Subgroups — C₂²×#382×C₁₀₉ #42152001

Order 27904.

4286 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₀₉	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₁₀₉	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₁₀₉	C₂²	1	4	yes	—
6	C₂⁴×C₄×C₁₀₉	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₀₉	—	2	16	no	—
9	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	24	yes	—
10	C₂²×C₄²×C₁₀₉	C₄	1	16	yes	complemented
11	C₂²×C₄²×C₁₀₉	C₂²	1	12	yes	—
12	#382×C₁₀₉	C₂²	1	64	yes	direct product factor
13	C₂×#95×C₁₀₉	C₂²	1	48	yes	—
14	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₁₀₉	—	2	16	no	—
16	C₂³×C₄×C₁₀₉	—	2	16	no	—
17	C₂³×C₄×C₁₀₉	—	2	32	no	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	16	yes	—
19	C₂³×C₄×C₁₀₉	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₁₀₉	D₄	1	8	yes	—
21	C₂³×C₄×C₁₀₉	Q₈	1	8	yes	—
22	C₂³×C₄×C₁₀₉	C₂³	1	24	yes	—
23	C₂³×C₄×C₁₀₉	C₂³	1	12	yes	—
24	C₂³×C₄×C₁₀₉	C₂³	1	6	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	96	no	—
26	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	32	yes	—
27	C₂×C₄²×C₁₀₉	C₂×C₄	1	96	yes	complemented
28	C₂×C₄²×C₁₀₉	C₂³	1	16	yes	—
29	#95×C₁₀₉	C₂³	1	64	yes	—
30	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
31	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
32	C₂⁴×C₁₀₉	C₂×Q₈	1	4	yes	—
33	C₂⁴×C₁₀₉	#41	1	16	yes	—
34	C₂⁴×C₁₀₉	C₂⁴	1	3	yes	—
35	C₂²×C₄×C₁₀₉	—	2	64	no	—
36	C₂²×C₄×C₁₀₉	—	2	48	no	—
37	C₂²×C₄×C₁₀₉	—	2	48	no	—
38	C₂²×C₄×C₁₀₉	—	2	96	no	—
39	C₂²×C₄×C₁₀₉	—	4	64	no	—
40	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	64	yes	—
41	C₂²×C₄×C₁₀₉	C₂²×C₄	1	48	yes	—
42	C₂²×C₄×C₁₀₉	C₂²×C₄	1	24	yes	—
43	C₂²×C₄×C₁₀₉	C₂×D₄	1	24	yes	—
44	C₂²×C₄×C₁₀₉	C₂×Q₈	1	24	yes	—
45	C₂²×C₄×C₁₀₉	C₂⁴	1	16	yes	—
46	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
47	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
48	C₄⋊C₄×C₁₀₉	—	2	64	no	—
49	C₄²×C₁₀₉	C₂²×C₄	1	64	yes	complemented
50	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	16	yes	—
51	C₂³×C₁₀₉	#117	1	32	yes	—
52	C₂³×C₁₀₉	#124	1	32	yes	—
53	C₂³×C₁₀₉	#125	1	32	yes	—
54	C₂³×C₁₀₉	C₂³×C₄	1	6	yes	—
55	C₂³×C₁₀₉	C₂²×D₄	1	6	yes	—
56	C₂³×C₁₀₉	C₂²×Q₈	1	6	yes	—
57	C₂³×C₁₀₉	C₂×#41	1	24	yes	—
58	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	—	2	96	no	—
60	C₂×C₄×C₁₀₉	—	2	16	no	—
61	C₂×C₄×C₁₀₉	—	2	16	no	—
62	C₂×C₄×C₁₀₉	—	2	32	no	—
63	C₂×C₄×C₁₀₉	—	4	96	no	—
64	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	96	yes	—
65	C₂×C₄×C₁₀₉	C₂³×C₄	1	16	yes	—
66	C₂×C₄×C₁₀₉	C₂³×C₄	1	8	yes	—
