Subgroups — C₁₀₉×#3294 #28360548

Order 13952.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#547×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#480×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#466×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#459×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#447×C₁₀₉	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#102×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₀₉	—	2	1	no	—
13	C₂²×D₄×C₁₀₉	—	2	1	no	—
14	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	complemented
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	#123×C₁₀₉	—	2	2	no	—
21	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
22	#121×C₁₀₉	—	2	1	no	—
23	#121×C₁₀₉	C₂²	1	2	yes	—
24	#121×C₁₀₉	C₂²	1	2	yes	complemented
25	#120×C₁₀₉	—	2	2	no	—
26	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
30	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	#107×C₁₀₉	C₂²	1	2	yes	complemented
32	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
33	#102×C₁₀₉	C₂²	1	1	yes	characteristic
34	#102×C₁₀₉	C₂²	1	2	yes	—
35	#102×C₁₀₉	C₂²	1	2	yes	complemented
36	#102×C₁₀₉	C₂²	1	2	yes	complemented
37	#102×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
38	#102×C₁₀₉	C₂²	1	2	yes	complemented
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₁₀₉	—	2	2	no	—
42	#41×C₁₀₉	—	4	4	no	—
43	C₂×Q₈×C₁₀₉	—	2	1	no	—
44	C₂×D₄×C₁₀₉	—	2	4	no	—
45	C₂×D₄×C₁₀₉	—	2	2	no	—
46	C₂×D₄×C₁₀₉	—	2	2	no	—
47	C₂×D₄×C₁₀₉	—	2	2	no	—
48	C₂×D₄×C₁₀₉	—	2	1	no	—
49	C₂×D₄×C₁₀₉	—	2	1	no	—
50	C₂×D₄×C₁₀₉	—	4	2	no	—
51	C₂×D₄×C₁₀₉	—	4	4	no	—
52	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	2	1	no	—
58	C₂²×C₄×C₁₀₉	—	2	1	no	—
59	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
60	D₈×C₁₀₉	—	2	2	no	—
61	D₈×C₁₀₉	—	4	2	no	—
62	C₂×C₈×C₁₀₉	—	4	2	no	—
63	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
71	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
77	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂³×C₁₀₉	—	2	1	no	—
79	C₂³×C₁₀₉	—	2	1	no	—
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	4	4	no	—
82	C₂³×C₁₀₉	—	4	4	no	—
83	C₂³×C₁₀₉	—	8	2	no	—
84	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
85	Q₈×C₁₀₉	—	4	2	no	—
86	D₄×C₁₀₉	—	2	1	no	—
87	D₄×C₁₀₉	—	2	2	no	—
88	D₄×C₁₀₉	—	2	1	no	—
89	D₄×C₁₀₉	—	2	1	no	—
90	D₄×C₁₀₉	—	2	2	no	—
91	D₄×C₁₀₉	—	4	4	no	—
92	D₄×C₁₀₉	—	4	4	no	—
93	D₄×C₁₀₉	—	4	4	no	—
94	D₄×C₁₀₉	—	4	2	no	—
95	D₄×C₁₀₉	—	8	1	no	—
96	D₄×C₁₀₉	D₈	1	4	yes	complemented
97	D₄×C₁₀₉	#41	1	2	yes	complemented
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	4	4	no	—
104	C₂×C₄×C₁₀₉	—	4	2	no	—
105	C₂×C₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
110	C₈×C₁₀₉	—	2	1	no	—
111	C₈×C₁₀₉	—	2	1	no	—
112	C₈×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	2	2	no	—
115	C₂²×C₁₀₉	—	2	2	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	—	4	1	no	—
118	C₂²×C₁₀₉	—	4	2	no	—
119	C₂²×C₁₀₉	—	4	1	no	—
120	C₂²×C₁₀₉	—	4	1	no	—
121	C₂²×C₁₀₉	—	4	2	no	—
122	C₂²×C₁₀₉	—	4	2	no	—
123	C₂²×C₁₀₉	—	4	1	no	—
124	C₂²×C₁₀₉	—	8	8	no	—
125	C₂²×C₁₀₉	C₂×D₈	1	4	yes	—
126	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	4	1	no	—
130	C₄×C₁₀₉	—	4	1	no	—
131	C₄×C₁₀₉	—	4	2	no	—
132	C₄×C₁₀₉	C₂×D₈	1	2	yes	—
133	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
134	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₁₀₉	—	2	4	no	—
136	C₂×C₁₀₉	—	4	2	no	—
137	C₂×C₁₀₉	—	8	1	no	—
138	C₂×C₁₀₉	—	8	2	no	—
139	C₂×C₁₀₉	#546	1	1	yes	characteristic
140	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×C₁₀₉	#577	1	1	yes	characteristic
142	#3294	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₀₉	#3294	1	1	yes	characteristic, direct product factor, Sylow
144	#547	C₂×C₁₀₉	1	1	yes	characteristic, complemented
145	D₄²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
146	#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
147	#480	C₂×C₁₀₉	1	2	yes	complemented
148	#474	C₂×C₁₀₉	1	1	yes	characteristic, complemented
149	#466	C₂×C₁₀₉	1	2	yes	complemented
150	#459	C₂×C₁₀₉	1	1	yes	characteristic, complemented
151	#447	C₂×C₁₀₉	1	2	yes	complemented
152	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
153	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
154	C₂×#102	C₂×C₁₀₉	1	2	yes	complemented
155	C₂×#41	—	2	1	no	—
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
158	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
159	C₂×D₈	C₂²×C₁₀₉	1	2	yes	complemented
