Subgroups — C₁₀₇×#33500 #41059464

Order 27392.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#2183	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₆×C₁₀₇	C₂	1	4	yes	maximal, complemented
4	C₁₀₇×#2164	C₂	1	4	yes	maximal
5	C₁₀₇×#2162	C₂	1	2	yes	maximal
6	#533×C₁₀₇	C₄	1	1	yes	characteristic
7	#446×C₁₀₇	C₄	1	4	yes	—
8	C₈×Q₈×C₁₀₇	C₂²	1	4	yes	—
9	#424×C₁₀₇	C₄	1	2	yes	—
10	#423×C₁₀₇	C₄	1	1	yes	characteristic
11	#405×C₁₀₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₄⋊C₁₆×C₁₀₇	C₂²	1	2	yes	—
15	C₄⋊C₁₆×C₁₀₇	C₂²	1	8	yes	—
16	C₂²⋊C₁₆×C₁₀₇	C₂²	1	4	yes	—
17	#346×C₁₀₇	C₂²	1	4	yes	—
18	C₄×C₁₆×C₁₀₇	C₂²	1	4	yes	—
19	C₄×C₁₆×C₁₀₇	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₀₇	C₈	1	2	yes	—
24	#125×C₁₀₇	C₈	1	2	yes	—
25	C₂²⋊Q₈×C₁₀₇	C₈	1	4	yes	—
26	C₄×Q₈×C₁₀₇	C₂×C₄	1	4	yes	—
27	#117×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₀₇	C₂³	1	4	yes	—
30	C₂×C₁₆×C₁₀₇	C₂³	1	4	yes	—
31	C₄⋊C₈×C₁₀₇	C₂×C₄	1	4	yes	—
32	C₄⋊C₈×C₁₀₇	C₂×C₄	1	8	yes	—
33	#97×C₁₀₇	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₀₇	C₂³	1	2	yes	—
36	C₄×C₈×C₁₀₇	C₂³	1	4	yes	—
37	C₂×Q₈×C₁₀₇	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₁₀₇	—	4	2	no	—
41	C₂×C₈×C₁₀₇	—	2	2	no	—
42	C₂×C₈×C₁₀₇	—	4	1	no	—
43	C₂×C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₁₀₇	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₇	C₂×C₈	1	4	yes	—
48	C₄⋊C₄×C₁₀₇	C₂×C₈	1	8	yes	—
49	C₄⋊C₄×C₁₀₇	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₁₀₇	C₂×C₈	1	4	yes	—
51	C₄²×C₁₀₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₁₀₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₀₇	C₂²×C₄	1	2	yes	—
54	C₄²×C₁₀₇	C₂²×C₄	1	4	yes	—
55	C₁₆×C₁₀₇	—	2	4	no	—
56	C₁₆×C₁₀₇	—	2	4	no	—
57	C₂³×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
58	Q₈×C₁₀₇	—	2	8	no	—
59	C₂×C₄×C₁₀₇	—	2	1	no	—
60	C₂×C₄×C₁₀₇	—	2	1	no	—
61	C₂×C₄×C₁₀₇	—	2	4	no	—
62	C₂×C₄×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₀₇	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₁₀₇	C₂²×C₈	1	8	yes	—
65	C₂×C₄×C₁₀₇	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
67	C₈×C₁₀₇	—	2	1	no	—
68	C₈×C₁₀₇	—	2	2	no	—
69	C₈×C₁₀₇	—	2	4	no	—
70	C₈×C₁₀₇	2₋¹⁺⁴	1	2	yes	—
71	C₂²×C₁₀₇	—	2	2	no	—
72	C₂²×C₁₀₇	—	2	1	no	—
73	C₂²×C₁₀₇	C₂³×C₈	1	1	yes	characteristic
74	C₄×C₁₀₇	—	2	2	no	—
75	C₄×C₁₀₇	—	2	8	no	—
76	C₄×C₁₀₇	—	2	1	no	—
77	C₄×C₁₀₇	#504	1	4	yes	—
78	C₄×C₁₀₇	#519	1	1	yes	characteristic
79	C₄×C₁₀₇	#568	1	1	yes	characteristic
80	#33500	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₁₀₇	—	4	1	no	—
82	C₂×C₁₀₇	#2982	1	1	yes	characteristic
83	C₂×C₁₀₇	C₂×#504	1	2	yes	—
84	#2960	C₂×C₁₀₇	1	1	yes	characteristic
85	#2183	C₂×C₁₀₇	1	4	yes	complemented
86	Q₈×C₁₆	C₂×C₁₀₇	1	4	yes	complemented
87	#2164	C₂×C₁₀₇	1	4	yes	—
88	#2162	C₂×C₁₀₇	1	2	yes	—
