Subgroups — C₂×C₈×Dic₁₉₄ #27150657

Order 12416.

440 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₉₄	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
4	C₂×#15690739	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₉₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₉₄	C₄	1	1	yes	characteristic
8	C₂²×#11715895	C₂²	1	2	yes	—
9	C₄×Dic₁₉₄	C₄	1	4	yes	—
10	C₄×Dic₁₉₄	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₉₇	C₄	1	1	yes	characteristic
12	C₂×#13095490	C₄	1	2	yes	—
13	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
14	C₄⋊#11715895	C₂²	1	4	yes	—
15	#15690739	C₂²	1	8	yes	—
16	C₈×Dic₉₇	C₂²	1	8	yes	—
17	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₉₇	C₂²	1	4	yes	—
21	C₂×Dic₁₉₄	C₈	1	8	yes	direct product factor
22	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
23	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
25	C₂×#11715895	—	2	4	no	—
26	C₂×#11715895	C₂³	1	4	yes	—
27	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
28	#13095490	C₂×C₄	1	8	yes	—
29	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₉₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₉₇	—	2	2	no	—
35	C₂×C₈×C₉₇	Q₈	1	4	yes	complemented
36	C₂×C₈×C₉₇	C₂³	1	2	yes	—
37	C₂×C₈×C₉₇	C₂³	1	2	yes	—
38	C₄²×C₉₇	C₂×C₄	1	2	yes	—
39	C₄²×C₉₇	C₂³	1	2	yes	—
40	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	—	2	2	no	—
42	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₉₇	#41	1	2	yes	—
48	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₉₇	—	2	4	no	—
50	C₂×Dic₉₇	C₂×C₈	1	8	yes	—
51	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
53	Dic₁₉₄	C₂×C₈	1	16	yes	direct product factor
54	C₈×C₉₇	—	2	2	no	—
55	C₈×C₉₇	C₂×Q₈	1	4	yes	complemented
56	#11715895	—	2	4	no	—
57	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₉₇	C₂²×C₈	1	2	yes	—
59	C₂²×C₉₇	#131	1	2	yes	—
60	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₉₇	—	2	2	no	—
62	C₄×C₉₇	C₄×Q₈	1	2	yes	—
63	C₄×C₉₇	C₂²×C₈	1	4	yes	—
64	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
66	Dic₉₇	—	2	4	no	—
67	Dic₉₇	C₂²×C₈	1	8	yes	—
68	C₂×C₉₇	C₈×Q₈	1	4	yes	complemented
69	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₉₇	C₂³×C₈	1	1	yes	characteristic
71	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
72	C₂×C₈×Q₈	—	97	1	no	maximal, Sylow
73	C₉₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	97	1	no	—
75	C₈×Q₈	—	97	8	no	—
76	C₂×C₄⋊C₈	—	97	2	no	—
77	C₂×C₄⋊C₈	—	97	1	no	—
78	C₂×C₄×C₈	—	97	2	no	—
79	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
80	C₂²×Q₈	—	97	1	no	—
81	C₂²×C₈	—	97	2	no	—
82	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
84	C₄×Q₈	—	97	4	no	—
85	C₄×Q₈	—	97	4	no	—
86	C₂×C₄⋊C₄	—	97	2	no	—
87	C₂×C₄⋊C₄	—	97	1	no	—
88	C₂×C₄²	—	97	2	no	—
89	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
90	C₄⋊C₈	—	97	8	no	—
91	C₄⋊C₈	—	97	4	no	—
92	C₄×C₈	—	97	8	no	—
93	C₄×C₈	C₂×D₉₇	1	4	yes	—
94	C₂×Q₈	—	97	8	no	—
95	C₂×Q₈	—	97	4	no	—
96	C₂²×C₄	—	97	2	no	—
97	C₂²×C₄	—	97	2	no	—
98	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	97	4	no	—
103	C₂×C₈	—	194	4	no	—
104	C₂×C₈	Dic₁₉₄	1	4	yes	direct product factor
105	C₂×C₈	C₂²×D₉₇	1	2	yes	—
106	C₂×C₈	C₂²×D₉₇	1	2	yes	—
107	C₄⋊C₄	—	97	8	no	—
108	C₄⋊C₄	—	97	4	no	—
109	C₄²	—	97	4	no	—
110	C₄²	—	97	4	no	—
111	C₄²	C₄×D₉₇	1	2	yes	—
112	C₄²	C₂²×D₉₇	1	2	yes	—
113	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
114	Q₈	—	97	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	97	8	no	—
117	C₂×C₄	—	97	4	no	—
118	C₂×C₄	—	97	4	no	—
119	C₂×C₄	—	194	4	no	—
120	C₂×C₄	C₈×D₉₇	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
125	C₂×C₄	#13095511	1	2	yes	—
126	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	194	4	no	—
129	C₈	C₂×Dic₁₉₄	1	4	yes	direct product factor
130	C₂²	C₄×Dic₁₉₄	1	2	yes	—
131	C₂²	C₂×C₈×D₉₇	1	2	yes	—
132	C₂²	#15690816	1	2	yes	—
133	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	97	8	no	—
136	C₄	—	194	4	no	—
137	C₄	C₄×Dic₁₉₄	1	2	yes	—
138	C₄	C₂×C₈×D₉₇	1	4	yes	—
139	C₄	C₂²×Dic₁₉₄	1	1	yes	characteristic
140	C₄	C₂×#13095511	1	1	yes	characteristic
141	C₂	C₈×Dic₁₉₄	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₁₉₄	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₉₇	1	1	yes	characteristic
144	C₂	C₂×#15690816	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₉₇
commutator	C₂×C₉₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	C₂×C₈×Q₈
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₉₄	C₂	direct (N × H)
21. C₂×Dic₁₉₄	C₈	direct (N × H)
35. C₂×C₈×C₉₇	Q₈	semidirect
53. Dic₁₉₄	C₂×C₈	direct (N × H)
55. C₈×C₉₇	C₂×Q₈	semidirect
68. C₂×C₉₇	C₈×Q₈	semidirect
73. C₉₇	C₂×C₈×Q₈	semidirect
104. C₂×C₈	Dic₁₉₄	direct (N × H)
129. C₈	C₂×Dic₁₉₄	direct (N × H)
141. C₂	C₈×Dic₁₉₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₉₄	2
C₈×Dic₁₉₄	2
C₂×C₄⋊#11715895	2
C₂×#15690739	2
C₂×C₈×Dic₉₇	2
C₂×C₄×C₈×C₉₇	2
C₂×C₈×Q₈	97

Subgroups — C2×C8×Dic194 #27150657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₁₉₄ #27150657