Subgroups — C₁₀₉×#29210 #42147204

Order 27904.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2059	C₂	1	4	yes	maximal
4	C₁₀₉×#1991	C₂	1	2	yes	maximal
5	C₁₀₉×#1948	C₂	1	2	yes	maximal
6	C₁₀₉×#1947	C₂	1	2	yes	maximal
7	C₁₀₉×#1846	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#125×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
12	C₂×#106×C₁₀₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
16	#389×C₁₀₉	—	2	2	no	—
17	#389×C₁₀₉	—	2	2	no	—
18	#389×C₁₀₉	C₂²	1	2	yes	—
19	#387×C₁₀₉	—	2	2	no	—
20	#384×C₁₀₉	C₂²	1	2	yes	—
21	#382×C₁₀₉	C₂²	1	2	yes	—
22	#382×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
26	#340×C₁₀₉	C₂²	1	4	yes	—
27	#340×C₁₀₉	C₂²	1	4	yes	—
28	#336×C₁₀₉	C₂²	1	2	yes	—
29	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
31	#125×C₁₀₉	—	2	2	no	—
32	#125×C₁₀₉	—	4	2	no	—
33	#125×C₁₀₉	—	4	2	no	—
34	#125×C₁₀₉	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₀₉	—	2	2	no	—
44	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
45	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
49	#107×C₁₀₉	—	4	4	no	—
50	#106×C₁₀₉	—	4	4	no	—
51	C₄⋊C₈×C₁₀₉	Q₈	1	4	yes	—
52	C₄⋊C₈×C₁₀₉	Q₈	1	4	yes	—
53	C₄×C₈×C₁₀₉	—	2	4	no	—
54	#95×C₁₀₉	—	2	2	no	—
55	#95×C₁₀₉	—	2	2	no	—
56	#95×C₁₀₉	—	2	2	no	—
57	#95×C₁₀₉	—	2	4	no	—
58	C₂²×C₄×C₁₀₉	—	2	2	no	—
59	C₂²×C₄×C₁₀₉	—	2	2	no	—
60	C₂²×C₄×C₁₀₉	—	2	4	no	—
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₀₉	—	2	2	no	—
67	C₂×C₈×C₁₀₉	—	2	2	no	—
68	C₂×C₈×C₁₀₉	—	2	2	no	—
69	C₂×C₈×C₁₀₉	—	2	2	no	—
70	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
72	C₄⋊C₄×C₁₀₉	—	2	8	no	—
73	C₄⋊C₄×C₁₀₉	—	2	2	no	—
74	C₄⋊C₄×C₁₀₉	—	2	2	no	—
75	C₄⋊C₄×C₁₀₉	—	2	4	no	—
76	C₄⋊C₄×C₁₀₉	—	2	2	no	—
77	C₄⋊C₄×C₁₀₉	—	2	4	no	—
78	C₄⋊C₄×C₁₀₉	—	4	4	no	—
79	C₄⋊C₄×C₁₀₉	—	4	4	no	—
80	C₄⋊C₄×C₁₀₉	—	4	8	no	—
81	C₄⋊C₄×C₁₀₉	—	4	4	no	—
82	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
83	C₄²×C₁₀₉	—	2	4	no	—
84	C₄²×C₁₀₉	—	2	1	no	—
85	C₄²×C₁₀₉	—	2	1	no	—
86	C₄²×C₁₀₉	—	4	2	no	—
87	C₄²×C₁₀₉	—	4	4	no	—
88	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
89	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
91	C₄²×C₁₀₉	#41	1	2	yes	—
92	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	4	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	—	4	4	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	—	4	2	no	—
105	C₂×C₄×C₁₀₉	—	4	4	no	—
106	C₂×C₄×C₁₀₉	—	4	2	no	—
107	C₂×C₄×C₁₀₉	—	4	4	no	—
108	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₀₉	#135	1	4	yes	—
112	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₈×C₁₀₉	—	2	4	no	—
119	C₈×C₁₀₉	—	2	4	no	—
120	C₂²×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₁₀₉	#535	1	1	yes	characteristic
123	C₂²×C₁₀₉	C₂×#135	1	2	yes	—
124	C₂²×C₁₀₉	#575	1	2	yes	—
125	C₄×C₁₀₉	—	2	4	no	—
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	4	4	no	—
128	C₄×C₁₀₉	—	4	2	no	—
129	C₄×C₁₀₉	—	8	4	no	—
130	C₄×C₁₀₉	—	8	2	no	—
131	C₄×C₁₀₉	#499	1	4	yes	—
132	C₄×C₁₀₉	#554	1	1	yes	characteristic
133	C₄×C₁₀₉	#554	1	1	yes	characteristic
134	C₄×C₁₀₉	Q₈²	1	1	yes	characteristic
135	C₄×C₁₀₉	Q₈²	1	1	yes	characteristic
136	#29210	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₀₉	#2567	1	1	yes	characteristic
