Subgroups — C₂×C₁₀₉×#2614 #42152592

Order 27904.

5194 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#123×C₁₀₉	C₂	1	3	yes	maximal
3	C₁₀₉×#2614	C₂	1	16	yes	maximal, direct product factor
4	C₂×#395×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	C₂×#393×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	C₂×#381×C₁₀₉	C₂	1	3	yes	maximal, complemented
7	C₂³×Q₈×C₁₀₉	—	2	24	no	—
8	C₂³×Q₈×C₁₀₉	C₂²	1	4	yes	—
9	C₂³×Q₈×C₁₀₉	C₂²	1	8	yes	complemented
10	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₀₉	C₂²	1	3	yes	—
12	C₂×#123×C₁₀₉	—	2	24	no	—
13	C₂×#123×C₁₀₉	C₂²	1	24	yes	—
14	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	6	yes	—
15	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
16	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	6	yes	—
17	#395×C₁₀₉	C₂²	1	32	yes	complemented
18	#393×C₁₀₉	C₂²	1	32	yes	complemented
19	#381×C₁₀₉	C₂²	1	24	yes	complemented
20	C₂×#95×C₁₀₉	C₂²	1	12	yes	—
21	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂²×Q₈×C₁₀₉	—	2	128	no	—
24	C₂²×Q₈×C₁₀₉	—	2	48	no	—
25	C₂²×Q₈×C₁₀₉	—	2	96	no	—
26	C₂²×Q₈×C₁₀₉	C₂³	1	16	yes	—
27	C₂²×D₄×C₁₀₉	—	2	12	no	—
28	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₀₉	—	2	24	no	—
30	C₂³×C₄×C₁₀₉	D₄	1	24	yes	—
31	C₂³×C₄×C₁₀₉	C₂³	1	6	yes	—
32	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₉	C₂³	1	12	yes	—
34	C₂³×C₄×C₁₀₉	C₂³	1	6	yes	—
35	#123×C₁₀₉	—	2	96	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	24	no	—
37	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	24	yes	—
38	C₂×C₂²⋊C₄×C₁₀₉	—	2	12	no	—
39	C₂×C₂²⋊C₄×C₁₀₉	—	2	24	no	—
40	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
41	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	24	yes	—
42	#95×C₁₀₉	C₂³	1	48	yes	—
43	C₂⁴×C₁₀₉	—	2	8	no	—
44	C₂⁴×C₁₀₉	—	2	6	no	—
45	C₂⁴×C₁₀₉	—	2	8	no	—
46	C₂⁴×C₁₀₉	C₂×D₄	1	12	yes	—
47	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
48	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
49	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₁₀₉	—	2	128	no	—
51	C₂×Q₈×C₁₀₉	—	2	256	no	—
52	C₂×Q₈×C₁₀₉	—	2	96	no	—
53	C₂×D₄×C₁₀₉	—	2	24	no	—
54	C₂×D₄×C₁₀₉	—	4	16	no	—
55	C₂²×C₄×C₁₀₉	—	2	96	no	—
56	C₂²×C₄×C₁₀₉	—	2	48	no	—
57	C₂²×C₄×C₁₀₉	—	2	24	no	—
58	C₂²×C₄×C₁₀₉	—	2	12	no	—
59	C₂²×C₄×C₁₀₉	—	2	6	no	—
60	C₂²×C₄×C₁₀₉	—	2	24	no	—
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	24	yes	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	48	yes	—
63	C₂²×C₄×C₁₀₉	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
65	C₂²×C₄×C₁₀₉	C₂⁴	1	12	yes	—
66	C₄⋊C₄×C₁₀₉	—	2	48	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	24	no	—
68	C₂²⋊C₄×C₁₀₉	—	2	48	no	—
69	C₂³×C₁₀₉	—	2	48	no	—
70	C₂³×C₁₀₉	—	2	8	no	—
71	C₂³×C₁₀₉	—	2	6	no	—
72	C₂³×C₁₀₉	—	2	8	no	—
73	C₂³×C₁₀₉	—	4	4	no	—
74	C₂³×C₁₀₉	—	4	24	no	—
75	C₂³×C₁₀₉	#120	1	16	yes	—
76	C₂³×C₁₀₉	C₂²×D₄	1	12	yes	—
