Subgroups — C₁₀₉×#9637 #42127631

Order 27904.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈×C₁₀₉	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₀₉	—	2	2	no	—
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
11	#391×C₁₀₉	C₄	1	2	yes	—
12	#389×C₁₀₉	—	2	2	no	—
13	#384×C₁₀₉	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₀₉	—	2	2	no	—
15	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
16	Q₈⋊C₈×C₁₀₉	—	2	8	no	—
17	C₂²×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₀₉	—	2	2	no	—
19	C₂²×C₈×C₁₀₉	—	2	2	no	—
20	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
21	C₄⋊Q₈×C₁₀₉	C₈	1	4	yes	complemented
22	C₄×Q₈×C₁₀₉	—	2	4	no	—
23	C₄×Q₈×C₁₀₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₉	—	2	2	no	—
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	—	2	4	no	—
35	C₄⋊C₈×C₁₀₉	—	4	4	no	—
36	C₄×C₈×C₁₀₉	—	2	8	no	—
37	C₄×C₈×C₁₀₉	—	2	2	no	—
38	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
39	#95×C₁₀₉	—	2	4	no	—
40	C₂×Q₈×C₁₀₉	—	2	2	no	—
41	C₂×Q₈×C₁₀₉	—	2	4	no	—
42	C₂×Q₈×C₁₀₉	—	2	4	no	—
43	C₂×Q₈×C₁₀₉	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₀₉	—	2	2	no	—
45	C₂²×C₄×C₁₀₉	—	2	2	no	—
46	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	—	2	8	no	—
52	C₂×C₈×C₁₀₉	—	2	4	no	—
53	C₂×C₈×C₁₀₉	—	4	2	no	—
54	C₂×C₈×C₁₀₉	—	4	4	no	—
55	C₄⋊C₄×C₁₀₉	—	2	8	no	—
56	C₄⋊C₄×C₁₀₉	—	2	4	no	—
57	C₄⋊C₄×C₁₀₉	—	4	4	no	—
58	C₄⋊C₄×C₁₀₉	—	4	4	no	—
59	C₄²×C₁₀₉	—	2	4	no	—
60	C₄²×C₁₀₉	—	2	2	no	—
61	C₄²×C₁₀₉	—	4	2	no	—
62	C₄²×C₁₀₉	—	4	2	no	—
63	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
64	C₄²×C₁₀₉	M₁₆	1	2	yes	—
65	C₄²×C₁₀₉	SD₁₆	1	4	yes	—
66	C₄²×C₁₀₉	Q₁₆	1	4	yes	—
67	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₄²×C₁₀₉	#41	1	2	yes	—
69	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	Q₈×C₁₀₉	—	2	8	no	—
71	Q₈×C₁₀₉	—	4	4	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	4	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	—	4	2	no	—
79	C₂×C₄×C₁₀₉	—	4	4	no	—
80	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₁₀₉	#103	1	4	yes	—
82	C₂×C₄×C₁₀₉	#103	1	4	yes	—
83	C₂×C₄×C₁₀₉	#104	1	4	yes	—
84	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	#121	1	2	yes	—
86	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	#131	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
93	C₈×C₁₀₉	—	2	8	no	—
94	C₈×C₁₀₉	—	8	2	no	—
95	C₂²×C₁₀₉	#390	1	1	yes	characteristic
96	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₀₉	#408	1	1	yes	characteristic
98	C₂²×C₁₀₉	C₂×#103	1	2	yes	—
99	C₂²×C₁₀₉	C₂×#104	1	2	yes	—
100	C₄×C₁₀₉	—	2	2	no	—
101	C₄×C₁₀₉	—	2	4	no	—
102	C₄×C₁₀₉	—	4	4	no	—
103	C₄×C₁₀₉	—	8	2	no	—
104	C₄×C₁₀₉	Q₈⋊C₈	1	8	yes	complemented
105	C₄×C₁₀₉	C₈×D₄	1	2	yes	—
106	C₄×C₁₀₉	#436	1	2	yes	—
107	C₄×C₁₀₉	#487	1	2	yes	—
108	C₄×C₁₀₉	#492	1	1	yes	characteristic
109	C₄×C₁₀₉	#494	1	1	yes	characteristic
110	#9637	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₁₀₉	C₂×Q₈⋊C₈	1	2	yes	—
112	C₂×C₁₀₉	#1493	1	2	yes	—
113	C₂×C₁₀₉	#1954	1	1	yes	characteristic
