Subgroups — C₂²×Q₈×Dic₁₃₄ #31762898

Order 17152.

7846 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₃₄	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14524043	C₂	1	3	yes	maximal
3	C₂²×#14524042	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₆₇	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₁₃₄	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	C₂²	1	6	yes	—
8	C₂³×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	Q₈×Dic₁₃₄	C₂²	1	256	yes	direct product factor
10	C₂×#14524043	C₂²	1	72	yes	—
11	C₂×#14524042	C₂²	1	144	yes	—
12	C₂×Q₈×Dic₆₇	C₂²	1	48	yes	—
13	C₂×C₄×Dic₁₃₄	C₂²	1	72	yes	—
14	C₂²×C₄⋊Dic₆₇	C₂²	1	3	yes	—
15	C₂²×C₄⋊Dic₆₇	C₂²	1	6	yes	—
16	C₂²×Dic₆₇⋊C₄	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₆₇	C₂²	1	6	yes	—
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	24	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
22	C₂²×Dic₁₃₄	—	4	24	no	—
23	C₂²×Dic₁₃₄	Q₈	1	16	yes	direct product factor
24	C₂²×Dic₁₃₄	C₂³	1	72	yes	—
25	C₂²×Dic₁₃₄	C₂³	1	12	yes	—
26	C₂³×Dic₆₇	C₂³	1	6	yes	—
27	C₂³×Dic₆₇	C₂³	1	2	yes	—
28	#14524043	C₂³	1	192	yes	—
29	#14524042	C₂³	1	384	yes	—
30	Q₈×Dic₆₇	C₂³	1	128	yes	—
31	C₄×Dic₁₃₄	C₂³	1	192	yes	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	36	yes	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	72	yes	—
34	C₂×Dic₆₇⋊C₄	C₂³	1	72	yes	—
35	C₂×C₄×Dic₆₇	C₂³	1	72	yes	—
36	C₂²×Q₈×C₆₇	Q₈	1	16	yes	complemented
37	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₇	C₂³	1	64	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	36	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	36	yes	—
44	C₂×Dic₁₃₄	—	4	144	no	—
45	C₂×Dic₁₃₄	C₂×Q₈	1	96	yes	direct product factor
46	C₂×Dic₁₃₄	C₂⁴	1	96	yes	—
47	C₂×Dic₁₃₄	C₂⁴	1	16	yes	—
48	C₂²×Dic₆₇	—	2	24	no	—
49	C₂²×Dic₆₇	C₂×Q₈	1	16	yes	—
50	C₂²×Dic₆₇	C₂⁴	1	36	yes	—
51	C₂²×Dic₆₇	C₂⁴	1	12	yes	—
52	C₄⋊Dic₆₇	C₂⁴	1	48	yes	—
53	C₄⋊Dic₆₇	C₂⁴	1	96	yes	—
54	Dic₆₇⋊C₄	C₂⁴	1	96	yes	—
55	C₄×Dic₆₇	C₂⁴	1	96	yes	—
56	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₆₇	C₂×Q₈	1	96	yes	complemented
58	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
59	C₂²×C₄×C₆₇	—	2	12	no	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	18	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	18	yes	—
65	C₄⋊C₄×C₆₇	C₂⁴	1	48	yes	—
66	C₄²×C₆₇	C₂⁴	1	48	yes	—
67	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
68	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
69	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
70	C₂³×C₆₇	C₂⁵	1	3	yes	—
71	Q₈×C₆₇	C₂²×Q₈	1	64	yes	complemented
72	C₂×C₄×C₆₇	—	2	36	no	—
73	C₂×C₄×C₆₇	C₂²×Q₈	1	24	yes	—
74	C₂×C₄×C₆₇	C₂²×Q₈	1	72	yes	—
75	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
76	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
77	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
78	C₂×Dic₆₇	—	2	72	no	—
79	C₂×Dic₆₇	C₂²×Q₈	1	48	yes	—
80	C₂×Dic₆₇	C₂⁵	1	24	yes	—
81	C₂×Dic₆₇	C₂⁵	1	8	yes	—
82	Dic₁₃₄	—	4	96	no	—
83	Dic₁₃₄	C₂²×Q₈	1	64	yes	direct product factor
84	C₂²×C₆₇	Q₈²	1	16	yes	complemented
85	C₂²×C₆₇	C₂³×Q₈	1	6	yes	—
86	C₂²×C₆₇	C₂³×Q₈	1	6	yes	—
87	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	6	yes	—
88	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
89	C₄×C₆₇	—	2	12	no	—
90	C₄×C₆₇	C₂³×Q₈	1	8	yes	—
91	C₄×C₆₇	C₂³×Q₈	1	24	yes	—
92	Dic₆₇	—	2	24	no	—
93	Dic₆₇	C₂³×Q₈	1	16	yes	—
94	C₂²×Q₈²	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	C₂×Q₈²	1	12	yes	complemented
96	C₂×C₆₇	C₂⁴×Q₈	1	1	yes	characteristic
97	C₂×C₆₇	C₂⁴×Q₈	1	1	yes	characteristic
98	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×Q₈²	—	67	48	no	—
100	C₂²×C₄⋊Q₈	—	67	3	no	—