77	C₂³×C₁₀₉	C₂²×D₄	1	6	yes	—
78	C₂³×C₁₀₉	2₋¹⁺⁴	1	8	yes	—
79	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
80	Q₈×C₁₀₉	—	2	128	no	—
81	D₄×C₁₀₉	—	4	16	no	—
82	C₂×C₄×C₁₀₉	—	2	96	no	—
83	C₂×C₄×C₁₀₉	—	2	48	no	—
84	C₂×C₄×C₁₀₉	—	2	24	no	—
85	C₂×C₄×C₁₀₉	—	2	12	no	—
86	C₂×C₄×C₁₀₉	—	2	6	no	—
87	C₂×C₄×C₁₀₉	—	2	24	no	—
88	C₂×C₄×C₁₀₉	—	4	4	no	—
89	C₂×C₄×C₁₀₉	—	4	24	no	—
90	C₂×C₄×C₁₀₉	C₂²×D₄	1	24	yes	—
91	C₂²×C₁₀₉	—	2	32	no	—
92	C₂²×C₁₀₉	—	2	24	no	—
93	C₂²×C₁₀₉	—	4	2	no	—
94	C₂²×C₁₀₉	—	4	12	no	—
95	C₂²×C₁₀₉	—	8	8	no	—
96	C₂²×C₁₀₉	C₂×#120	1	8	yes	—
97	C₂²×C₁₀₉	C₂×#120	1	4	yes	—
98	C₂²×C₁₀₉	#536	1	24	yes	—
99	C₂²×C₁₀₉	C₂³×D₄	1	3	yes	—
100	C₂²×C₁₀₉	C₂×2₋¹⁺⁴	1	4	yes	—
101	C₄×C₁₀₉	—	2	24	no	—
102	C₄×C₁₀₉	—	4	2	no	—
103	C₄×C₁₀₉	—	4	12	no	—
104	C₂×#2614	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂×C₁₀₉	—	2	8	no	—
106	C₂×C₁₀₉	—	8	2	no	—
107	C₂×C₁₀₉	#2614	1	8	yes	direct product factor
108	C₂×C₁₀₉	C₂²×#120	1	1	yes	characteristic
109	C₂×C₁₀₉	C₂×#536	1	6	yes	—
110	C₂⁴×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
111	C₂²×#123	C₂×C₁₀₉	1	3	yes	—
112	#2614	C₂×C₁₀₉	1	16	yes	direct product factor
113	C₂×#395	C₂×C₁₀₉	1	4	yes	complemented
114	C₂×#393	C₂×C₁₀₉	1	4	yes	complemented
115	C₂×#381	C₂×C₁₀₉	1	3	yes	complemented
116	C₁₀₉	C₂×#2614	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×Q₈	—	2	24	no	—
118	C₂³×Q₈	C₂²×C₁₀₉	1	4	yes	—
119	C₂³×Q₈	C₂²×C₁₀₉	1	8	yes	complemented
120	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
121	C₂⁴×C₄	C₂²×C₁₀₉	1	3	yes	—
122	C₂×#123	—	2	24	no	—
123	C₂×#123	C₂²×C₁₀₉	1	24	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	6	yes	—
125	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	6	yes	—
126	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	3	yes	—
127	#395	C₂²×C₁₀₉	1	32	yes	complemented
128	#393	C₂²×C₁₀₉	1	32	yes	complemented
129	#381	C₂²×C₁₀₉	1	24	yes	complemented
130	C₂×#95	C₂²×C₁₀₉	1	12	yes	—
131	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
132	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
133	C₂²×Q₈	—	2	128	no	—
134	C₂²×Q₈	—	2	48	no	—
135	C₂²×Q₈	—	2	96	no	—
136	C₂²×Q₈	C₂³×C₁₀₉	1	16	yes	—
137	C₂²×D₄	—	2	12	no	—
138	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
139	C₂³×C₄	—	2	24	no	—
140	C₂³×C₄	D₄×C₁₀₉	1	24	yes	—
141	C₂³×C₄	C₂³×C₁₀₉	1	6	yes	—
142	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
143	C₂³×C₄	C₂³×C₁₀₉	1	6	yes	—
144	C₂³×C₄	C₂³×C₁₀₉	1	12	yes	—
145	#123	—	2	96	no	—
146	C₂×C₄⋊C₄	—	2	24	no	—
147	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	24	yes	—
148	C₂×C₂²⋊C₄	—	2	12	no	—
149	C₂×C₂²⋊C₄	—	2	24	no	—
150	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
151	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	24	yes	—
152	#95	C₂³×C₁₀₉	1	48	yes	—
153	C₂⁴	—	2	8	no	—
154	C₂⁴	—	2	6	no	—
155	C₂⁴	—	2	8	no	—
156	C₂⁴	C₂×D₄×C₁₀₉	1	12	yes	—
