Subgroups — #28371554

Order 13952.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812745	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812705	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295506	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295030	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295029	C₂	1	2	yes	maximal, complemented
6	#20295027	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295026	C₂	1	2	yes	maximal, complemented
8	#20295021	C₂	1	2	yes	maximal, complemented
9	C₂×#16812621	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#416×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812745	—	2	4	no	—
13	#16812745	C₄	1	8	yes	complemented
14	C₂×#13182030	C₂²	1	2	yes	—
15	C₂×#13182029	—	2	1	no	—
16	C₂×#13182029	C₂²	1	1	yes	characteristic
17	C₂²×D₄₃₆	—	2	1	no	—
18	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₁₀₉	—	2	2	no	—
20	C₂×D₄×D₁₀₉	C₂²	1	2	yes	—
21	#16812705	—	2	4	no	—
22	C₂×#13182013	C₂²	1	1	yes	characteristic, complemented
23	C₂×#13182012	C₂²	1	2	yes	complemented
24	C₂×C₈×D₁₀₉	C₂²	1	2	yes	complemented
25	C₄×Dic₂₁₈	—	2	1	no	—
26	C₄×D₄₃₆	—	2	1	no	—
27	#16812659	—	2	2	no	—
28	#16812657	C₂²	1	2	yes	complemented
29	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	complemented
31	#16812623	C₂²	1	2	yes	complemented
32	#16812621	C₂²	1	2	yes	complemented
33	#16812621	C₂²	1	2	yes	complemented
34	#16812620	C₂²	1	2	yes	complemented
35	#16812619	C₂²	1	2	yes	complemented
36	#16812619	C₂²	1	2	yes	complemented
37	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
40	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	#103×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
44	C₂³×D₁₀₉	—	2	1	no	—
45	C₂³×D₁₀₉	—	2	2	no	—
46	#13182030	—	2	4	no	—
47	#13182030	C₂×C₄	1	8	yes	complemented
48	#13182029	—	2	8	no	—
49	#13182029	C₂×C₄	1	4	yes	—
50	#13182029	D₄	1	4	yes	—
51	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
52	C₂×D₄₃₆	—	2	2	no	—
53	C₂×D₄₃₆	—	2	1	no	—
54	C₂×D₄₃₆	—	2	2	no	—
55	C₂×D₄₃₆	—	2	8	no	—
56	C₂×D₄₃₆	C₂×C₄	1	4	yes	complemented
57	C₂×D₄₃₆	D₄	1	1	yes	characteristic
58	C₂×D₄₃₆	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
60	C₂×C₂²⋊D₁₀₉	D₄	1	2	yes	—
61	D₄×D₁₀₉	—	2	16	no	—
62	D₄×D₁₀₉	—	2	4	no	—
63	D₄×D₁₀₉	C₂×C₄	1	8	yes	complemented
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	2	no	—
66	C₂×C₄×D₁₀₉	—	2	2	no	—
67	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
68	#13182013	—	2	2	no	—
69	#13182012	—	2	4	no	—
70	C₂×#11724784	C₂³	1	2	yes	—
71	C₈×D₁₀₉	—	2	4	no	—
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
74	#13182008	—	2	2	no	—
75	#13182006	—	2	2	no	—
76	C₄×Dic₁₀₉	—	2	2	no	—
77	#41×C₁₀₉	—	2	2	no	—
78	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
79	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₀₉	—	2	1	no	—
81	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	—	2	1	no	—
83	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
90	C₄²×C₁₀₉	—	2	1	no	—
91	C₂³×C₁₀₉	—	2	1	no	—
92	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×D₁₀₉	—	2	2	no	—
94	C₂²×D₁₀₉	—	2	4	no	—
95	C₂²×D₁₀₉	—	2	8	no	—
96	C₂²×D₁₀₉	—	8	2	no	—
97	C₂²×D₁₀₉	—	8	2	no	—
98	C₂²×D₁₀₉	—	8	1	no	—
99	C₂²×D₁₀₉	—	8	1	no	—
100	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
101	Q₈×C₁₀₉	—	2	1	no	—
102	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
103	D₄×C₁₀₉	—	2	4	no	—
104	D₄×C₁₀₉	—	2	1	no	—
105	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	4	no	—
108	C₂×C₄×C₁₀₉	—	2	1	no	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₁₀₉	—	2	8	no	—
