Subgroups — C₁₀₉×#29143 #42147137

Order 27904.

890 conjugacy classes of subgroups.

The subgroup lattice has 282 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, complemented
2	C₄×#127×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2025	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#1957	C₂	1	2	yes	maximal
5	C₁₀₉×#1834	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1804	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₀₉	—	2	2	no	—
11	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂×#102×C₁₀₉	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
15	#390×C₁₀₉	—	2	2	no	—
16	#390×C₁₀₉	C₂²	1	2	yes	—
17	#389×C₁₀₉	—	2	2	no	—
18	#389×C₁₀₉	—	2	2	no	—
19	#384×C₁₀₉	C₂²	1	2	yes	—
20	#384×C₁₀₉	C₂²	1	1	yes	characteristic
21	#382×C₁₀₉	C₂²	1	2	yes	—
22	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
23	#340×C₁₀₉	C₂²	1	8	yes	—
24	#336×C₁₀₉	C₂²	1	2	yes	—
25	C₂²×D₄×C₁₀₉	—	2	2	no	—
26	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₀₉	—	2	2	no	—
28	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
29	#127×C₁₀₉	—	2	2	no	—
30	#127×C₁₀₉	Q₈	1	4	yes	—
31	#125×C₁₀₉	—	4	4	no	—
32	C₄×D₄×C₁₀₉	—	2	8	no	—
33	C₄×D₄×C₁₀₉	—	4	8	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
43	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
44	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₁₀₉	—	2	4	no	—
48	#102×C₁₀₉	—	4	8	no	—
49	C₄×C₈×C₁₀₉	—	2	4	no	—
50	#95×C₁₀₉	—	2	2	no	—
51	#95×C₁₀₉	—	2	2	no	—
52	C₂⁴×C₁₀₉	—	2	2	no	—
53	C₂×D₄×C₁₀₉	—	2	8	no	—
54	C₂×D₄×C₁₀₉	—	2	4	no	—
55	C₂×D₄×C₁₀₉	—	2	2	no	—
56	C₂×D₄×C₁₀₉	—	2	4	no	—
57	C₂×D₄×C₁₀₉	—	4	4	no	—
58	C₂×D₄×C₁₀₉	—	4	4	no	—
59	C₂×D₄×C₁₀₉	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₁₀₉	—	2	4	no	—
61	C₂²×C₄×C₁₀₉	—	2	2	no	—
62	C₂²×C₄×C₁₀₉	—	4	8	no	—
63	C₂²×C₄×C₁₀₉	—	4	4	no	—
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	—	2	4	no	—
70	C₂×C₈×C₁₀₉	—	2	4	no	—
71	C₂×C₈×C₁₀₉	—	2	4	no	—
72	C₄⋊C₄×C₁₀₉	—	2	4	no	—
73	C₄⋊C₄×C₁₀₉	—	2	8	no	—
74	C₄⋊C₄×C₁₀₉	—	4	8	no	—
75	C₄⋊C₄×C₁₀₉	—	4	4	no	—
76	C₄⋊C₄×C₁₀₉	—	4	4	no	—
77	C₄⋊C₄×C₁₀₉	—	4	4	no	—
78	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
79	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
80	C₄²×C₁₀₉	—	2	4	no	—
81	C₄²×C₁₀₉	—	2	4	no	—
82	C₄²×C₁₀₉	—	2	2	no	—
83	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₀₉	#41	1	2	yes	—
85	C₄²×C₁₀₉	#41	1	4	yes	—
86	C₂³×C₁₀₉	—	2	4	no	—
87	C₂³×C₁₀₉	—	4	2	no	—
88	C₂³×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	—	4	4	no	—
90	C₂³×C₁₀₉	—	4	4	no	—
91	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
92	D₄×C₁₀₉	—	2	8	no	—
93	D₄×C₁₀₉	—	4	4	no	—
94	D₄×C₁₀₉	—	4	16	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	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	4	no	—
102	C₂×C₄×C₁₀₉	—	4	4	no	—
103	C₂×C₄×C₁₀₉	—	4	4	no	—
104	C₂×C₄×C₁₀₉	—	4	4	no	—
105	C₂×C₄×C₁₀₉	—	8	8	no	—
106	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₁₀₉	#124	1	2	yes	—
108	C₂×C₄×C₁₀₉	#124	1	2	yes	—
109	C₂×C₄×C₁₀₉	#135	1	4	yes	—
110	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
115	C₈×C₁₀₉	—	4	4	no	—
116	C₂²×C₁₀₉	—	4	8	no	—
117	C₂²×C₁₀₉	—	4	4	no	—
118	C₂²×C₁₀₉	—	8	2	no	—
119	C₂²×C₁₀₉	—	8	2	no	—
120	C₂²×C₁₀₉	—	8	4	no	—
121	C₂²×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₁₀₉	C₂×#124	1	1	yes	characteristic
123	C₂²×C₁₀₉	#535	1	1	yes	characteristic
124	C₂²×C₁₀₉	C₂×#135	1	2	yes	—
125	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	—	4	2	no	—
129	C₄×C₁₀₉	—	8	2	no	—
130	C₄×C₁₀₉	—	8	4	no	—
131	C₄×C₁₀₉	#478	1	8	yes	—
132	C₄×C₁₀₉	#554	1	2	yes	—
133	C₄×C₁₀₉	#555	1	2	yes	—
