Subgroups — C₁₀₉×#8655 #42126649

Order 27904.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1834	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1803	C₂	1	2	yes	maximal
4	C₂×#335×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	#446×C₁₀₉	—	2	2	no	—
8	#446×C₁₀₉	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#103×C₁₀₉	—	2	2	no	—
13	C₂×#97×C₁₀₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#384×C₁₀₉	C₂²	1	2	yes	—
16	#340×C₁₀₉	—	2	1	no	—
17	#340×C₁₀₉	—	2	1	no	—
18	#340×C₁₀₉	—	2	2	no	—
19	#335×C₁₀₉	—	2	2	no	—
20	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	—	2	1	no	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₀₉	—	2	1	no	—
26	C₄×Q₈×C₁₀₉	—	2	1	no	—
27	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₁₀₉	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	—	2	1	no	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₉	—	2	4	no	—
37	C₄⋊C₈×C₁₀₉	—	2	1	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
41	#103×C₁₀₉	—	4	4	no	—
42	#97×C₁₀₉	—	2	4	no	—
43	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂×Q₈×C₁₀₉	—	2	1	no	—
46	C₂×Q₈×C₁₀₉	—	2	2	no	—
47	C₂×Q₈×C₁₀₉	—	2	2	no	—
48	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	—	2	1	no	—
51	C₂²×C₄×C₁₀₉	—	4	2	no	—
52	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	—	2	2	no	—
56	C₂×C₈×C₁₀₉	—	2	2	no	—
57	C₂×C₈×C₁₀₉	—	2	2	no	—
58	C₂×C₈×C₁₀₉	—	4	2	no	—
59	C₂×C₈×C₁₀₉	—	4	2	no	—
60	C₂×C₈×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	—	4	2	no	—
62	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₄⋊C₄×C₁₀₉	—	4	2	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄⋊C₄×C₁₀₉	—	4	2	no	—
70	C₄⋊C₄×C₁₀₉	—	4	2	no	—
71	C₄⋊C₄×C₁₀₉	—	8	2	no	—
72	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	—	2	2	no	—
75	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₀₉	#41	1	1	yes	characteristic
79	C₂³×C₁₀₉	#121	1	1	yes	characteristic
80	Q₈×C₁₀₉	—	2	4	no	—
81	Q₈×C₁₀₉	—	4	1	no	—
82	Q₈×C₁₀₉	—	4	1	no	—
83	C₂×C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	4	2	no	—
88	C₂×C₄×C₁₀₉	—	4	1	no	—
89	C₂×C₄×C₁₀₉	—	4	1	no	—
90	C₂×C₄×C₁₀₉	—	8	2	no	—
91	C₂×C₄×C₁₀₉	—	8	2	no	—
92	C₂×C₄×C₁₀₉	—	8	2	no	—
93	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#135	1	2	yes	—
100	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₁₀₉	#137	1	2	yes	—
102	C₂×C₄×C₁₀₉	#137	1	2	yes	—
103	C₈×C₁₀₉	—	2	2	no	—
104	C₈×C₁₀₉	—	2	2	no	—
105	C₈×C₁₀₉	—	4	2	no	—
106	C₈×C₁₀₉	—	8	2	no	—
107	C₂²×C₁₀₉	#380	1	1	yes	characteristic
108	C₂²×C₁₀₉	#464	1	2	yes	—
109	C₂²×C₁₀₉	#468	1	2	yes	—
110	C₂²×C₁₀₉	#472	1	1	yes	characteristic
111	C₂²×C₁₀₉	#473	1	1	yes	characteristic
112	C₄×C₁₀₉	—	2	1	no	—
113	C₄×C₁₀₉	—	2	1	no	—
114	C₄×C₁₀₉	—	4	2	no	—
115	C₄×C₁₀₉	—	4	2	no	—
116	C₄×C₁₀₉	—	16	2	no	—
117	C₄×C₁₀₉	#416	1	1	yes	characteristic
118	C₄×C₁₀₉	#419	1	1	yes	characteristic
119	C₄×C₁₀₉	#428	1	2	yes	—
120	C₄×C₁₀₉	#469	1	1	yes	characteristic
121	C₄×C₁₀₉	#470	1	1	yes	characteristic
122	C₄×C₁₀₉	#479	1	2	yes	—
123	#8655	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₁₀₉	#1698	1	2	yes	—
125	C₂×C₁₀₉	#1700	1	2	yes	—
126	C₂×C₁₀₉	#1805	1	1	yes	characteristic
127	C₂×C₁₀₉	#1809	1	1	yes	characteristic
128	C₂×C₁₀₉	#1816	1	1	yes	characteristic
129	C₂×#446	C₂×C₁₀₉	1	1	yes	characteristic
130	#1834	C₂×C₁₀₉	1	1	yes	characteristic
131	#1803	C₂×C₁₀₉	1	2	yes	—
132	C₂×#335	C₂×C₁₀₉	1	1	yes	characteristic
133	#1296	C₂×C₁₀₉	1	2	yes	—
134	C₁₀₉	#8655	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
136	#446	—	2	2	no	—
137	#446	C₄×C₁₀₉	1	4	yes	complemented
138	C₂×C₄⋊C₈	—	2	1	no	—
139	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	C₂×#103	—	2	2	no	—
142	C₂×#97	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	#384	C₂²×C₁₀₉	1	2	yes	—
145	#340	—	2	1	no	—
146	#340	—	2	1	no	—
147	#340	—	2	2	no	—
148	#335	—	2	2	no	—
149	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
153	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
157	C₄×Q₈	Q₈×C₁₀₉	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
170	#103	—	4	4	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
173	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
183	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
192	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	8	2	no	—
201	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
205	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
207	C₄²	#41×C₁₀₉	1	1	yes	characteristic
208	C₂³	#121×C₁₀₉	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	1	no	—
211	Q₈	—	4	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄	#135×C₁₀₉	1	2	yes	—
229	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
230	C₂×C₄	#137×C₁₀₉	1	2	yes	—
231	C₂×C₄	#137×C₁₀₉	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	8	2	no	—
236	C₂²	#380×C₁₀₉	1	1	yes	characteristic
237	C₂²	#464×C₁₀₉	1	2	yes	—
238	C₂²	#468×C₁₀₉	1	2	yes	—
239	C₂²	#472×C₁₀₉	1	1	yes	characteristic
240	C₂²	#473×C₁₀₉	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	16	2	no	—
246	C₄	#416×C₁₀₉	1	1	yes	characteristic
247	C₄	#419×C₁₀₉	1	1	yes	characteristic
248	C₄	#428×C₁₀₉	1	2	yes	—
249	C₄	#469×C₁₀₉	1	1	yes	characteristic
250	C₄	#470×C₁₀₉	1	1	yes	characteristic
251	C₄	#479×C₁₀₉	1	2	yes	—
252	C₂	C₁₀₉×#1698	1	2	yes	—
253	C₂	C₁₀₉×#1700	1	2	yes	—
254	C₂	C₁₀₉×#1805	1	1	yes	characteristic
255	C₂	C₁₀₉×#1809	1	1	yes	characteristic
256	C₂	C₁₀₉×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	#121
commutator	C₄²	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. #446×C₁₀₉	C₄	semidirect
123. #8655	C₁₀₉	direct (N × H)
134. C₁₀₉	#8655	direct (N × H)
137. #446	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₀₉	2
C₁₀₉×#1834	2
C₁₀₉×#1803	2
C₂×#335×C₁₀₉	2
C₁₀₉×#1296	2
#8655	109

Subgroups — C109×#8655 #42126649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8655 #42126649