	C₄×#11724784	C₂²	1	2	yes	complemented
32	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
34	#123×C₁₀₉	C₄	1	2	yes	—
35	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
37	#121×C₁₀₉	C₄	1	1	yes	characteristic
38	#121×C₁₀₉	C₄	1	1	yes	characteristic
39	#120×C₁₀₉	C₄	1	2	yes	—
40	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
41	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
42	#117×C₁₀₉	C₂²	1	1	yes	characteristic
43	#13182013	—	2	2	no	—
44	#13182013	—	2	4	no	—
45	#13182013	—	2	4	no	—
46	#13182013	D₄	1	4	yes	complemented
47	C₂×#11724784	—	2	2	no	—
48	C₂×#11724784	—	2	2	no	—
49	C₂×#11724784	—	2	4	no	—
50	C₂×#11724784	—	2	4	no	—
51	C₂×#11724784	C₂³	1	1	yes	characteristic
52	C₂×#11724784	C₂³	1	1	yes	characteristic
53	C₂×#11724784	C₂³	1	1	yes	characteristic
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₂×#11724784	C₂³	1	2	yes	—
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	#41×C₁₀₉	—	2	2	no	—
59	#41×C₁₀₉	—	2	2	no	—
60	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
62	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
63	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₉	—	2	2	no	—
66	C₂²×C₄×C₁₀₉	—	2	2	no	—
67	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
77	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
78	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
79	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
80	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
81	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
83	C₄²×C₁₀₉	C₂³	1	2	yes	—
84	C₂³×C₁₀₉	—	2	2	no	—
85	C₂³×C₁₀₉	—	2	2	no	—
86	C₂³×C₁₀₉	—	4	1	no	—
87	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₁₀₉	—	2	2	no	—
92	D₄×C₁₀₉	—	2	4	no	—
93	D₄×C₁₀₉	—	2	4	no	—
94	D₄×C₁₀₉	—	2	2	no	—
95	D₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	2	1	no	—
103	C₂×C₄×C₁₀₉	—	2	1	no	—
104	C₂×C₄×C₁₀₉	—	2	2	no	—
105	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₉	#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
114	#11724784	—	2	1	no	—
115	#11724784	—	2	1	no	—
116	#11724784	—	2	2	no	—
117	#11724784	—	2	1	no	—
118	#11724784	—	2	1	no	—
119	#11724784	C₂×D₄	1	4	yes	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	2	1	no	—
122	C₂²×C₁₀₉	—	2	1	no	—
123	C₂²×C₁₀₉	—	2	1	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	2	no	—
126	C₂²×C₁₀₉	—	2	2	no	—
127	C₂²×C₁₀₉	—	2	1	no	—
128	C₂²×C₁₀₉	—	2	1	no	—
129	C₂²×C₁₀₉	—	2	2	no	—
130	C₂²×C₁₀₉	—	4	4	no	—
131	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
132	C₂²×C₁₀₉	#131	1	2	yes	complemented
133	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	1	no	—
136	C₄×C₁₀₉	—	2	2	no	—
137	C₄×C₁₀₉	—	2	2	no	—
138	C₄×C₁₀₉	—	2	2	no	—
139	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
140	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₁₀₉	—	2	2	no	—
143	C₂×C₁₀₉	—	2	2	no	—
144	C₂×C₁₀₉	—	4	2	no	—
145	C₂×C₁₀₉	—	4	1	no	—
146	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
148	C₂×C₁₀₉	#568	1	1	yes	characteristic
149	#2933	—	109	1	no	maximal, Sylow
150	C₁₀₉	#2933	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	109	1	no	—
152	C₂²×M₁₆	—	109	1	no	—
153	#525	D₁₀₉	1	1	yes	characteristic
154	#436	—	109	2	no	—
155	#435	—	109	2	no	—
156	#435	—	109	2	no	—
157	C₈×D₄	—	109	2	no	—
158	#424	—	109	1	no	—
159	#408	—	109	1	no	—
160	C₂×C₂²⋊C₈	—	109	1	no	—
161	#405	—	109	1	no	—
162	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
164	#131	—	218	4	no	—
165	C₂×M₁₆	—	109	1	no	—
166	C₂×M₁₆	—	109	1	no	—
167	C₂×M₁₆	—	109	2	no	—
168	C₂×M₁₆	—	109	2	no	—
169	C₂×M₁₆	—	218	4	no	—
170	C₂²×C₈	—	109	2	no	—
171	C₂²×C₈	—	109	1	no	—
172	C₂²×C₈	—	109	2	no	—
173	C₂²×C₈	—	109	1	no	—
174	#123	Dic₁₀₉	1	2	yes	—
175	C₂²⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
