Subgroups — #28376702

Order 13952.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 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, complemented
2	C₂×#41×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295698	C₂	1	2	yes	maximal, complemented
4	#20295688	C₂	1	2	yes	maximal, complemented
5	#20295671	C₂	1	6	yes	maximal, complemented
6	D₄×D₄₃₆	C₂	1	6	yes	maximal, complemented
7	#20295607	C₂	1	3	yes	maximal, complemented
8	#20295595	C₂	1	3	yes	maximal, complemented
9	#20295538	C₂	1	3	yes	maximal, complemented
10	C₂×#16812674	C₂	1	3	yes	maximal, complemented
11	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16812745	—	2	8	no	—
13	C₂×#13182030	C₂²	1	2	yes	complemented
14	C₂×#13182030	C₂²	1	1	yes	characteristic, complemented
15	C₂×#13182029	C₂²	1	3	yes	complemented
16	C₂×#13182029	C₂²	1	3	yes	complemented
17	C₂×#13182028	C₂²	1	3	yes	complemented
18	#41×D₁₀₉	—	2	8	no	—
19	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄₃₆	C₂²	1	3	yes	complemented
21	C₂²×D₄₃₆	C₂²	1	6	yes	complemented
22	C₂×D₄×D₁₀₉	C₂²	1	3	yes	complemented
23	C₂×D₄×D₁₀₉	C₂²	1	6	yes	complemented
24	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	complemented
25	#16812696	C₂²	1	6	yes	complemented
26	#16812687	C₂²	1	6	yes	complemented
27	#16812683	C₂²	1	6	yes	complemented
28	#16812679	C₂²	1	6	yes	complemented
29	#16812679	C₂²	1	1	yes	characteristic, complemented
30	#16812679	C₂²	1	3	yes	complemented
31	#16812674	C₂²	1	6	yes	complemented
32	#16812674	C₂²	1	6	yes	complemented
33	#16812674	C₂²	1	6	yes	complemented
34	#16812674	C₂²	1	3	yes	complemented
35	#16812674	C₂²	1	6	yes	complemented
36	#16812674	C₂²	1	3	yes	complemented
37	#16812673	C₂²	1	12	yes	complemented
38	#16812669	C₂²	1	6	yes	complemented
39	#16812669	C₂²	1	12	yes	complemented
40	C₄×D₄₃₆	C₂²	1	6	yes	complemented
41	C₄×D₄₃₆	C₂²	1	6	yes	complemented
42	C₂×#13182006	C₂²	1	6	yes	complemented
43	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	complemented
45	C₄×D₄×C₁₀₉	C₂²	1	6	yes	complemented
46	#117×C₁₀₉	C₂²	1	3	yes	complemented
47	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	complemented
48	C₂³×D₁₀₉	C₂³	1	3	yes	—
49	C₂³×D₁₀₉	C₂³	1	6	yes	—
50	#13182030	—	2	4	no	—
51	#13182030	—	2	8	no	—
52	#13182029	—	2	12	no	—
53	#13182029	—	2	12	no	—
54	#13182028	—	2	12	no	—
55	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₁₈	C₂³	1	3	yes	—
57	Q₈×D₁₀₉	—	2	4	no	—
58	C₂×D₄₃₆	—	2	12	no	—
59	C₂×D₄₃₆	—	4	12	no	—
60	C₂×D₄₃₆	C₂³	1	6	yes	—
61	C₂×D₄₃₆	C₂³	1	3	yes	—
62	C₂×D₄₃₆	C₂³	1	6	yes	—
63	C₂×D₄₃₆	C₂³	1	3	yes	—
64	C₂×D₄₃₆	C₂³	1	6	yes	—
65	C₂×D₄₃₆	C₂³	1	12	yes	—
66	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
67	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
68	D₄×D₁₀₉	—	2	12	no	—
69	D₄×D₁₀₉	—	2	24	no	—
70	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
71	C₂×C₄×D₁₀₉	—	2	12	no	—
72	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
73	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
74	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
75	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