160	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	complemented
161	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
162	#127	C₂²×C₁₀₉	1	1	yes	characteristic
163	#123	—	2	2	no	—
164	C₂²⋊Q₈	—	2	1	no	—
165	#121	—	2	1	no	—
166	#121	C₂²×C₁₀₉	1	2	yes	—
167	#121	C₂²×C₁₀₉	1	2	yes	complemented
168	#120	—	2	2	no	—
169	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
170	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
171	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
172	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
173	#107	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
174	#107	C₂²×C₁₀₉	1	2	yes	complemented
175	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
176	#102	C₂²×C₁₀₉	1	1	yes	characteristic
177	#102	C₂²×C₁₀₉	1	2	yes	—
178	#102	C₂²×C₁₀₉	1	2	yes	complemented
179	#102	C₂²×C₁₀₉	1	2	yes	complemented
180	#102	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
181	#102	C₂²×C₁₀₉	1	2	yes	complemented
182	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
183	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
184	C₂⁴	—	2	2	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
196	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
197	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
198	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
203	D₈	—	2	2	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
208	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
213	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
214	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
220	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
228	Q₈	—	4	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	1	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	1	no	—
239	D₄	D₈×C₁₀₉	1	4	yes	complemented
240	D₄	#41×C₁₀₉	1	2	yes	complemented
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	8	no	—
268	C₂²	C₂×D₈×C₁₀₉	1	4	yes	—
269	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	C₂×D₈×C₁₀₉	1	2	yes	—
276	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
277	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	8	2	no	—
281	C₂	—	8	1	no	—
282	C₂	#546×C₁₀₉	1	1	yes	characteristic
283	C₂	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
284	C₂	#577×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #547×C₁₀₉	C₂	semidirect
2. D₄²×C₁₀₉	C₂	semidirect
3. #486×C₁₀₉	C₂	semidirect
4. #480×C₁₀₉	C₂	semidirect
5. #474×C₁₀₉	C₂	semidirect
6. #466×C₁₀₉	C₂	semidirect
7. #459×C₁₀₉	C₂	semidirect
8. #447×C₁₀₉	C₂	semidirect
9. C₄×D₈×C₁₀₉	C₂	semidirect
10. C₈×D₄×C₁₀₉	C₂	semidirect
11. C₂×#102×C₁₀₉	C₂	semidirect
16. C₂×D₈×C₁₀₉	C₂²	semidirect
17. C₂²×C₈×C₁₀₉	C₂²	semidirect
18. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
24. #121×C₁₀₉	C₂²	semidirect
28. C₄×D₄×C₁₀₉	C₂²	semidirect
29. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
30. #107×C₁₀₉	C₂²	semidirect
31. #107×C₁₀₉	C₂²	semidirect
32. C₄⋊C₈×C₁₀₉	C₂²	semidirect
35. #102×C₁₀₉	C₂²	semidirect
36. #102×C₁₀₉	C₂²	semidirect
37. #102×C₁₀₉	C₂²	semidirect
38. #102×C₁₀₉	C₂²	semidirect
39. C₂²⋊C₈×C₁₀₉	C₂²	semidirect
40. C₄×C₈×C₁₀₉	C₂²	semidirect
96. D₄×C₁₀₉	D₈	semidirect
97. D₄×C₁₀₉	#41	semidirect
142. #3294	C₁₀₉	direct (N × H)
143. C₁₀₉	#3294	direct (N × H)
144. #547	C₂×C₁₀₉	semidirect
145. D₄²	C₂×C₁₀₉	semidirect
146. #486	C₂×C₁₀₉	semidirect
147. #480	C₂×C₁₀₉	semidirect
148. #474	C₂×C₁₀₉	semidirect
149. #466	C₂×C₁₀₉	semidirect
150. #459	C₂×C₁₀₉	semidirect
151. #447	C₂×C₁₀₉	semidirect
152. C₄×D₈	C₂×C₁₀₉	semidirect
153. C₈×D₄	C₂×C₁₀₉	semidirect
154. C₂×#102	C₂×C₁₀₉	semidirect
159. C₂×D₈	C₂²×C₁₀₉	semidirect
160. C₂²×C₈	C₂²×C₁₀₉	semidirect
161. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
167. #121	C₂²×C₁₀₉	semidirect
171. C₄×D₄	C₂²×C₁₀₉	semidirect
172. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
173. #107	C₂²×C₁₀₉	semidirect
174. #107	C₂²×C₁₀₉	semidirect
175. C₄⋊C₈	C₂²×C₁₀₉	semidirect
178. #102	C₂²×C₁₀₉	semidirect
179. #102	C₂²×C₁₀₉	semidirect
180. #102	C₂²×C₁₀₉	semidirect
181. #102	C₂²×C₁₀₉	semidirect
182. C₂²⋊C₈	C₂²×C₁₀₉	semidirect
183. C₄×C₈	C₂²×C₁₀₉	semidirect
239. D₄	D₈×C₁₀₉	semidirect
240. D₄	#41×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#547×C₁₀₉	2
D₄²×C₁₀₉	2
#486×C₁₀₉	2
#480×C₁₀₉	2
#474×C₁₀₉	2
#466×C₁₀₉	2
#459×C₁₀₉	2
#447×C₁₀₉	2
C₄×D₈×C₁₀₉	2
C₈×D₄×C₁₀₉	2
C₂×#102×C₁₀₉	2
#3294	109

Subgroups — C109×#3294 #28360548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3294 #28360548