176	C₂²⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
177	#121	Dic₁₀₉	1	1	yes	characteristic
178	#121	Dic₁₀₉	1	1	yes	characteristic
179	#120	Dic₁₀₉	1	2	yes	—
180	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
181	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
182	#117	C₂×D₁₀₉	1	1	yes	characteristic
183	C₄⋊C₈	—	109	2	no	—
184	C₄⋊C₈	—	109	2	no	—
185	C₂²⋊C₈	—	109	1	no	—
186	C₂²⋊C₈	—	109	1	no	—
187	C₂²⋊C₈	—	109	1	no	—
188	C₂²⋊C₈	—	109	1	no	—
189	C₂²⋊C₈	—	109	2	no	—
190	C₂²⋊C₈	—	109	2	no	—
191	#97	—	109	2	no	—
192	C₄×C₈	—	109	2	no	—
193	C₂⁴	C₂×Dic₁₀₉	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×Dic₁₀₉	1	1	yes	characteristic
197	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
198	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
199	C₂×D₄	C₂×Dic₁₀₉	1	1	yes	characteristic
200	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
210	M₁₆	—	109	4	no	—
211	M₁₆	—	218	4	no	—
212	M₁₆	—	218	4	no	—
213	M₁₆	—	218	2	no	—
214	C₂×C₈	—	109	2	no	—
215	C₂×C₈	—	109	1	no	—
216	C₂×C₈	—	109	1	no	—
217	C₂×C₈	—	109	1	no	—
218	C₂×C₈	—	109	1	no	—
219	C₂×C₈	—	109	2	no	—
220	C₂×C₈	—	218	4	no	—
221	C₂×C₈	—	218	4	no	—
222	C₂×C₈	—	218	2	no	—
223	C₂×C₈	—	218	2	no	—
224	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
225	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
227	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
228	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
229	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
230	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
231	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
232	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
233	C₄²	C₂²×D₁₀₉	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
238	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
239	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
240	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
256	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
257	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
260	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
262	C₂×C₄	#13182028	1	2	yes	—
263	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
264	C₈	—	109	4	no	—
265	C₈	—	218	1	no	—
266	C₈	—	218	2	no	—
267	C₈	—	218	1	no	—
268	C₈	—	218	1	no	—
269	C₈	—	218	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	D₄×Dic₁₀₉	1	2	yes	—
282	C₂²	#16812706	1	2	yes	complemented
283	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	D₄×Dic₁₀₉	1	2	yes	—
290	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
291	C₄	C₂×#13182028	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
297	C₂	C₂×#16812706	1	1	yes	characteristic
298	C₂	#20295793	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting, prime radical	#525×C₁₀₉	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#525	D₁₀₉
hypocenter	C₁₀₉	#2933
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16812706	C₂	semidirect
2. C₂²×#13182013	C₂	semidirect
3. #20295104	C₂	semidirect
4. #20295098	C₂	semidirect
5. #20295098	C₂	semidirect
6. D₄×#11724784	C₂	semidirect
7. #20295053	C₂	semidirect
8. #20294993	C₂	semidirect
9. C₂×C₂²⋊#11724784	C₂	semidirect
10. #20294982	C₂	semidirect
17. C₂×#13182013	C₂²	semidirect
21. C₂²×#11724784	C₂²	semidirect
22. C₄⋊#11724784	C₂²	semidirect
23. C₄⋊#11724784	C₂²	semidirect
28. C₂²⋊#11724784	C₂²	semidirect
29. C₂²⋊#11724784	C₂²	semidirect
30. #16812610	C₂²	semidirect
31. C₄×#11724784	C₂²	semidirect
46. #13182013	D₄	semidirect
132. C₂²×C₁₀₉	#131	semidirect
150. C₁₀₉	#2933	semidirect
282. C₂²	#16812706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812706	2
C₂²×#13182013	2
#20295104	2
#20295098	2
D₄×#11724784	2
#20295053	2
#20294993	2
C₂×C₂²⋊#11724784	2
#20294982	2
#525×C₁₀₉	2
#2933	109