Subgroups — C₁₀₉×#9647 #42127641

Order 27904.

558 conjugacy classes of subgroups.

The subgroup lattice has 216 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×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×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	C₂²	1	2	yes	—
8	C₂×#97×C₁₀₉	—	2	2	no	—
9	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
10	#391×C₁₀₉	C₄	1	2	yes	—
11	#389×C₁₀₉	—	2	2	no	—
12	#384×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₀₉	—	2	2	no	—
14	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
15	#336×C₁₀₉	—	2	2	no	—
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	4	no	—
21	C₄×Q₈×C₁₀₉	—	2	4	no	—
22	C₄×Q₈×C₁₀₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₀₉	—	2	2	no	—
28	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₁₀₉	—	2	4	no	—
34	#97×C₁₀₉	—	2	8	no	—
35	C₄×C₈×C₁₀₉	—	2	2	no	—
36	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
37	#95×C₁₀₉	—	2	4	no	—
38	C₂×Q₈×C₁₀₉	—	2	2	no	—
39	C₂×Q₈×C₁₀₉	—	2	4	no	—
40	C₂×Q₈×C₁₀₉	—	2	4	no	—
41	C₂×Q₈×C₁₀₉	M₁₆	1	4	yes	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	—	2	8	no	—
50	C₂×C₈×C₁₀₉	—	2	4	no	—
51	C₂×C₈×C₁₀₉	—	4	2	no	—
52	C₂×C₈×C₁₀₉	—	4	4	no	—
53	C₄⋊C₄×C₁₀₉	—	2	8	no	—
54	C₄⋊C₄×C₁₀₉	—	2	4	no	—
55	C₄⋊C₄×C₁₀₉	—	4	4	no	—
56	C₄⋊C₄×C₁₀₉	—	4	4	no	—
57	C₄²×C₁₀₉	—	2	4	no	—
58	C₄²×C₁₀₉	—	2	2	no	—
59	C₄²×C₁₀₉	—	2	2	no	—
60	C₄²×C₁₀₉	—	2	4	no	—
61	C₄²×C₁₀₉	—	4	2	no	—
62	C₄²×C₁₀₉	—	4	2	no	—
63	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₄²×C₁₀₉	#41	1	2	yes	—
65	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	Q₈×C₁₀₉	—	2	8	no	—
67	Q₈×C₁₀₉	—	4	4	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	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	—	4	2	no	—
76	C₂×C₄×C₁₀₉	—	4	2	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	—	4	4	no	—
79	C₂×C₄×C₁₀₉	#104	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₀₉	#121	1	2	yes	—
82	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₁₀₉	#131	1	2	yes	—
86	C₂×C₄×C₁₀₉	#137	1	4	yes	—
87	C₈×C₁₀₉	—	2	8	no	—
88	C₈×C₁₀₉	—	8	2	no	—
89	C₂²×C₁₀₉	#390	1	1	yes	characteristic
90	C₂²×C₁₀₉	#407	1	1	yes	characteristic
91	C₂²×C₁₀₉	#408	1	1	yes	characteristic
92	C₂²×C₁₀₉	#419	1	2	yes	—
93	C₂²×C₁₀₉	C₂×#104	1	2	yes	—
94	C₄×C₁₀₉	—	2	2	no	—
95	C₄×C₁₀₉	—	2	4	no	—
96	C₄×C₁₀₉	—	4	4	no	—
97	C₄×C₁₀₉	—	4	2	no	—
98	C₄×C₁₀₉	—	8	2	no	—
99	C₄×C₁₀₉	#435	1	4	yes	—
100	C₄×C₁₀₉	#491	1	2	yes	—
101	C₄×C₁₀₉	#497	1	2	yes	—
102	#9647	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₁₀₉	#1485	1	2	yes	—
104	C₂×C₁₀₉	#1491	1	2	yes	—
105	C₂×C₁₀₉	#1955	1	1	yes	characteristic
106	C₂×C₁₀₉	#1960	1	1	yes	characteristic
107	C₂×C₁₀₉	#1962	1	1	yes	characteristic
108	C₄×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
109	#1831	C₂×C₁₀₉	1	1	yes	characteristic
110	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic
111	C₂×Q₈⋊C₈	C₂×C₁₀₉	1	4	yes	—
112	C₁₀₉	#9647	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
116	C₂×#97	—	2	2	no	—
117	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
118	#391	C₄×C₁₀₉	1	2	yes	—
119	#389	—	2	2	no	—
120	#384	C₄×C₁₀₉	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
123	#336	—	2	2	no	—
124	Q₈⋊C₈	—	2	8	no	—
125	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	C₄×Q₈	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
138	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
139	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
140	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	#97	—	2	8	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
145	#95	—	2	4	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	M₁₆×C₁₀₉	1	4	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
153	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	4	no	—
161	C₄⋊C₄	—	2	8	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
172	C₄²	#41×C₁₀₉	1	2	yes	—
173	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	#104×C₁₀₉	1	4	yes	—
188	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
189	C₂×C₄	#121×C₁₀₉	1	2	yes	—
190	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
191	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
192	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
193	C₂×C₄	#131×C₁₀₉	1	2	yes	—
194	C₂×C₄	#137×C₁₀₉	1	4	yes	—
195	C₈	—	2	8	no	—
196	C₈	—	8	2	no	—
197	C₂²	#390×C₁₀₉	1	1	yes	characteristic
198	C₂²	#407×C₁₀₉	1	1	yes	characteristic
199	C₂²	#408×C₁₀₉	1	1	yes	characteristic
200	C₂²	#419×C₁₀₉	1	2	yes	—
201	C₂²	C₂×#104×C₁₀₉	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	8	2	no	—
207	C₄	#435×C₁₀₉	1	4	yes	—
208	C₄	#491×C₁₀₉	1	2	yes	—
209	C₄	#497×C₁₀₉	1	2	yes	—
210	C₂	C₁₀₉×#1485	1	2	yes	—
211	C₂	C₁₀₉×#1491	1	2	yes	—
212	C₂	C₁₀₉×#1955	1	1	yes	characteristic
213	C₂	C₁₀₉×#1960	1	1	yes	characteristic
214	C₂	C₁₀₉×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×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
102. #9647	C₁₀₉	direct (N × H)
112. C₁₀₉	#9647	direct (N × H)

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×#1831	2
C₄×#97×C₁₀₉	2
C₂×Q₈⋊C₈×C₁₀₉	2
#9647	109

Subgroups — C109×#9647 #42127641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9647 #42127641