Subgroups — C₁₀₉×#29327 #42147321

Order 27904.

606 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	C₁₀₉×#2647	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2330	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2084	C₂	1	2	yes	maximal
4	C₁₀₉×#2059	C₂	1	2	yes	maximal
5	C₁₀₉×#1949	C₂	1	2	yes	maximal
6	C₁₀₉×#1948	C₂	1	2	yes	maximal
7	C₁₀₉×#1943	C₂	1	2	yes	maximal
8	C₁₀₉×#1941	C₂	1	2	yes	maximal
9	C₁₀₉×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
13	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
14	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#400×C₁₀₉	—	2	2	no	—
19	#398×C₁₀₉	—	2	2	no	—
20	#389×C₁₀₉	—	2	2	no	—
21	#389×C₁₀₉	C₂²	1	2	yes	—
22	#389×C₁₀₉	C₂²	1	2	yes	—
23	#389×C₁₀₉	C₂²	1	2	yes	—
24	#387×C₁₀₉	—	2	2	no	—
25	#384×C₁₀₉	C₂²	1	1	yes	characteristic
26	#384×C₁₀₉	C₂²	1	1	yes	characteristic
27	#382×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	#376×C₁₀₉	C₂²	1	1	yes	characteristic
30	#340×C₁₀₉	C₂²	1	2	yes	—
31	#340×C₁₀₉	C₂²	1	2	yes	—
32	#340×C₁₀₉	C₂²	1	2	yes	—
33	#340×C₁₀₉	C₂²	1	2	yes	—
34	#336×C₁₀₉	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
37	#125×C₁₀₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₀₉	—	2	2	no	—
49	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
52	#107×C₁₀₉	—	2	4	no	—
53	#106×C₁₀₉	—	2	4	no	—
54	#106×C₁₀₉	—	4	2	no	—
55	#106×C₁₀₉	—	4	2	no	—
56	C₄⋊C₈×C₁₀₉	Q₈	1	8	yes	—
57	#97×C₁₀₉	—	4	1	no	—
58	C₄×C₈×C₁₀₉	—	4	1	no	—
59	#95×C₁₀₉	—	2	2	no	—
60	#95×C₁₀₉	—	2	2	no	—
61	#95×C₁₀₉	—	2	2	no	—
62	#95×C₁₀₉	—	2	2	no	—
63	#95×C₁₀₉	—	2	2	no	—
64	#95×C₁₀₉	D₄	1	2	yes	—
65	#95×C₁₀₉	D₄	1	2	yes	—
66	C₂²×C₄×C₁₀₉	—	2	2	no	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	—	2	2	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	—	2	2	no	—
78	C₂×C₈×C₁₀₉	—	2	2	no	—
79	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₁₀₉	—	2	8	no	—
82	C₄⋊C₄×C₁₀₉	—	2	2	no	—
83	C₄⋊C₄×C₁₀₉	—	2	2	no	—
84	C₄⋊C₄×C₁₀₉	—	2	2	no	—
85	C₄⋊C₄×C₁₀₉	—	2	2	no	—
86	C₄⋊C₄×C₁₀₉	—	2	2	no	—
87	C₄⋊C₄×C₁₀₉	—	2	2	no	—
88	C₄⋊C₄×C₁₀₉	—	2	2	no	—
89	C₄⋊C₄×C₁₀₉	—	8	2	no	—
90	C₄⋊C₄×C₁₀₉	—	8	2	no	—
91	C₄⋊C₄×C₁₀₉	—	8	2	no	—
92	C₄⋊C₄×C₁₀₉	—	8	2	no	—
93	C₄⋊C₄×C₁₀₉	—	8	2	no	—
94	C₄⋊C₄×C₁₀₉	#41	1	4	yes	—
95	C₄²×C₁₀₉	—	2	2	no	—
96	C₄²×C₁₀₉	—	2	1	no	—
97	C₄²×C₁₀₉	—	2	1	no	—
98	C₄²×C₁₀₉	—	8	2	no	—
99	C₄²×C₁₀₉	C₂×Q₈	1	4	yes	—
100	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	—	2	2	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	2	2	no	—
104	C₂×C₄×C₁₀₉	—	2	4	no	—
105	C₂×C₄×C₁₀₉	—	2	1	no	—
106	C₂×C₄×C₁₀₉	—	2	1	no	—
107	C₂×C₄×C₁₀₉	—	2	2	no	—
108	C₂×C₄×C₁₀₉	—	2	2	no	—
109	C₂×C₄×C₁₀₉	—	2	2	no	—
110	C₂×C₄×C₁₀₉	—	2	2	no	—
111	C₂×C₄×C₁₀₉	—	4	2	no	—
112	C₂×C₄×C₁₀₉	—	4	2	no	—
113	C₂×C₄×C₁₀₉	—	4	2	no	—
114	C₂×C₄×C₁₀₉	—	4	2	no	—
115	C₂×C₄×C₁₀₉	—	4	2	no	—
116	C₂×C₄×C₁₀₉	—	4	2	no	—
117	C₂×C₄×C₁₀₉	—	4	2	no	—
118	C₂×C₄×C₁₀₉	—	4	2	no	—
119	C₂×C₄×C₁₀₉	—	4	2	no	—
120	C₂×C₄×C₁₀₉	—	4	2	no	—
121	C₂×C₄×C₁₀₉	—	4	2	no	—
122	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
127	C₈×C₁₀₉	—	2	4	no	—
128	C₈×C₁₀₉	—	2	4	no	—
129	C₂²×C₁₀₉	#525	1	1	yes	characteristic
130	C₂²×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₁₀₉	#537	1	1	yes	characteristic
132	C₂²×C₁₀₉	#577	1	2	yes	—
133	C₂²×C₁₀₉	#577	1	2	yes	—
134	C₄×C₁₀₉	—	2	4	no	—
135	C₄×C₁₀₉	—	4	2	no	—
136	C₄×C₁₀₉	—	4	4	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	—	4	1	no	—