138	C₂×C₁₀₉	#3076	1	2	yes	—
139	C₂×C₁₀₉	C₂×#499	1	1	yes	characteristic
140	C₂×C₁₀₉	#3166	1	1	yes	characteristic
141	C₂×C₁₀₉	#3212	1	2	yes	—
142	#2859	C₂×C₁₀₉	1	1	yes	characteristic
143	C₄×#125	C₂×C₁₀₉	1	1	yes	characteristic
144	#2059	C₂×C₁₀₉	1	4	yes	—
145	#1991	C₂×C₁₀₉	1	2	yes	—
146	#1948	C₂×C₁₀₉	1	2	yes	—
147	#1947	C₂×C₁₀₉	1	2	yes	—
148	#1846	C₂×C₁₀₉	1	2	yes	—
149	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
150	C₁₀₉	#29210	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₂×#125	C₂²×C₁₀₉	1	2	yes	—
153	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
154	C₂×#106	C₂²×C₁₀₉	1	4	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
158	#389	—	2	2	no	—
159	#389	—	2	2	no	—
160	#389	C₂²×C₁₀₉	1	2	yes	—
161	#387	—	2	2	no	—
162	#384	C₂²×C₁₀₉	1	2	yes	—
163	#382	C₂²×C₁₀₉	1	1	yes	characteristic
164	#382	C₂²×C₁₀₉	1	2	yes	—
165	C₄×C₄⋊C₄	—	2	2	no	—
166	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
167	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
168	#340	C₂²×C₁₀₉	1	4	yes	—
169	#340	C₂²×C₁₀₉	1	4	yes	—
170	#336	C₂²×C₁₀₉	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
172	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
173	#125	—	2	2	no	—
174	#125	—	4	2	no	—
175	#125	—	4	2	no	—
176	#125	Q₈×C₁₀₉	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
187	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
188	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
191	#107	—	4	4	no	—
192	#106	—	4	4	no	—
193	C₄⋊C₈	Q₈×C₁₀₉	1	4	yes	—
194	C₄⋊C₈	Q₈×C₁₀₉	1	4	yes	—
195	C₄×C₈	—	2	4	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
231	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
232	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
233	C₄²	#41×C₁₀₉	1	2	yes	—
234	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
251	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
252	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
253	C₂×C₄	#135×C₁₀₉	1	4	yes	—
254	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	4	no	—
262	C₂²	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
264	C₂²	#535×C₁₀₉	1	1	yes	characteristic
265	C₂²	C₂×#135×C₁₀₉	1	2	yes	—
266	C₂²	#575×C₁₀₉	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	4	no	—
271	C₄	—	8	4	no	—
272	C₄	—	8	2	no	—
273	C₄	#499×C₁₀₉	1	4	yes	—
274	C₄	#554×C₁₀₉	1	1	yes	characteristic
275	C₄	#554×C₁₀₉	1	1	yes	characteristic
276	C₄	Q₈²×C₁₀₉	1	1	yes	characteristic
277	C₄	Q₈²×C₁₀₉	1	1	yes	characteristic
278	C₂	C₁₀₉×#2567	1	1	yes	characteristic
279	C₂	C₁₀₉×#3076	1	2	yes	—
280	C₂	C₂×#499×C₁₀₉	1	1	yes	characteristic
281	C₂	C₁₀₉×#3166	1	1	yes	characteristic
282	C₂	C₁₀₉×#3212	1	2	yes	—

	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 ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
136. #29210	C₁₀₉	direct (N × H)
150. C₁₀₉	#29210	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×#2859	2
C₄×#125×C₁₀₉	2
C₁₀₉×#2059	2
C₁₀₉×#1991	2
C₁₀₉×#1948	2
C₁₀₉×#1947	2
C₁₀₉×#1846	2
C₄×C₄⋊C₈×C₁₀₉	2
#29210	109

Subgroups — C109×#29210 #42147204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#29210 #42147204