115	C₂²⋊D₁₀₉	C₂²⋊C₄	1	8	yes	—
116	C₂×Dic₁₀₉	—	2	2	no	—
117	C₂×Dic₁₀₉	—	8	1	no	—
118	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
119	D₄₃₆	—	2	8	no	—
120	D₄₃₆	—	2	4	no	—
121	D₄₃₆	—	2	2	no	—
122	D₄₃₆	C₂²⋊C₄	1	4	yes	—
123	D₄₃₆	C₂²×C₄	1	4	yes	—
124	C₄×D₁₀₉	—	2	8	no	—
125	C₄×D₁₀₉	—	8	2	no	—
126	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
127	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
128	Dic₂₁₈	C₂²⋊C₄	1	4	yes	—
129	C₈×C₁₀₉	—	2	1	no	—
130	C₈×C₁₀₉	—	2	1	no	—
131	#11724784	—	2	2	no	—
132	C₂²×C₁₀₉	—	2	1	no	—
133	C₂²×C₁₀₉	—	2	1	no	—
134	C₂²×C₁₀₉	—	2	2	no	—
135	C₂²×C₁₀₉	—	4	1	no	—
136	C₂²×C₁₀₉	—	4	1	no	—
137	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	2	2	no	—
140	C₂×D₁₀₉	—	2	2	no	—
141	C₂×D₁₀₉	—	2	4	no	—
142	C₂×D₁₀₉	—	4	2	no	—
143	C₂×D₁₀₉	—	4	2	no	—
144	C₂×D₁₀₉	—	8	8	no	—
145	C₂×D₁₀₉	—	8	4	no	—
146	C₂×D₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₁₀₉	—	2	2	no	—
148	C₄×C₁₀₉	—	4	1	no	—
149	C₄×C₁₀₉	—	4	1	no	—
150	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
151	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
152	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
153	Dic₁₀₉	—	4	2	no	—
154	Dic₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₁₀₉	—	2	2	no	—
156	C₂×C₁₀₉	—	4	2	no	—
157	C₂×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₁₀₉	#576	1	1	yes	characteristic
159	C₂×C₁₀₉	#577	1	1	yes	characteristic
160	D₁₀₉	—	2	4	no	—
161	D₁₀₉	—	4	4	no	—
162	#2897	—	109	1	no	maximal, Sylow
163	C₁₀₉	#2897	1	1	yes	characteristic, complemented, Sylow
164	C₂×H(5,𝔽₂)	—	109	1	no	—
165	C₂×#131	—	109	1	no	—
166	#520	—	109	1	no	—
167	#418	—	109	1	no	—
168	#418	—	109	2	no	—
169	#417	—	109	1	no	—
170	#417	—	109	2	no	—
171	#416	—	109	2	no	—
172	#416	D₁₀₉	1	1	yes	characteristic, complemented
173	C₂×#102	—	109	1	no	—
174	C₂×#102	—	109	2	no	—
175	H(5,𝔽₂)	—	109	8	no	—
176	H(5,𝔽₂)	—	218	4	no	—
177	C₂×#41	—	109	1	no	—
178	C₂×#41	—	109	2	no	—
179	C₂×#41	—	218	1	no	—
180	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
181	C₂²×D₄	—	109	2	no	—
182	C₂²×D₄	—	109	1	no	—
183	C₂²×D₄	—	218	1	no	—
184	C₂²×D₄	—	218	2	no	—
185	#131	—	218	4	no	—
186	C₂×M₁₆	—	109	1	no	—
187	C₂×M₁₆	—	109	2	no	—
188	C₂²×C₈	—	109	2	no	—
189	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
190	C₄×Q₈	—	218	1	no	—
191	C₄×D₄	—	218	1	no	—
192	C₄×D₄	—	218	2	no	—
193	#117	—	109	2	no	—
194	#117	C₂×D₁₀₉	1	1	yes	characteristic, complemented
195	C₂×C₄⋊C₄	—	109	1	no	—
196	C₂×C₄⋊C₄	—	109	2	no	—
197	#103	—	109	2	no	—
198	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
199	#103	C₂×D₁₀₉	1	1	yes	characteristic, complemented
200	#102	—	109	2	no	—
201	#102	—	109	2	no	—
202	#102	—	109	2	no	—
203	#102	—	109	2	no	—
204	#102	—	109	2	no	—
205	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
206	#102	C₂×D₁₀₉	1	1	yes	characteristic, complemented
207	C₂⁴	—	218	2	no	—
208	C₂⁴	—	218	1	no	—
209	#41	—	2	2	no	—
210	#41	—	109	4	no	—
211	#41	—	109	4	no	—
212	#41	—	109	8	no	—
213	#41	—	218	8	no	—
214	#41	—	218	4	no	—
215	#41	C₄×D₁₀₉	1	4	yes	complemented
216	C₂×Q₈	—	109	1	no	—
217	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	109	2	no	—
220	C₂×D₄	—	109	8	no	—
221	C₂×D₄	—	109	4	no	—
222	C₂×D₄	—	109	1	no	—
223	C₂×D₄	—	109	2	no	—
224	C₂×D₄	—	218	8	no	—
225	C₂×D₄	—	218	2	no	—
226	C₂×D₄	—	218	4	no	—
227	C₂×D₄	—	218	2	no	—
228	C₂×D₄	—	218	16	no	—
229	C₂×D₄	—	218	1	no	—
230	C₂×D₄	—	218	2	no	—
231	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	109	2	no	—
234	C₂²×C₄	—	218	2	no	—
235	C₂²×C₄	—	218	2	no	—
236	C₂²×C₄	—	218	1	no	—
237	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	M₁₆	—	218	4	no	—