77	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
78	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
79	#13182006	C₂³	1	6	yes	—
80	#13182006	C₂³	1	12	yes	—
81	#13182006	C₂³	1	2	yes	—
82	#13182006	C₂³	1	6	yes	—
83	#13182006	C₂³	1	6	yes	—
84	#41×C₁₀₉	D₄	1	8	yes	complemented
85	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
87	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
88	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
89	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
90	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
91	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
93	C₄²×C₁₀₉	C₂³	1	6	yes	—
94	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
95	C₂²×D₁₀₉	—	2	6	no	—
96	C₂²×D₁₀₉	—	2	12	no	—
97	C₂²×D₁₀₉	—	2	6	no	—
98	C₂²×D₁₀₉	—	4	3	no	—
99	C₂²×D₁₀₉	—	4	6	no	—
100	C₂²×D₁₀₉	—	4	6	no	—
101	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
102	C₂²×D₁₀₉	C₂⁴	1	3	yes	—
103	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
104	Q₈×C₁₀₉	C₂×D₄	1	4	yes	complemented
105	D₄×C₁₀₉	C₂×D₄	1	12	yes	complemented
106	C₂×C₄×C₁₀₉	—	4	6	no	—
107	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	complemented
108	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
110	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
111	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₀₉	—	2	12	no	—
113	C₂²⋊D₁₀₉	—	2	12	no	—
114	C₂×Dic₁₀₉	—	2	6	no	—
115	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
117	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
118	D₄₃₆	—	2	12	no	—
119	D₄₃₆	—	2	12	no	—
120	D₄₃₆	—	2	6	no	—
121	D₄₃₆	—	2	6	no	—
122	D₄₃₆	—	4	12	no	—
123	D₄₃₆	—	4	6	no	—
124	C₄×D₁₀₉	—	2	2	no	—
125	C₄×D₁₀₉	—	2	6	no	—
126	C₄×D₁₀₉	—	2	6	no	—
127	C₄×D₁₀₉	—	2	6	no	—
128	C₄×D₁₀₉	—	2	12	no	—
129	Dic₂₁₈	—	2	6	no	—
130	Dic₂₁₈	—	2	2	no	—
131	C₂²×C₁₀₉	—	2	3	no	—
132	C₂²×C₁₀₉	—	2	3	no	—
133	C₂²×C₁₀₉	C₂²×D₄	1	6	yes	—
134	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₀₉	—	2	1	no	—
136	C₂×D₁₀₉	—	2	1	no	—
137	C₂×D₁₀₉	—	2	3	no	—
138	C₂×D₁₀₉	—	2	6	no	—
139	C₂×D₁₀₉	—	2	6	no	—
140	C₂×D₁₀₉	—	2	3	no	—
141	C₂×D₁₀₉	—	2	3	no	—
142	C₂×D₁₀₉	—	2	6	no	—
143	C₂×D₁₀₉	—	4	12	no	—
144	C₂×D₁₀₉	—	8	12	no	—
145	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	complemented
146	C₄×C₁₀₉	—	2	6	no	—
147	C₄×C₁₀₉	—	2	2	no	—
148	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
149	C₄×C₁₀₉	C₂²×D₄	1	6	yes	—
150	Dic₁₀₉	—	2	3	no	—
151	Dic₁₀₉	—	2	2	no	—
152	Dic₁₀₉	—	2	1	no	—
153	C₂×C₁₀₉	—	2	6	no	—
154	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
155	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₁₀₉	#585	1	1	yes	characteristic
157	D₁₀₉	—	2	2	no	—
158	D₁₀₉	—	4	3	no	—
159	D₁₀₉	—	4	6	no	—
160	#3488	—	109	1	no	maximal, Sylow
161	C₁₀₉	#3488	1	1	yes	characteristic, complemented, Sylow
162	C₂×H(5,𝔽₂)	—	109	1	no	—
163	C₂²×#41	—	109	1	no	—
164	#550	—	109	2	no	—
165	#548	—	109	2	no	—
166	#546	—	109	6	no	—
167	D₄²	—	109	6	no	—
168	#537	—	109	3	no	—
169	#535	—	109	3	no	—
170	#525	—	109	3	no	—
171	C₂×#121	—	109	3	no	—
172	#520	D₁₀₉	1	1	yes	characteristic, complemented
173	H(5,𝔽₂)	—	218	8	no	—