67	C₂×C₄×C₁₀₉	C₂²×D₄	1	8	yes	—
68	C₂×C₄×C₁₀₉	C₂²×Q₈	1	8	yes	—
69	C₂²×C₁₀₉	#382	1	64	yes	direct product factor
70	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	12	yes	—
71	C₂²×C₁₀₉	C₂×#117	1	24	yes	—
72	C₂²×C₁₀₉	C₂×#124	1	24	yes	—
73	C₂²×C₁₀₉	C₂×#125	1	24	yes	—
74	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
75	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
76	C₂²×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
77	C₂²×C₁₀₉	C₂²×#41	1	4	yes	—
78	C₄×C₁₀₉	—	2	16	no	—
79	C₄×C₁₀₉	—	4	16	no	—
80	C₄×C₁₀₉	C₂²×C₄⋊C₄	1	16	yes	—
81	C₂²×#382	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₀₉	C₂×#382	1	24	yes	direct product factor
83	C₂×C₁₀₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂²×#117	1	2	yes	—
85	C₂×C₁₀₉	C₂²×#124	1	2	yes	—
86	C₂×C₁₀₉	C₂²×#125	1	2	yes	—
87	C₂³×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	—
88	C₂³×C₄²	C₂×C₁₀₉	1	1	yes	characteristic
89	C₂×#382	C₂×C₁₀₉	1	24	yes	direct product factor
90	C₂²×#95	C₂×C₁₀₉	1	4	yes	—
91	C₁₀₉	C₂²×#382	1	1	yes	characteristic, direct product factor, Sylow
92	C₂⁴×C₄	C₂²×C₁₀₉	1	4	yes	—
93	C₂⁴×C₄	C₂²×C₁₀₉	1	2	yes	—
94	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
95	C₂²×C₄⋊C₄	—	2	16	no	—
96	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	24	yes	—
97	C₂²×C₄²	C₄×C₁₀₉	1	16	yes	complemented
98	C₂²×C₄²	C₂²×C₁₀₉	1	12	yes	—
99	#382	C₂²×C₁₀₉	1	64	yes	direct product factor
100	C₂×#95	C₂²×C₁₀₉	1	48	yes	—
101	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
102	C₂³×C₄	—	2	16	no	—
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	2	32	no	—
105	C₂³×C₄	C₂×C₄×C₁₀₉	1	16	yes	—
106	C₂³×C₄	C₂×C₄×C₁₀₉	1	8	yes	—
107	C₂³×C₄	D₄×C₁₀₉	1	8	yes	—
108	C₂³×C₄	Q₈×C₁₀₉	1	8	yes	—
109	C₂³×C₄	C₂³×C₁₀₉	1	24	yes	—
110	C₂³×C₄	C₂³×C₁₀₉	1	12	yes	—
111	C₂³×C₄	C₂³×C₁₀₉	1	6	yes	—
112	C₂×C₄⋊C₄	—	2	96	no	—
113	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	32	yes	—
114	C₂×C₄²	C₂×C₄×C₁₀₉	1	96	yes	complemented
115	C₂×C₄²	C₂³×C₁₀₉	1	16	yes	—
116	#95	C₂³×C₁₀₉	1	64	yes	—
117	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
118	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
119	C₂⁴	C₂×Q₈×C₁₀₉	1	4	yes	—
120	C₂⁴	#41×C₁₀₉	1	16	yes	—
121	C₂⁴	C₂⁴×C₁₀₉	1	3	yes	—
122	C₂²×C₄	—	2	64	no	—
123	C₂²×C₄	—	2	48	no	—
124	C₂²×C₄	—	2	48	no	—
125	C₂²×C₄	—	2	96	no	—
126	C₂²×C₄	—	4	64	no	—
127	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	64	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₀₉	1	48	yes	—
129	C₂²×C₄	C₂²×C₄×C₁₀₉	1	24	yes	—