89	C₁₀₇	#33500	1	1	yes	characteristic, direct product factor, Sylow
90	#533	C₄×C₁₀₇	1	1	yes	characteristic
91	#446	C₄×C₁₀₇	1	4	yes	—
92	C₈×Q₈	C₂²×C₁₀₇	1	4	yes	—
93	#424	C₄×C₁₀₇	1	2	yes	—
94	#423	C₄×C₁₀₇	1	1	yes	characteristic
95	#405	C₂²×C₁₀₇	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
97	C₄⋊C₁₆	C₂²×C₁₀₇	1	2	yes	—
98	C₄⋊C₁₆	C₂²×C₁₀₇	1	2	yes	—
99	C₄⋊C₁₆	C₂²×C₁₀₇	1	8	yes	—
100	C₂²⋊C₁₆	C₂²×C₁₀₇	1	4	yes	—
101	#346	C₂²×C₁₀₇	1	4	yes	—
102	C₄×C₁₆	C₂²×C₁₀₇	1	4	yes	—
103	C₄×C₁₆	C₂²×C₁₀₇	1	4	yes	—
104	C₂×M₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
105	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
107	C₄⋊Q₈	C₈×C₁₀₇	1	2	yes	—
108	#125	C₈×C₁₀₇	1	2	yes	—
109	C₂²⋊Q₈	C₈×C₁₀₇	1	4	yes	—
110	C₄×Q₈	C₂×C₄×C₁₀₇	1	4	yes	—
111	#117	C₂×C₄×C₁₀₇	1	2	yes	—
112	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
113	C₂×C₁₆	C₂³×C₁₀₇	1	4	yes	—
114	C₂×C₁₆	C₂³×C₁₀₇	1	4	yes	—
115	C₄⋊C₈	C₂×C₄×C₁₀₇	1	4	yes	—
116	C₄⋊C₈	C₂×C₄×C₁₀₇	1	8	yes	—
117	#97	C₂×C₄×C₁₀₇	1	4	yes	—
118	C₄×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
119	C₄×C₈	C₂³×C₁₀₇	1	2	yes	—
120	C₄×C₈	C₂³×C₁₀₇	1	4	yes	—
121	C₂×Q₈	C₂×C₈×C₁₀₇	1	4	yes	—
122	C₂²×C₄	C₂×C₈×C₁₀₇	1	2	yes	—
123	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₄×C₁₀₇	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₁₀₇	1	4	yes	—
130	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
131	C₄⋊C₄	C₂×C₈×C₁₀₇	1	4	yes	—
132	C₄⋊C₄	C₂×C₈×C₁₀₇	1	8	yes	—
133	C₄⋊C₄	C₂×C₈×C₁₀₇	1	4	yes	—
134	C₂²⋊C₄	C₂×C₈×C₁₀₇	1	4	yes	—
135	C₄²	C₂×C₈×C₁₀₇	1	1	yes	characteristic
136	C₄²	C₂×C₈×C₁₀₇	1	1	yes	characteristic
137	C₄²	C₂²×C₄×C₁₀₇	1	2	yes	—
138	C₄²	C₂²×C₄×C₁₀₇	1	4	yes	—
139	C₁₆	—	2	4	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₈×C₁₀₇	1	2	yes	—
148	C₂×C₄	C₂²×C₈×C₁₀₇	1	8	yes	—
149	C₂×C₄	C₂²×C₈×C₁₀₇	1	2	yes	—
150	C₂×C₄	C₂³×C₄×C₁₀₇	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	2₋¹⁺⁴×C₁₀₇	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂³×C₈×C₁₀₇	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	2	1	no	—
161	C₄	#504×C₁₀₇	1	4	yes	—
162	C₄	#519×C₁₀₇	1	1	yes	characteristic
163	C₄	#568×C₁₀₇	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	C₁₀₇×#2982	1	1	yes	characteristic
166	C₂	C₂×#504×C₁₀₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₀₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
socle	C₂²×C₁₀₇	C₂³×C₈
prime radical	C₂³×C₁₀₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#2183	C₂	semidirect
3. Q₈×C₁₆×C₁₀₇	C₂	semidirect
80. #33500	C₁₀₇	direct (N × H)
85. #2183	C₂×C₁₀₇	semidirect
86. Q₈×C₁₆	C₂×C₁₀₇	semidirect
89. C₁₀₇	#33500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₇×#2960	2
C₁₀₇×#2183	2
Q₈×C₁₆×C₁₀₇	2
C₁₀₇×#2164	2
C₁₀₇×#2162	2
#33500	107

Subgroups — C107×#33500 #41059464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#33500 #41059464