174	C₂×#41	—	109	2	no	—
175	C₂×#41	—	109	1	no	—
176	C₂×#41	—	109	3	no	—
177	C₂×#41	—	109	3	no	—
178	C₂×#41	—	109	3	no	—
179	C₂×#41	—	218	8	no	—
180	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
181	C₂²×Q₈	—	109	1	no	—
182	C₂²×D₄	—	109	3	no	—
183	C₂²×D₄	—	109	6	no	—
184	C₂²×D₄	—	109	6	no	—
185	C₂²×D₄	—	109	3	no	—
186	C₂³×C₄	—	109	3	no	—
187	#127	—	109	6	no	—
188	#124	—	109	6	no	—
189	#123	—	109	6	no	—
190	C₂²⋊Q₈	—	109	1	no	—
191	C₂²⋊Q₈	—	109	3	no	—
192	C₂²⋊Q₈	—	109	6	no	—
193	#121	—	109	12	no	—
194	#121	—	109	3	no	—
195	#121	—	109	6	no	—
196	#121	—	109	6	no	—
197	#121	—	109	6	no	—
198	#121	—	109	3	no	—
199	#121	—	109	6	no	—
200	#120	—	109	6	no	—
201	#120	—	109	12	no	—
202	C₄×Q₈	C₂×D₁₀₉	1	2	yes	complemented
203	C₄×D₄	—	109	6	no	—
204	C₄×D₄	—	109	6	no	—
205	C₄×D₄	C₂×D₁₀₉	1	6	yes	complemented
206	#117	C₂×D₁₀₉	1	3	yes	complemented
207	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	complemented
208	C₂×C₂²⋊C₄	—	109	6	no	—
209	C₂⁴	—	109	6	no	—
210	C₂⁴	—	109	3	no	—
211	#41	—	218	4	no	—
212	#41	—	218	8	no	—
213	#41	—	218	12	no	—
214	#41	—	218	12	no	—
215	#41	—	218	12	no	—
216	#41	D₄₃₆	1	8	yes	complemented
217	C₂×Q₈	—	109	1	no	—
218	C₂×Q₈	—	109	3	no	—
219	C₂×Q₈	—	218	4	no	—
220	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
221	C₂×D₄	—	109	6	no	—
222	C₂×D₄	—	109	6	no	—
223	C₂×D₄	—	109	6	no	—
224	C₂×D₄	—	109	12	no	—
225	C₂×D₄	—	109	6	no	—
226	C₂×D₄	—	109	3	no	—
227	C₂×D₄	—	109	6	no	—
228	C₂×D₄	—	109	3	no	—
229	C₂×D₄	—	436	12	no	—
230	C₂×D₄	—	218	12	no	—
231	C₂×D₄	—	218	24	no	—
232	C₂×D₄	—	218	12	no	—
233	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
234	C₂²×C₄	—	109	3	no	—
235	C₂²×C₄	—	109	3	no	—
236	C₂²×C₄	—	109	3	no	—
237	C₂²×C₄	—	109	6	no	—
238	C₂²×C₄	—	109	1	no	—
239	C₂²×C₄	—	109	3	no	—
240	C₂²×C₄	—	218	12	no	—
241	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
242	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
243	C₄⋊C₄	—	109	2	no	—
244	C₄⋊C₄	—	109	6	no	—
245	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
246	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
247	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	109	6	no	—
249	C₂²⋊C₄	—	109	12	no	—
250	C₂²⋊C₄	—	109	2	no	—
251	C₂²⋊C₄	—	109	6	no	—
252	C₂²⋊C₄	—	109	6	no	—
253	C₂²⋊C₄	C₂²×D₁₀₉	1	6	yes	—
254	C₄²	C₂²×D₁₀₉	1	6	yes	—
255	C₂³	—	109	1	no	—
256	C₂³	—	109	6	no	—
257	C₂³	—	109	3	no	—
258	C₂³	—	436	6	no	—
259	C₂³	—	436	6	no	—
260	C₂³	—	436	3	no	—
261	C₂³	—	218	6	no	—
262	C₂³	—	218	12	no	—
263	C₂³	—	218	6	no	—
264	C₂³	C₂³×D₁₀₉	1	3	yes	—
265	Q₈	—	218	6	no	—
266	Q₈	—	218	2	no	—
267	Q₈	C₂×D₄₃₆	1	4	yes	complemented
268	D₄	—	436	12	no	—
269	D₄	—	436	6	no	—
270	D₄	—	218	12	no	—
271	D₄	—	218	12	no	—
272	D₄	—	218	12	no	—
273	D₄	—	218	6	no	—
274	D₄	—	218	12	no	—
275	D₄	—	218	6	no	—
276	D₄	C₂×D₄₃₆	1	12	yes	complemented
277	C₂×C₄	—	4	6	no	—
278	C₂×C₄	—	109	1	no	—
279	C₂×C₄	—	109	3	no	—
280	C₂×C₄	—	109	2	no	—
281	C₂×C₄	—	218	6	no	—
282	C₂×C₄	—	218	2	no	—
283	C₂×C₄	—	218	6	no	—
284	C₂×C₄	—	218	6	no	—
285	C₂×C₄	—	218	6	no	—
286	C₂×C₄	—	218	12	no	—
287	C₂×C₄	C₂×D₄₃₆	1	12	yes	complemented
