Subgroups — C₂×#555×C₁₀₇ #28334156

Order 13696.

1240 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₀₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124×C₁₀₇	C₂	1	4	yes	maximal, complemented
4	C₂×#121×C₁₀₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×D₄×C₁₀₇	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₁₀₇	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂²×D₄×C₁₀₇	C₂²	1	2	yes	—
9	C₂²×D₄×C₁₀₇	C₂²	1	4	yes	—
10	C₂³×C₄×C₁₀₇	C₂²	1	4	yes	—
11	C₄⋊Q₈×C₁₀₇	C₂²	1	8	yes	complemented
12	#124×C₁₀₇	C₂²	1	32	yes	complemented
13	#121×C₁₀₇	C₂²	1	32	yes	complemented
14	C₄×D₄×C₁₀₇	C₂²	1	16	yes	complemented
15	#117×C₁₀₇	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₁₀₇	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₁₀₇	C₂³	1	4	yes	—
23	C₂×Q₈×C₁₀₇	—	4	4	no	—
24	C₂×Q₈×C₁₀₇	C₂³	1	8	yes	—
25	C₂×D₄×C₁₀₇	—	2	8	no	—
26	C₂×D₄×C₁₀₇	—	4	8	no	—
27	C₂×D₄×C₁₀₇	C₂³	1	8	yes	—
28	C₂×D₄×C₁₀₇	C₂³	1	16	yes	—
29	C₂²×C₄×C₁₀₇	—	2	16	no	—
30	C₂²×C₄×C₁₀₇	C₂³	1	16	yes	—
31	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₇	C₂³	1	8	yes	—
36	C₄⋊C₄×C₁₀₇	C₂³	1	16	yes	—
37	C₂²⋊C₄×C₁₀₇	C₂³	1	32	yes	—
38	C₂²⋊C₄×C₁₀₇	C₂³	1	16	yes	—
39	C₄²×C₁₀₇	C₂³	1	16	yes	—
40	C₄²×C₁₀₇	C₂³	1	4	yes	—
41	C₂³×C₁₀₇	—	2	8	no	—
42	C₂³×C₁₀₇	—	2	8	no	—
43	C₂³×C₁₀₇	—	2	8	no	—
44	C₂³×C₁₀₇	C₂⁴	1	8	yes	—
45	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
46	Q₈×C₁₀₇	—	4	8	no	—
47	D₄×C₁₀₇	—	2	16	no	—
48	D₄×C₁₀₇	—	4	16	no	—
49	C₂×C₄×C₁₀₇	—	2	8	no	—
50	C₂×C₄×C₁₀₇	—	2	8	no	—
51	C₂×C₄×C₁₀₇	—	2	2	no	—
52	C₂×C₄×C₁₀₇	—	2	32	no	—
53	C₂×C₄×C₁₀₇	#41	1	8	yes	—
54	C₂×C₄×C₁₀₇	C₂⁴	1	8	yes	—
55	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
56	C₂×C₄×C₁₀₇	C₂⁴	1	8	yes	—
57	C₂×C₄×C₁₀₇	C₂⁴	1	4	yes	—
58	C₂²×C₁₀₇	—	2	8	no	—
59	C₂²×C₁₀₇	—	2	8	no	—
60	C₂²×C₁₀₇	—	2	8	no	—
61	C₂²×C₁₀₇	—	4	16	no	—
62	C₂²×C₁₀₇	C₂×#41	1	4	yes	—
63	C₂²×C₁₀₇	H(5,𝔽₂)	1	2	yes	—
64	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
65	C₄×C₁₀₇	—	2	8	no	—
66	C₄×C₁₀₇	—	2	8	no	—
67	C₄×C₁₀₇	—	2	2	no	—
68	C₄×C₁₀₇	C₂×#41	1	8	yes	—
69	C₂×C₁₀₇	—	4	8	no	—
70	C₂×C₁₀₇	#555	1	4	yes	direct product factor
71	C₂×C₁₀₇	C₂²×#41	1	2	yes	—
72	C₂×C₁₀₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×#555	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₀₇	C₂×#555	1	1	yes	characteristic, direct product factor, Sylow
75	#555	C₂×C₁₀₇	1	16	yes	direct product factor
76	C₂×C₄⋊Q₈	C₂×C₁₀₇	1	1	yes	characteristic, complemented
77	C₂×#124	C₂×C₁₀₇	1	4	yes	complemented
78	C₂×#121	C₂×C₁₀₇	1	4	yes	complemented
79	C₂×C₄×D₄	C₂×C₁₀₇	1	2	yes	complemented
80	C₂×#117	C₂×C₁₀₇	1	4	yes	complemented
81	C₂²×Q₈	C₂²×C₁₀₇	1	2	yes	—
82	C₂²×D₄	C₂²×C₁₀₇	1	2	yes	—
83	C₂²×D₄	C₂²×C₁₀₇	1	4	yes	—
84	C₂³×C₄	C₂²×C₁₀₇	1	4	yes	—
85	C₄⋊Q₈	C₂²×C₁₀₇	1	8	yes	complemented
86	#124	C₂²×C₁₀₇	1	32	yes	complemented
87	#121	C₂²×C₁₀₇	1	32	yes	complemented