114	C₂×C₁₀₉	#1956	1	1	yes	characteristic
115	C₂×C₁₀₉	#1962	1	1	yes	characteristic
116	C₄×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
117	#1842	C₂×C₁₀₉	1	1	yes	characteristic
118	C₄²×C₈	C₂×C₁₀₉	1	1	yes	characteristic
119	C₂×Q₈⋊C₈	C₂×C₁₀₉	1	4	yes	—
120	C₁₀₉	#9637	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
122	C₂×C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
125	C₂×C₄×C₈	—	2	2	no	—
126	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
127	#391	C₄×C₁₀₉	1	2	yes	—
128	#389	—	2	2	no	—
129	#384	C₄×C₁₀₉	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	2	2	no	—
131	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
132	Q₈⋊C₈	—	2	8	no	—
133	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₈×C₁₀₉	1	4	yes	complemented
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
148	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
149	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	4	4	no	—
152	C₄×C₈	—	2	8	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
155	#95	—	2	4	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	C₂×C₈×C₁₀₉	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
164	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
180	C₄²	M₁₆×C₁₀₉	1	2	yes	—
181	C₄²	SD₁₆×C₁₀₉	1	4	yes	—
182	C₄²	Q₁₆×C₁₀₉	1	4	yes	—
183	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
184	C₄²	#41×C₁₀₉	1	2	yes	—
185	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	Q₈	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	4	yes	—
197	C₂×C₄	#103×C₁₀₉	1	4	yes	—
198	C₂×C₄	#103×C₁₀₉	1	4	yes	—
199	C₂×C₄	#104×C₁₀₉	1	4	yes	—
200	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
201	C₂×C₄	#121×C₁₀₉	1	2	yes	—
202	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄	#131×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
208	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
209	C₈	—	2	8	no	—
210	C₈	—	8	2	no	—
211	C₂²	#390×C₁₀₉	1	1	yes	characteristic
212	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
213	C₂²	#408×C₁₀₉	1	1	yes	characteristic
214	C₂²	C₂×#103×C₁₀₉	1	2	yes	—
215	C₂²	C₂×#104×C₁₀₉	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	Q₈⋊C₈×C₁₀₉	1	8	yes	complemented
221	C₄	C₈×D₄×C₁₀₉	1	2	yes	—
222	C₄	#436×C₁₀₉	1	2	yes	—
223	C₄	#487×C₁₀₉	1	2	yes	—
224	C₄	#492×C₁₀₉	1	1	yes	characteristic
225	C₄	#494×C₁₀₉	1	1	yes	characteristic
226	C₂	C₂×Q₈⋊C₈×C₁₀₉	1	2	yes	—
227	C₂	C₁₀₉×#1493	1	2	yes	—
228	C₂	C₁₀₉×#1954	1	1	yes	characteristic
229	C₂	C₁₀₉×#1956	1	1	yes	characteristic
230	C₂	C₁₀₉×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₄⋊Q₈×C₁₀₉	C₈	semidirect
104. C₄×C₁₀₉	Q₈⋊C₈	semidirect
110. #9637	C₁₀₉	direct (N × H)
120. C₁₀₉	#9637	direct (N × H)
137. C₄⋊Q₈	C₈×C₁₀₉	semidirect
220. C₄	Q₈⋊C₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₀₉	2
C₁₀₉×#1842	2
C₄²×C₈×C₁₀₉	2
C₂×Q₈⋊C₈×C₁₀₉	2
#9637	109

Subgroups — C109×#9637 #42127631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9637 #42127631