Subgroups — C₁₀₉×#32815 #42150809

Order 27904.

746 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2642	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#2641	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2055	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2026	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#2025	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2014	C₂	1	2	yes	maximal
7	C₁₀₉×#2013	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1832	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#125×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂×#103×C₁₀₉	C₂²	1	4	yes	—
16	C₂×#102×C₁₀₉	C₂²	1	4	yes	—
17	#400×C₁₀₉	—	2	4	no	—
18	#394×C₁₀₉	—	2	4	no	—
19	#391×C₁₀₉	C₂²	1	2	yes	—
20	#391×C₁₀₉	C₂²	1	2	yes	—
21	#390×C₁₀₉	C₂²	1	2	yes	—
22	#390×C₁₀₉	C₂²	1	2	yes	—
23	#376×C₁₀₉	C₂²	1	1	yes	characteristic
24	#340×C₁₀₉	C₂²	1	4	yes	—
25	#336×C₁₀₉	C₂²	1	2	yes	—
26	#336×C₁₀₉	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₀₉	C₂³	1	2	yes	—
28	C₂²×D₄×C₁₀₉	—	2	2	no	—
29	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
30	C₂×SD₁₆×C₁₀₉	—	4	4	no	—
31	C₂×SD₁₆×C₁₀₉	—	4	4	no	—
32	C₂×SD₁₆×C₁₀₉	—	4	4	no	—
33	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
35	#127×C₁₀₉	—	4	2	no	—
36	#125×C₁₀₉	—	4	2	no	—
37	#125×C₁₀₉	—	4	2	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	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₁₀₉	—	4	2	no	—
49	#103×C₁₀₉	—	4	4	no	—
50	#102×C₁₀₉	—	4	4	no	—
51	#95×C₁₀₉	—	2	4	no	—
52	#95×C₁₀₉	—	2	4	no	—
53	#95×C₁₀₉	D₄	1	4	yes	—
54	C₂⁴×C₁₀₉	—	2	2	no	—
55	C₂×Q₈×C₁₀₉	—	2	4	no	—
56	C₂×Q₈×C₁₀₉	—	2	4	no	—
57	C₂×Q₈×C₁₀₉	—	2	2	no	—
58	C₂×Q₈×C₁₀₉	—	2	2	no	—
59	C₂×D₄×C₁₀₉	—	2	4	no	—
60	C₂×D₄×C₁₀₉	—	2	4	no	—
61	C₂×D₄×C₁₀₉	—	2	2	no	—
62	C₂×D₄×C₁₀₉	—	2	2	no	—
63	C₂×D₄×C₁₀₉	—	4	4	no	—
64	C₂×D₄×C₁₀₉	—	8	4	no	—
65	C₂²×C₄×C₁₀₉	—	2	2	no	—
66	C₂²×C₄×C₁₀₉	—	2	2	no	—
67	C₂²×C₄×C₁₀₉	—	2	2	no	—
68	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
72	SD₁₆×C₁₀₉	—	4	16	no	—
73	C₂×C₈×C₁₀₉	—	2	4	no	—
74	C₂×C₈×C₁₀₉	—	2	4	no	—
75	C₂×C₈×C₁₀₉	—	2	4	no	—
76	C₄⋊C₄×C₁₀₉	—	2	4	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	4	no	—
81	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
82	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
83	C₄²×C₁₀₉	—	2	1	no	—
84	C₄²×C₁₀₉	—	2	2	no	—
85	C₄²×C₁₀₉	—	2	2	no	—
86	C₄²×C₁₀₉	—	2	1	no	—
87	C₂³×C₁₀₉	—	4	2	no	—
88	C₂³×C₁₀₉	—	4	2	no	—
89	C₂³×C₁₀₉	—	4	2	no	—
90	C₂³×C₁₀₉	—	4	4	no	—
91	C₂³×C₁₀₉	—	4	4	no	—
92	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₁₀₉	—	4	4	no	—
94	Q₈×C₁₀₉	—	4	4	no	—
95	D₄×C₁₀₉	—	4	4	no	—
96	D₄×C₁₀₉	—	4	4	no	—
97	D₄×C₁₀₉	—	8	8	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	2	no	—
100	C₂×C₄×C₁₀₉	—	2	2	no	—
101	C₂×C₄×C₁₀₉	—	2	4	no	—
102	C₂×C₄×C₁₀₉	—	2	4	no	—
103	C₂×C₄×C₁₀₉	—	2	4	no	—
104	C₂×C₄×C₁₀₉	—	4	2	no	—
105	C₂×C₄×C₁₀₉	—	4	2	no	—
106	C₂×C₄×C₁₀₉	—	4	2	no	—
107	C₂×C₄×C₁₀₉	—	4	2	no	—
108	C₂×C₄×C₁₀₉	—	4	2	no	—
109	C₂×C₄×C₁₀₉	—	4	4	no	—
110	C₂×C₄×C₁₀₉	—	4	4	no	—
111	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
113	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
114	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
115	C₈×C₁₀₉	—	4	4	no	—
116	C₂²×C₁₀₉	—	8	2	no	—
117	C₂²×C₁₀₉	—	8	4	no	—
118	C₂²×C₁₀₉	—	8	2	no	—
119	C₂²×C₁₀₉	—	8	2	no	—
120	C₂²×C₁₀₉	—	8	4	no	—
121	C₂²×C₁₀₉	#535	1	1	yes	characteristic
122	C₂²×C₁₀₉	#535	1	1	yes	characteristic
123	C₂²×C₁₀₉	#537	1	1	yes	characteristic
124	C₂²×C₁₀₉	#577	1	4	yes	—
125	C₄×C₁₀₉	—	4	2	no	—
126	C₄×C₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	—	4	2	no	—
128	C₄×C₁₀₉	—	8	2	no	—