239	M₁₆	—	218	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	109	2	no	—
242	C₂×C₈	—	218	4	no	—
243	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₄⋊C₄	—	109	2	no	—
246	C₄⋊C₄	—	218	2	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	218	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	218	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	872	1	no	—
256	C₂³	—	872	1	no	—
257	C₂³	—	872	2	no	—
258	C₂³	—	872	2	no	—
259	C₂³	—	109	2	no	—
260	C₂³	—	218	8	no	—
261	C₂³	—	218	4	no	—
262	C₂³	—	218	2	no	—
263	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	109	4	no	—
266	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	1	no	—
269	D₄	—	109	8	no	—
270	D₄	—	109	4	no	—
271	D₄	—	109	4	no	—
272	D₄	—	218	8	no	—
273	D₄	—	218	8	no	—
274	D₄	—	218	4	no	—
275	D₄	—	218	2	no	—
276	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	872	2	no	—
282	C₂×C₄	—	872	1	no	—
283	C₂×C₄	—	109	2	no	—
284	C₂×C₄	—	109	4	no	—
285	C₂×C₄	—	109	4	no	—
286	C₂×C₄	—	218	8	no	—
287	C₂×C₄	—	218	2	no	—
288	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
289	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
291	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	218	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	872	4	no	—
301	C₂²	—	872	8	no	—
302	C₂²	—	109	4	no	—
303	C₂²	—	436	2	no	—
304	C₂²	—	436	2	no	—
305	C₂²	—	218	4	no	—
306	C₂²	—	218	2	no	—
307	C₂²	—	218	2	no	—
308	C₂²	C₂²⋊C₄×D₁₀₉	1	2	yes	—
309	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	109	4	no	—
314	C₄	—	436	2	no	—
315	C₄	C₂²⋊C₄×D₁₀₉	1	2	yes	—
316	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
317	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	436	4	no	—
321	C₂	—	218	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
323	C₂	#20295828	1	1	yes	characteristic
324	C₂	#20295833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#416×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812745	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#416	D₁₀₉
hypocenter	C₁₀₉	#2897
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. C₂×#16812705	C₂	semidirect
3. #20295506	C₂	semidirect
4. #20295030	C₂	semidirect
5. #20295029	C₂	semidirect
6. #20295027	C₂	semidirect
7. #20295026	C₂	semidirect
8. #20295021	C₂	semidirect
9. C₂×#16812621	C₂	semidirect
10. #102×D₁₀₉	C₂	semidirect
11. #416×C₁₀₉	C₂	semidirect
13. #16812745	C₄	semidirect
22. C₂×#13182013	C₂²	semidirect
23. C₂×#13182012	C₂²	semidirect
24. C₂×C₈×D₁₀₉	C₂²	semidirect
28. #16812657	C₂²	semidirect
29. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
30. C₄⋊C₄×D₁₀₉	C₂²	semidirect
31. #16812623	C₂²	semidirect
32. #16812621	C₂²	semidirect
33. #16812621	C₂²	semidirect
34. #16812620	C₂²	semidirect
35. #16812619	C₂²	semidirect
36. #16812619	C₂²	semidirect
38. C₂²×C₈×C₁₀₉	C₂²	semidirect
39. #117×C₁₀₉	C₂²	semidirect
40. #103×C₁₀₉	C₂²	semidirect
41. #103×C₁₀₉	C₂²	semidirect
42. #102×C₁₀₉	C₂²	semidirect
43. #102×C₁₀₉	C₂²	semidirect
47. #13182030	C₂×C₄	semidirect
56. C₂×D₄₃₆	C₂×C₄	semidirect
63. D₄×D₁₀₉	C₂×C₄	semidirect
78. #41×C₁₀₉	C₂×C₄	semidirect
163. C₁₀₉	#2897	semidirect
172. #416	D₁₀₉	semidirect
189. C₂²×C₈	C₂×D₁₀₉	semidirect
194. #117	C₂×D₁₀₉	semidirect
198. #103	C₂×D₁₀₉	semidirect
199. #103	C₂×D₁₀₉	semidirect
205. #102	C₂×D₁₀₉	semidirect
206. #102	C₂×D₁₀₉	semidirect
215. #41	C₄×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812745	2
C₂×#16812705	2
#20295506	2
#20295030	2
#20295029	2
#20295027	2
#20295026	2
#20295021	2
C₂×#16812621	2
#102×D₁₀₉	2
#416×C₁₀₉	2
#2897	109