88	C₄×D₄	C₂²×C₁₀₇	1	16	yes	complemented
89	#117	C₂²×C₁₀₇	1	32	yes	complemented
90	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	4	yes	—
92	C₂×C₂²⋊C₄	C₂²×C₁₀₇	1	8	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₁₀₇	1	4	yes	—
94	C₂×C₄²	C₂²×C₁₀₇	1	4	yes	—
95	C₂×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
96	C₂⁴	C₂³×C₁₀₇	1	4	yes	—
97	C₂×Q₈	—	4	4	no	—
98	C₂×Q₈	C₂³×C₁₀₇	1	8	yes	—
99	C₂×D₄	—	2	8	no	—
100	C₂×D₄	—	4	8	no	—
101	C₂×D₄	C₂³×C₁₀₇	1	8	yes	—
102	C₂×D₄	C₂³×C₁₀₇	1	16	yes	—
103	C₂²×C₄	—	2	16	no	—
104	C₂²×C₄	C₂³×C₁₀₇	1	4	yes	—
105	C₂²×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₁₀₇	1	4	yes	—
107	C₂²×C₄	C₂³×C₁₀₇	1	16	yes	—
108	C₂²×C₄	C₂³×C₁₀₇	1	2	yes	—
109	C₄⋊C₄	C₂³×C₁₀₇	1	8	yes	—
110	C₄⋊C₄	C₂³×C₁₀₇	1	16	yes	—
111	C₂²⋊C₄	C₂³×C₁₀₇	1	32	yes	—
112	C₂²⋊C₄	C₂³×C₁₀₇	1	16	yes	—
113	C₄²	C₂³×C₁₀₇	1	16	yes	—
114	C₄²	C₂³×C₁₀₇	1	4	yes	—
115	C₂³	—	2	8	no	—
116	C₂³	—	2	8	no	—
117	C₂³	—	2	8	no	—
118	C₂³	C₂⁴×C₁₀₇	1	8	yes	—
119	C₂³	C₂⁴×C₁₀₇	1	1	yes	characteristic
120	Q₈	—	4	8	no	—
121	D₄	—	2	16	no	—
122	D₄	—	4	16	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	8	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	32	no	—
127	C₂×C₄	#41×C₁₀₇	1	8	yes	—
128	C₂×C₄	C₂⁴×C₁₀₇	1	8	yes	—
129	C₂×C₄	C₂⁴×C₁₀₇	1	2	yes	—
130	C₂×C₄	C₂⁴×C₁₀₇	1	8	yes	—
131	C₂×C₄	C₂⁴×C₁₀₇	1	4	yes	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	8	no	—
134	C₂²	—	2	8	no	—
135	C₂²	—	4	16	no	—
136	C₂²	C₂×#41×C₁₀₇	1	4	yes	—
137	C₂²	H(5,𝔽₂)×C₁₀₇	1	2	yes	—
138	C₂²	C₂⁵×C₁₀₇	1	1	yes	characteristic
139	C₄	—	2	8	no	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×#41×C₁₀₇	1	8	yes	—
143	C₂	—	4	8	no	—
144	C₂	#555×C₁₀₇	1	4	yes	direct product factor
145	C₂	C₂²×#41×C₁₀₇	1	2	yes	—
146	C₂	C₂×H(5,𝔽₂)×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #555×C₁₀₇	C₂	direct (N × H)
2. C₂×C₄⋊Q₈×C₁₀₇	C₂	semidirect
3. C₂×#124×C₁₀₇	C₂	semidirect
4. C₂×#121×C₁₀₇	C₂	semidirect
5. C₂×C₄×D₄×C₁₀₇	C₂	semidirect
6. C₂×#117×C₁₀₇	C₂	semidirect
11. C₄⋊Q₈×C₁₀₇	C₂²	semidirect
12. #124×C₁₀₇	C₂²	semidirect
13. #121×C₁₀₇	C₂²	semidirect
14. C₄×D₄×C₁₀₇	C₂²	semidirect
15. #117×C₁₀₇	C₂²	semidirect
70. C₂×C₁₀₇	#555	direct (N × H)
73. C₂×#555	C₁₀₇	direct (N × H)
74. C₁₀₇	C₂×#555	direct (N × H)
75. #555	C₂×C₁₀₇	direct (N × H)
76. C₂×C₄⋊Q₈	C₂×C₁₀₇	semidirect
77. C₂×#124	C₂×C₁₀₇	semidirect
78. C₂×#121	C₂×C₁₀₇	semidirect
79. C₂×C₄×D₄	C₂×C₁₀₇	semidirect
80. C₂×#117	C₂×C₁₀₇	semidirect
85. C₄⋊Q₈	C₂²×C₁₀₇	semidirect
86. #124	C₂²×C₁₀₇	semidirect
87. #121	C₂²×C₁₀₇	semidirect
88. C₄×D₄	C₂²×C₁₀₇	semidirect
89. #117	C₂²×C₁₀₇	semidirect
144. C₂	#555×C₁₀₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#555×C₁₀₇	2
C₂×C₄⋊Q₈×C₁₀₇	2
C₂×#124×C₁₀₇	2
C₂×#121×C₁₀₇	2
C₂×C₄×D₄×C₁₀₇	2
C₂×#117×C₁₀₇	2
C₂×#555	107

Subgroups — C2×#555×C107 #28334156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#555×C₁₀₇ #28334156