129	C₄×C₁₀₉	—	8	2	no	—
130	C₄×C₁₀₉	—	8	2	no	—
131	C₄×C₁₀₉	H(3,𝔽₄)	1	2	yes	—
132	C₄×C₁₀₉	#562	1	2	yes	—
133	#32815	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₀₉	—	16	2	no	—
135	C₂×C₁₀₉	#2819	1	1	yes	characteristic
136	C₂×C₁₀₉	#3221	1	2	yes	—
137	C₂×C₁₀₉	#3222	1	2	yes	—
138	C₂×C₁₀₉	#3260	1	2	yes	—
139	#2642	C₂×C₁₀₉	1	1	yes	characteristic, complemented
140	#2641	C₂×C₁₀₉	1	1	yes	characteristic
141	#2055	C₂×C₁₀₉	1	4	yes	complemented
142	#2026	C₂×C₁₀₉	1	2	yes	complemented
143	#2025	C₂×C₁₀₉	1	2	yes	complemented
144	#2014	C₂×C₁₀₉	1	2	yes	—
145	#2013	C₂×C₁₀₉	1	2	yes	complemented
146	#1832	C₂×C₁₀₉	1	1	yes	characteristic, complemented
147	C₁₀₉	#32815	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×SD₁₆	C₂²×C₁₀₉	1	4	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
150	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic
152	C₂×#125	C₂²×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
154	C₂×#103	C₂²×C₁₀₉	1	4	yes	—
155	C₂×#102	C₂²×C₁₀₉	1	4	yes	—
156	#400	—	2	4	no	—
157	#394	—	2	4	no	—
158	#391	C₂²×C₁₀₉	1	2	yes	—
159	#391	C₂²×C₁₀₉	1	2	yes	—
160	#390	C₂²×C₁₀₉	1	2	yes	—
161	#390	C₂²×C₁₀₉	1	2	yes	—
162	#376	C₂²×C₁₀₉	1	1	yes	characteristic
163	#340	C₂²×C₁₀₉	1	4	yes	—
164	#336	C₂²×C₁₀₉	1	2	yes	—
165	#336	C₂²×C₁₀₉	1	2	yes	—
166	C₂²×Q₈	C₂³×C₁₀₉	1	2	yes	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
169	C₂×SD₁₆	—	4	4	no	—
170	C₂×SD₁₆	—	4	4	no	—
171	C₂×SD₁₆	—	4	4	no	—
172	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
173	C₄⋊Q₈	—	4	2	no	—
174	#127	—	4	2	no	—
175	#125	—	4	2	no	—
176	#125	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
187	C₄⋊C₈	—	4	2	no	—
188	#103	—	4	4	no	—
189	#102	—	4	4	no	—
190	#95	—	2	4	no	—
191	#95	—	2	4	no	—
192	#95	D₄×C₁₀₉	1	4	yes	—
193	C₂⁴	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	8	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
211	SD₁₆	—	4	16	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₂²⋊C₄	—	8	4	no	—
221	C₂²⋊C₄	—	8	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
232	Q₈	—	4	4	no	—
233	Q₈	—	4	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	8	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
251	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
252	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
253	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
254	C₈	—	4	4	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	8	2	no	—
260	C₂²	#535×C₁₀₉	1	1	yes	characteristic
261	C₂²	#535×C₁₀₉	1	1	yes	characteristic
262	C₂²	#537×C₁₀₉	1	1	yes	characteristic
263	C₂²	#577×C₁₀₉	1	4	yes	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	H(3,𝔽₄)×C₁₀₉	1	2	yes	—
271	C₄	#562×C₁₀₉	1	2	yes	—
272	C₂	—	16	2	no	—
273	C₂	C₁₀₉×#2819	1	1	yes	characteristic
274	C₂	C₁₀₉×#3221	1	2	yes	—
275	C₂	C₁₀₉×#3222	1	2	yes	—
276	C₂	C₁₀₉×#3260	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 ◁ #376×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#2642	C₂	semidirect
3. C₁₀₉×#2055	C₂	semidirect
4. C₁₀₉×#2026	C₂	semidirect
5. C₁₀₉×#2025	C₂	semidirect
7. C₁₀₉×#2013	C₂	semidirect
8. C₁₀₉×#1832	C₂	semidirect
133. #32815	C₁₀₉	direct (N × H)
139. #2642	C₂×C₁₀₉	semidirect
141. #2055	C₂×C₁₀₉	semidirect
142. #2026	C₂×C₁₀₉	semidirect
143. #2025	C₂×C₁₀₉	semidirect
145. #2013	C₂×C₁₀₉	semidirect
146. #1832	C₂×C₁₀₉	semidirect
147. C₁₀₉	#32815	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×#2642	2
C₁₀₉×#2641	2
C₁₀₉×#2055	2
C₁₀₉×#2026	2
C₁₀₉×#2025	2
C₁₀₉×#2014	2
C₁₀₉×#2013	2
C₁₀₉×#1832	2
#32815	109

Subgroups — C109×#32815 #42150809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32815 #42150809