288	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
289	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
290	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
291	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
292	C₂²	—	2	3	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	872	12	no	—
295	C₂²	—	109	2	no	—
296	C₂²	—	436	12	no	—
297	C₂²	—	218	3	no	—
298	C₂²	—	218	6	no	—
299	C₂²	—	218	3	no	—
300	C₂²	—	218	6	no	—
301	C₂²	—	218	3	no	—
302	C₂²	—	218	6	no	—
303	C₂²	—	218	1	no	—
304	C₂²	—	218	1	no	—
305	C₂²	C₂²×D₄₃₆	1	6	yes	—
306	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
307	C₄	—	2	6	no	—
308	C₄	—	2	2	no	—
309	C₄	—	218	1	no	—
310	C₄	—	218	2	no	—
311	C₄	—	218	3	no	—
312	C₄	C₂²×D₄₃₆	1	2	yes	—
313	C₄	C₂²×D₄₃₆	1	6	yes	—
314	C₂	—	2	6	no	—
315	C₂	—	436	6	no	—
316	C₂	—	436	3	no	—
317	C₂	—	218	2	no	—
318	C₂	C₂³×D₄₃₆	1	1	yes	characteristic
319	C₂	H(5,𝔽₂)×D₁₀₉	1	1	yes	characteristic
320	C₂	#20295863	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#520×C₁₀₉	C₂
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#3488
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#16812745	C₂	semidirect
2. C₂×#41×D₁₀₉	C₂	semidirect
3. #20295698	C₂	semidirect
4. #20295688	C₂	semidirect
5. #20295671	C₂	semidirect
6. D₄×D₄₃₆	C₂	semidirect
7. #20295607	C₂	semidirect
8. #20295595	C₂	semidirect
9. #20295538	C₂	semidirect
10. C₂×#16812674	C₂	semidirect
11. #520×C₁₀₉	C₂	semidirect
13. C₂×#13182030	C₂²	semidirect
14. C₂×#13182030	C₂²	semidirect
15. C₂×#13182029	C₂²	semidirect
16. C₂×#13182029	C₂²	semidirect
17. C₂×#13182028	C₂²	semidirect
19. C₂×Q₈×D₁₀₉	C₂²	semidirect
20. C₂²×D₄₃₆	C₂²	semidirect
21. C₂²×D₄₃₆	C₂²	semidirect
22. C₂×D₄×D₁₀₉	C₂²	semidirect
23. C₂×D₄×D₁₀₉	C₂²	semidirect
24. C₂²×C₄×D₁₀₉	C₂²	semidirect
25. #16812696	C₂²	semidirect
26. #16812687	C₂²	semidirect
27. #16812683	C₂²	semidirect
28. #16812679	C₂²	semidirect
29. #16812679	C₂²	semidirect
30. #16812679	C₂²	semidirect
31. #16812674	C₂²	semidirect
32. #16812674	C₂²	semidirect
33. #16812674	C₂²	semidirect
34. #16812674	C₂²	semidirect
35. #16812674	C₂²	semidirect
36. #16812674	C₂²	semidirect
37. #16812673	C₂²	semidirect
38. #16812669	C₂²	semidirect
39. #16812669	C₂²	semidirect
40. C₄×D₄₃₆	C₂²	semidirect
41. C₄×D₄₃₆	C₂²	semidirect
42. C₂×#13182006	C₂²	semidirect
44. C₄×Q₈×C₁₀₉	C₂²	semidirect
45. C₄×D₄×C₁₀₉	C₂²	semidirect
46. #117×C₁₀₉	C₂²	semidirect
47. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
84. #41×C₁₀₉	D₄	semidirect
104. Q₈×C₁₀₉	C₂×D₄	semidirect
105. D₄×C₁₀₉	C₂×D₄	semidirect
107. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
145. C₂×D₁₀₉	H(5,𝔽₂)	semidirect
161. C₁₀₉	#3488	semidirect
172. #520	D₁₀₉	semidirect
202. C₄×Q₈	C₂×D₁₀₉	semidirect
205. C₄×D₄	C₂×D₁₀₉	semidirect
206. #117	C₂×D₁₀₉	semidirect
207. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
216. #41	D₄₃₆	semidirect
267. Q₈	C₂×D₄₃₆	semidirect
276. D₄	C₂×D₄₃₆	semidirect
287. C₂×C₄	C₂×D₄₃₆	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×#41×D₁₀₉	2
#20295698	2
#20295688	2
#20295671	2
D₄×D₄₃₆	2
#20295607	2
#20295595	2
#20295538	2
C₂×#16812674	2
#520×C₁₀₉	2
#3488	109