Subgroups — C₁₀₉×#8184 #42126178

Order 27904.

530 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1476	C₂	1	2	yes	maximal
3	C₁₀₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1278	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₀₉	C₄	1	2	yes	—
7	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
8	#431×C₁₀₉	C₂²	1	2	yes	—
9	#405×C₁₀₉	—	2	2	no	—
10	#405×C₁₀₉	C₄	1	2	yes	—
11	C₄×M₁₆×C₁₀₉	—	2	1	no	—
12	C₄×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	Q₈⋊C₈×C₁₀₉	C₄	1	4	yes	—
17	D₄⋊C₈×C₁₀₉	C₄	1	4	yes	—
18	#322×C₁₀₉	—	2	1	no	—
19	C₈²×C₁₀₉	—	2	2	no	—
20	C₈²×C₁₀₉	—	2	1	no	—
21	C₂×#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	#131×C₁₀₉	—	2	4	no	—
23	#131×C₁₀₉	C₈	1	8	yes	complemented
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₀₉	—	2	2	no	—
27	C₂²×C₈×C₁₀₉	—	2	2	no	—
28	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₁₀₉	—	2	2	no	—
31	C₄×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	#117×C₁₀₉	—	2	1	no	—
33	#117×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	—	2	1	no	—
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
38	#97×C₁₀₉	—	2	2	no	—
39	#97×C₁₀₉	—	2	1	no	—
40	C₄×C₈×C₁₀₉	—	2	2	no	—
41	C₄×C₈×C₁₀₉	—	2	2	no	—
42	C₄×C₈×C₁₀₉	—	2	2	no	—
43	C₄×C₈×C₁₀₉	—	2	1	no	—
44	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
46	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
48	#41×C₁₀₉	—	2	2	no	—
49	#41×C₁₀₉	C₂×C₈	1	4	yes	—
50	C₂×Q₈×C₁₀₉	C₄²	1	1	yes	characteristic
51	C₂×D₄×C₁₀₉	—	2	1	no	—
52	C₂×D₄×C₁₀₉	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	—	2	1	no	—
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
57	M₁₆×C₁₀₉	—	2	2	no	—
58	M₁₆×C₁₀₉	—	2	8	no	—
59	M₁₆×C₁₀₉	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₁₀₉	—	2	8	no	—
61	C₂×C₈×C₁₀₉	—	2	2	no	—
62	C₂×C₈×C₁₀₉	—	2	4	no	—
63	C₂×C₈×C₁₀₉	—	2	8	no	—
64	C₂×C₈×C₁₀₉	—	2	4	no	—
65	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
69	C₄⋊C₄×C₁₀₉	—	2	2	no	—
70	C₄⋊C₄×C₁₀₉	C₄²	1	2	yes	—
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	2	no	—
75	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₁₀₉	—	2	1	no	—
79	C₂³×C₁₀₉	#117	1	1	yes	characteristic
80	Q₈×C₁₀₉	—	2	1	no	—
81	Q₈×C₁₀₉	C₄×C₈	1	2	yes	—
82	D₄×C₁₀₉	—	2	4	no	—
83	D₄×C₁₀₉	—	2	1	no	—
84	D₄×C₁₀₉	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	4	no	—
87	C₂×C₄×C₁₀₉	—	2	4	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	4	4	no	—
90	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
96	C₈×C₁₀₉	—	2	8	no	—
97	C₈×C₁₀₉	—	2	4	no	—
98	C₈×C₁₀₉	—	4	4	no	—
99	C₈×C₁₀₉	C₂²⋊C₈	1	4	yes	—
100	C₈×C₁₀₉	C₂²⋊C₈	1	4	yes	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	4	2	no	—
104	C₂²×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₀₉	#431	1	2	yes	—
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	2	2	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	2	yes	—
112	C₄×C₁₀₉	#443	1	4	yes	—
113	#8184	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₁₀₉	—	2	2	no	—
115	C₂×C₁₀₉	—	4	2	no	—
116	C₂×C₁₀₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₁₀₉	#1764	1	2	yes	—
118	C₄×#131	C₂×C₁₀₉	1	1	yes	characteristic
119	#1476	C₂×C₁₀₉	1	2	yes	—
120	#1450	C₂×C₁₀₉	1	1	yes	characteristic, complemented
121	C₂×C₈²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
122	#1278	C₂×C₁₀₉	1	2	yes	complemented
123	C₁₀₉	#8184	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×#131	C₄×C₁₀₉	1	2	yes	—
125	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
126	#431	C₂²×C₁₀₉	1	2	yes	—
127	#405	—	2	2	no	—
128	#405	C₄×C₁₀₉	1	2	yes	—
129	C₄×M₁₆	—	2	1	no	—
130	C₄×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
134	Q₈⋊C₈	C₄×C₁₀₉	1	4	yes	—
135	D₄⋊C₈	C₄×C₁₀₉	1	4	yes	—
136	#322	—	2	1	no	—
137	C₈²	—	2	2	no	—
138	C₈²	—	2	1	no	—
139	C₂×#41	C₂×C₄×C₁₀₉	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	C₈×C₁₀₉	1	8	yes	complemented
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
147	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
150	#117	—	2	1	no	—
151	#117	C₂×C₄×C₁₀₉	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
155	C₂²⋊C₈	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	1	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
163	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
164	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
165	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
166	#41	—	2	2	no	—
167	#41	C₂×C₈×C₁₀₉	1	4	yes	—
168	C₂×Q₈	C₄²×C₁₀₉	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₄²×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
174	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	C₂×C₈×C₁₀₉	1	4	yes	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
184	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
185	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
186	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	C₄²×C₁₀₉	1	2	yes	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
194	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	#117×C₁₀₉	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	C₄×C₈×C₁₀₉	1	2	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	1	no	—
202	D₄	C₄×C₈×C₁₀₉	1	2	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
211	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
212	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
213	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
214	C₈	—	2	8	no	—
215	C₈	—	2	4	no	—
216	C₈	—	4	4	no	—
217	C₈	C₂²⋊C₈×C₁₀₉	1	4	yes	—
218	C₈	C₂²⋊C₈×C₁₀₉	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
223	C₂²	#431×C₁₀₉	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
228	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
229	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	2	yes	—
230	C₄	#443×C₁₀₉	1	4	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₄×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
234	C₂	C₁₀₉×#1764	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₀₉	D₄
commutator	C₄	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1450	C₂	semidirect
4. C₂×C₈²×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1278	C₂	semidirect
23. #131×C₁₀₉	C₈	semidirect
113. #8184	C₁₀₉	direct (N × H)
120. #1450	C₂×C₁₀₉	semidirect
121. C₂×C₈²	C₂×C₁₀₉	semidirect
122. #1278	C₂×C₁₀₉	semidirect
123. C₁₀₉	#8184	direct (N × H)
141. #131	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×#131×C₁₀₉	2
C₁₀₉×#1476	2
C₁₀₉×#1450	2
C₂×C₈²×C₁₀₉	2
C₁₀₉×#1278	2
#8184	109

Subgroups — C109×#8184 #42126178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8184 #42126178