157	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
158	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
159	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
160	C₂×Q₈	—	2	128	no	—
161	C₂×Q₈	—	2	256	no	—
162	C₂×Q₈	—	2	96	no	—
163	C₂×D₄	—	2	24	no	—
164	C₂×D₄	—	4	16	no	—
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	48	no	—
167	C₂²×C₄	—	2	24	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	6	no	—
170	C₂²×C₄	—	2	24	no	—
171	C₂²×C₄	C₂×D₄×C₁₀₉	1	24	yes	—
172	C₂²×C₄	C₂×D₄×C₁₀₉	1	48	yes	—
173	C₂²×C₄	C₂⁴×C₁₀₉	1	12	yes	—
174	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
175	C₂²×C₄	C₂⁴×C₁₀₉	1	12	yes	—
176	C₄⋊C₄	—	2	48	no	—
177	C₂²⋊C₄	—	2	24	no	—
178	C₂²⋊C₄	—	2	48	no	—
179	C₂³	—	2	48	no	—
180	C₂³	—	2	8	no	—
181	C₂³	—	2	6	no	—
182	C₂³	—	2	8	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	24	no	—
185	C₂³	#120×C₁₀₉	1	16	yes	—
186	C₂³	C₂²×D₄×C₁₀₉	1	12	yes	—
187	C₂³	C₂²×D₄×C₁₀₉	1	6	yes	—
188	C₂³	2₋¹⁺⁴×C₁₀₉	1	8	yes	—
189	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
190	Q₈	—	2	128	no	—
191	D₄	—	4	16	no	—
192	C₂×C₄	—	2	96	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	2	24	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	24	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	24	no	—
200	C₂×C₄	C₂²×D₄×C₁₀₉	1	24	yes	—
201	C₂²	—	2	32	no	—
202	C₂²	—	2	24	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	12	no	—
205	C₂²	—	8	8	no	—
206	C₂²	C₂×#120×C₁₀₉	1	8	yes	—
207	C₂²	C₂×#120×C₁₀₉	1	4	yes	—
208	C₂²	#536×C₁₀₉	1	24	yes	—
209	C₂²	C₂³×D₄×C₁₀₉	1	3	yes	—
210	C₂²	C₂×2₋¹⁺⁴×C₁₀₉	1	4	yes	—
211	C₄	—	2	24	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	12	no	—
214	C₂	—	2	8	no	—
215	C₂	—	8	2	no	—
216	C₂	C₁₀₉×#2614	1	8	yes	direct product factor
217	C₂	C₂²×#120×C₁₀₉	1	1	yes	characteristic
218	C₂	C₂×#536×C₁₀₉	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂³×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2614	C₂	direct (N × H)
4. C₂×#395×C₁₀₉	C₂	semidirect
5. C₂×#393×C₁₀₉	C₂	semidirect
6. C₂×#381×C₁₀₉	C₂	semidirect
9. C₂³×Q₈×C₁₀₉	C₂²	semidirect
17. #395×C₁₀₉	C₂²	semidirect
18. #393×C₁₀₉	C₂²	semidirect
19. #381×C₁₀₉	C₂²	semidirect
104. C₂×#2614	C₁₀₉	direct (N × H)
107. C₂×C₁₀₉	#2614	direct (N × H)
110. C₂⁴×Q₈	C₂×C₁₀₉	semidirect
112. #2614	C₂×C₁₀₉	direct (N × H)
113. C₂×#395	C₂×C₁₀₉	semidirect
114. C₂×#393	C₂×C₁₀₉	semidirect
115. C₂×#381	C₂×C₁₀₉	semidirect
116. C₁₀₉	C₂×#2614	direct (N × H)
119. C₂³×Q₈	C₂²×C₁₀₉	semidirect
127. #395	C₂²×C₁₀₉	semidirect
128. #393	C₂²×C₁₀₉	semidirect
129. #381	C₂²×C₁₀₉	semidirect
216. C₂	C₁₀₉×#2614	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₀₉	2
C₂²×#123×C₁₀₉	2
C₁₀₉×#2614	2
C₂×#395×C₁₀₉	2
C₂×#393×C₁₀₉	2
C₂×#381×C₁₀₉	2
C₂×#2614	109

Subgroups — C2×C109×#2614 #42152592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#2614 #42152592