134	#29143	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂×C₁₀₉	—	8	4	no	—
136	C₂×C₁₀₉	#2563	1	1	yes	characteristic
137	C₂×C₁₀₉	C₂×#478	1	2	yes	—
138	C₂×C₁₀₉	#3131	1	1	yes	characteristic
139	C₂×C₁₀₉	#3138	1	1	yes	characteristic
140	C₂×C₁₀₉	#3207	1	2	yes	—
141	#2859	C₂×C₁₀₉	1	1	yes	characteristic, complemented
142	C₄×#127	C₂×C₁₀₉	1	1	yes	characteristic
143	#2025	C₂×C₁₀₉	1	4	yes	complemented
144	#1957	C₂×C₁₀₉	1	2	yes	—
145	#1834	C₂×C₁₀₉	1	2	yes	complemented
146	#1804	C₂×C₁₀₉	1	4	yes	complemented
147	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
148	C₁₀₉	#29143	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₂×#125	C₂²×C₁₀₉	1	2	yes	—
151	C₂×C₄×D₄	—	2	2	no	—
152	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
154	C₂×#102	C₂²×C₁₀₉	1	8	yes	—
155	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
156	#390	—	2	2	no	—
157	#390	C₂²×C₁₀₉	1	2	yes	—
158	#389	—	2	2	no	—
159	#389	—	2	2	no	—
160	#384	C₂²×C₁₀₉	1	1	yes	characteristic
161	#384	C₂²×C₁₀₉	1	2	yes	—
162	#382	C₂²×C₁₀₉	1	2	yes	—
163	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
164	#340	C₂²×C₁₀₉	1	8	yes	—
165	#336	C₂²×C₁₀₉	1	2	yes	—
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
168	C₂³×C₄	—	2	2	no	—
169	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
170	#127	—	2	2	no	—
171	#127	Q₈×C₁₀₉	1	4	yes	—
172	#125	—	4	4	no	—
173	C₄×D₄	—	2	8	no	—
174	C₄×D₄	—	4	8	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	4	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
184	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	#102	—	4	8	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂⁴	—	2	2	no	—
194	C₂×D₄	—	2	8	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	C₂×Q₈×C₁₀₉	1	4	yes	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	2	no	—
224	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
225	C₄²	#41×C₁₀₉	1	2	yes	—
226	C₄²	#41×C₁₀₉	1	4	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
233	D₄	—	2	8	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	16	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	8	no	—
247	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
248	C₂×C₄	#124×C₁₀₉	1	2	yes	—
249	C₂×C₄	#124×C₁₀₉	1	2	yes	—
250	C₂×C₄	#135×C₁₀₉	1	4	yes	—
251	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
252	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
254	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	4	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
263	C₂²	C₂×#124×C₁₀₉	1	1	yes	characteristic
264	C₂²	#535×C₁₀₉	1	1	yes	characteristic
265	C₂²	C₂×#135×C₁₀₉	1	2	yes	—
266	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	2	no	—
272	C₄	#478×C₁₀₉	1	8	yes	—
273	C₄	#554×C₁₀₉	1	2	yes	—
274	C₄	#555×C₁₀₉	1	2	yes	—
275	C₂	—	8	4	no	—
276	C₂	C₁₀₉×#2563	1	1	yes	characteristic
277	C₂	C₂×#478×C₁₀₉	1	2	yes	—
278	C₂	C₁₀₉×#3131	1	1	yes	characteristic
279	C₂	C₁₀₉×#3138	1	1	yes	characteristic
280	C₂	C₁₀₉×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
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
1. C₁₀₉×#2859	C₂	semidirect
3. C₁₀₉×#2025	C₂	semidirect
5. C₁₀₉×#1834	C₂	semidirect
6. C₁₀₉×#1804	C₂	semidirect
7. C₄×C₄⋊C₈×C₁₀₉	C₂	semidirect
134. #29143	C₁₀₉	direct (N × H)
141. #2859	C₂×C₁₀₉	semidirect
143. #2025	C₂×C₁₀₉	semidirect
145. #1834	C₂×C₁₀₉	semidirect
146. #1804	C₂×C₁₀₉	semidirect
147. C₄×C₄⋊C₈	C₂×C₁₀₉	semidirect
148. C₁₀₉	#29143	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×#127×C₁₀₉	2
C₁₀₉×#2025	2
C₁₀₉×#1957	2
C₁₀₉×#1834	2
C₁₀₉×#1804	2
C₄×C₄⋊C₈×C₁₀₉	2
#29143	109

Subgroups — C109×#29143 #42147137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#29143 #42147137