139	C₄×C₁₀₉	—	8	2	no	—
140	C₄×C₁₀₉	—	8	2	no	—
141	C₄×C₁₀₉	—	8	2	no	—
142	C₄×C₁₀₉	#554	1	2	yes	—
143	C₄×C₁₀₉	#557	1	2	yes	—
144	#29327	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₁₀₉	#2584	1	1	yes	characteristic
146	C₂×C₁₀₉	#3125	1	2	yes	—
147	C₂×C₁₀₉	#3163	1	2	yes	—
148	C₂×C₁₀₉	#3260	1	1	yes	characteristic
149	C₂×C₁₀₉	#3260	1	1	yes	characteristic
150	#2647	C₂×C₁₀₉	1	1	yes	characteristic
151	#2330	C₂×C₁₀₉	1	1	yes	characteristic
152	#2084	C₂×C₁₀₉	1	2	yes	—
153	#2059	C₂×C₁₀₉	1	2	yes	—
154	#1949	C₂×C₁₀₉	1	2	yes	—
155	#1948	C₂×C₁₀₉	1	2	yes	—
156	#1943	C₂×C₁₀₉	1	2	yes	—
157	#1941	C₂×C₁₀₉	1	2	yes	—
158	#1768	C₂×C₁₀₉	1	1	yes	characteristic
159	C₁₀₉	#29327	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic
161	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
162	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
163	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
164	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
166	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
168	#400	—	2	2	no	—
169	#398	—	2	2	no	—
170	#389	—	2	2	no	—
171	#389	C₂²×C₁₀₉	1	2	yes	—
172	#389	C₂²×C₁₀₉	1	2	yes	—
173	#389	C₂²×C₁₀₉	1	2	yes	—
174	#387	—	2	2	no	—
175	#384	C₂²×C₁₀₉	1	1	yes	characteristic
176	#384	C₂²×C₁₀₉	1	1	yes	characteristic
177	#382	C₂²×C₁₀₉	1	1	yes	characteristic
178	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
179	#376	C₂²×C₁₀₉	1	1	yes	characteristic
180	#340	C₂²×C₁₀₉	1	2	yes	—
181	#340	C₂²×C₁₀₉	1	2	yes	—
182	#340	C₂²×C₁₀₉	1	2	yes	—
183	#340	C₂²×C₁₀₉	1	2	yes	—
184	#336	C₂²×C₁₀₉	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
187	#125	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	#106	—	2	4	no	—
204	#106	—	4	2	no	—
205	#106	—	4	2	no	—
206	C₄⋊C₈	Q₈×C₁₀₉	1	8	yes	—
207	#97	—	4	1	no	—
208	C₄×C₈	—	4	1	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	D₄×C₁₀₉	1	2	yes	—
215	#95	D₄×C₁₀₉	1	2	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
230	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	#41×C₁₀₉	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	8	2	no	—
249	C₄²	C₂×Q₈×C₁₀₉	1	4	yes	—
250	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	4	yes	—
273	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
274	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
275	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
276	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	2	4	no	—
279	C₂²	#525×C₁₀₉	1	1	yes	characteristic
280	C₂²	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
281	C₂²	#537×C₁₀₉	1	1	yes	characteristic
282	C₂²	#577×C₁₀₉	1	2	yes	—
283	C₂²	#577×C₁₀₉	1	2	yes	—
284	C₄	—	2	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	#554×C₁₀₉	1	2	yes	—
293	C₄	#557×C₁₀₉	1	2	yes	—
294	C₂	C₁₀₉×#2584	1	1	yes	characteristic
295	C₂	C₁₀₉×#3125	1	2	yes	—
296	C₂	C₁₀₉×#3163	1	2	yes	—
297	C₂	C₁₀₉×#3260	1	1	yes	characteristic
298	C₂	C₁₀₉×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ #376×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
144. #29327	C₁₀₉	direct (N × H)
159. C₁₀₉	#29327	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2647	2
C₁₀₉×#2330	2
C₁₀₉×#2084	2
C₁₀₉×#2059	2
C₁₀₉×#1949	2
C₁₀₉×#1948	2
C₁₀₉×#1943	2
C₁₀₉×#1941	2
C₁₀₉×#1768	2
#29327	109

Subgroups — C109×#29327 #42147321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#29327 #42147321