130	C₂²×C₄	C₂×D₄×C₁₀₉	1	24	yes	—
131	C₂²×C₄	C₂×Q₈×C₁₀₉	1	24	yes	—
132	C₂²×C₄	C₂⁴×C₁₀₉	1	16	yes	—
133	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
134	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
135	C₄⋊C₄	—	2	64	no	—
136	C₄²	C₂²×C₄×C₁₀₉	1	64	yes	complemented
137	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	16	yes	—
138	C₂³	#117×C₁₀₉	1	32	yes	—
139	C₂³	#124×C₁₀₉	1	32	yes	—
140	C₂³	#125×C₁₀₉	1	32	yes	—
141	C₂³	C₂³×C₄×C₁₀₉	1	6	yes	—
142	C₂³	C₂²×D₄×C₁₀₉	1	6	yes	—
143	C₂³	C₂²×Q₈×C₁₀₉	1	6	yes	—
144	C₂³	C₂×#41×C₁₀₉	1	24	yes	—
145	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄	—	2	96	no	—
147	C₂×C₄	—	2	16	no	—
148	C₂×C₄	—	2	16	no	—
149	C₂×C₄	—	2	32	no	—
150	C₂×C₄	—	4	96	no	—
151	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	96	yes	—
152	C₂×C₄	C₂³×C₄×C₁₀₉	1	16	yes	—
153	C₂×C₄	C₂³×C₄×C₁₀₉	1	8	yes	—
154	C₂×C₄	C₂²×D₄×C₁₀₉	1	8	yes	—
155	C₂×C₄	C₂²×Q₈×C₁₀₉	1	8	yes	—
156	C₂²	#382×C₁₀₉	1	64	yes	direct product factor
157	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	12	yes	—
158	C₂²	C₂×#117×C₁₀₉	1	24	yes	—
159	C₂²	C₂×#124×C₁₀₉	1	24	yes	—
160	C₂²	C₂×#125×C₁₀₉	1	24	yes	—
161	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
162	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
163	C₂²	C₂³×Q₈×C₁₀₉	1	1	yes	characteristic
164	C₂²	C₂²×#41×C₁₀₉	1	4	yes	—
165	C₄	—	2	16	no	—
166	C₄	—	4	16	no	—
167	C₄	C₂²×C₄⋊C₄×C₁₀₉	1	16	yes	—
168	C₂	C₂×#382×C₁₀₉	1	24	yes	direct product factor
169	C₂	C₂³×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
170	C₂	C₂²×#117×C₁₀₉	1	2	yes	—
171	C₂	C₂²×#124×C₁₀₉	1	2	yes	—
172	C₂	C₂²×#125×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₁₀₉	C₂³
commutator	C₂²	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#382×C₁₀₉	C₂	direct (N × H)
10. C₂²×C₄²×C₁₀₉	C₄	semidirect
12. #382×C₁₀₉	C₂²	direct (N × H)
27. C₂×C₄²×C₁₀₉	C₂×C₄	semidirect
49. C₄²×C₁₀₉	C₂²×C₄	semidirect
69. C₂²×C₁₀₉	#382	direct (N × H)
81. C₂²×#382	C₁₀₉	direct (N × H)
82. C₂×C₁₀₉	C₂×#382	direct (N × H)
89. C₂×#382	C₂×C₁₀₉	direct (N × H)
91. C₁₀₉	C₂²×#382	direct (N × H)
97. C₂²×C₄²	C₄×C₁₀₉	semidirect
99. #382	C₂²×C₁₀₉	direct (N × H)
114. C₂×C₄²	C₂×C₄×C₁₀₉	semidirect
136. C₄²	C₂²×C₄×C₁₀₉	semidirect
156. C₂²	#382×C₁₀₉	direct (N × H)
168. C₂	C₂×#382×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₁₀₉	2
C₂³×C₄²×C₁₀₉	2
C₂×#382×C₁₀₉	2
C₂²×#95×C₁₀₉	2
C₂²×#382	109

Subgroups — C22×#382×C109 #42152001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#382×C₁₀₉ #42152001