101	C₂²×C₄⋊Q₈	—	67	6	no	—
102	C₂²×C₄×Q₈	—	67	3	no	—
103	C₂²×C₄×Q₈	—	67	2	no	—
104	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
105	C₆₇	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
106	C₂³×Q₈	—	67	6	no	—
107	C₂³×Q₈	—	67	1	no	—
108	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
109	Q₈²	—	67	256	no	—
110	C₂×C₄⋊Q₈	—	67	144	no	—
111	C₂×C₄⋊Q₈	—	67	72	no	—
112	C₂×C₄×Q₈	—	67	48	no	—
113	C₂×C₄×Q₈	—	67	72	no	—
114	C₂×C₄×Q₈	C₂×D₆₇	1	24	yes	—
115	C₂²×C₄⋊C₄	—	67	6	no	—
116	C₂²×C₄⋊C₄	—	67	3	no	—
117	C₂²×C₄⋊C₄	—	67	6	no	—
118	C₂²×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
119	C₂²×C₄²	—	67	6	no	—
120	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
121	C₂²×Q₈	—	268	24	no	—
122	C₂²×Q₈	—	67	16	no	—
123	C₂²×Q₈	—	67	72	no	—
124	C₂²×Q₈	—	67	12	no	—
125	C₂²×Q₈	Dic₁₃₄	1	16	yes	direct product factor
126	C₂²×Q₈	C₂²×D₆₇	1	12	yes	—
127	C₂³×C₄	—	67	2	no	—
128	C₂³×C₄	—	67	6	no	—
129	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
130	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
132	C₄⋊Q₈	—	67	384	no	—
133	C₄⋊Q₈	—	67	192	no	—
134	C₄×Q₈	—	67	128	no	—
135	C₄×Q₈	—	67	192	no	—
136	C₄×Q₈	C₂²×D₆₇	1	64	yes	—
137	C₂×C₄⋊C₄	—	67	72	no	—
138	C₂×C₄⋊C₄	—	67	36	no	—
139	C₂×C₄⋊C₄	—	67	72	no	—
140	C₂×C₄⋊C₄	C₂²×D₆₇	1	36	yes	—
141	C₂×C₄²	—	67	72	no	—
142	C₂×C₄²	C₂²×D₆₇	1	36	yes	—
143	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
144	C₂×Q₈	—	268	144	no	—
145	C₂×Q₈	—	67	96	no	—
146	C₂×Q₈	—	67	96	no	—
147	C₂×Q₈	—	67	16	no	—
148	C₂×Q₈	C₂×Dic₁₃₄	1	96	yes	direct product factor
149	C₂×Q₈	C₂³×D₆₇	1	16	yes	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	67	16	no	—
152	C₂²×C₄	—	67	12	no	—
153	C₂²×C₄	—	67	36	no	—
154	C₂²×C₄	—	134	24	no	—
155	C₂²×C₄	Q₈×D₆₇	1	8	yes	—
156	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
157	C₂²×C₄	C₂³×D₆₇	1	18	yes	—
158	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
159	C₂²×C₄	C₂³×D₆₇	1	18	yes	—
160	C₄⋊C₄	—	67	96	no	—
161	C₄⋊C₄	—	67	48	no	—
162	C₄⋊C₄	—	67	96	no	—
163	C₄⋊C₄	C₂³×D₆₇	1	48	yes	—
164	C₄²	—	67	96	no	—
165	C₄²	C₂³×D₆₇	1	48	yes	—
166	C₂³	C₂×Q₈×D₆₇	1	4	yes	—
167	C₂³	C₂²×Dic₁₃₄	1	4	yes	—
168	C₂³	#14524089	1	4	yes	—
169	C₂³	C₂⁴×D₆₇	1	3	yes	—
170	Q₈	—	268	96	no	—
171	Q₈	—	67	64	no	—
172	Q₈	C₂²×Dic₁₃₄	1	64	yes	direct product factor
173	C₂×C₄	—	2	36	no	—
174	C₂×C₄	—	67	48	no	—
175	C₂×C₄	—	67	8	no	—
176	C₂×C₄	—	67	24	no	—
177	C₂×C₄	—	134	72	no	—
178	C₂×C₄	C₂×Q₈×D₆₇	1	24	yes	—
179	C₂×C₄	C₂²×Dic₁₃₄	1	72	yes	—
180	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
181	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
182	C₂×C₄	C₂⁴×D₆₇	1	12	yes	—
183	C₂²	Q₈×Dic₁₃₄	1	16	yes	direct product factor
184	C₂²	C₂²×Q₈×D₆₇	1	6	yes	—
185	C₂²	C₂³×Dic₁₃₄	1	6	yes	—
186	C₂²	C₂×#14524089	1	6	yes	—
187	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
188	C₄	—	2	12	no	—
189	C₄	—	67	16	no	—
190	C₄	—	134	24	no	—
191	C₄	C₂²×Q₈×D₆₇	1	8	yes	—
192	C₄	C₂³×Dic₁₃₄	1	24	yes	—
193	C₂	C₂×Q₈×Dic₁₃₄	1	12	yes	direct product factor
194	C₂	C₂³×Q₈×D₆₇	1	1	yes	characteristic
195	C₂	C₂⁴×Dic₁₃₄	1	1	yes	characteristic
196	C₂	C₂²×#14524089	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂²×Q₈²
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₃₄	C₂	direct (N × H)
9. Q₈×Dic₁₃₄	C₂²	direct (N × H)
23. C₂²×Dic₁₃₄	Q₈	direct (N × H)
36. C₂²×Q₈×C₆₇	Q₈	semidirect
45. C₂×Dic₁₃₄	C₂×Q₈	direct (N × H)
57. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
71. Q₈×C₆₇	C₂²×Q₈	semidirect
83. Dic₁₃₄	C₂²×Q₈	direct (N × H)
84. C₂²×C₆₇	Q₈²	semidirect
95. C₂×C₆₇	C₂×Q₈²	semidirect
105. C₆₇	C₂²×Q₈²	semidirect
125. C₂²×Q₈	Dic₁₃₄	direct (N × H)
148. C₂×Q₈	C₂×Dic₁₃₄	direct (N × H)
172. Q₈	C₂²×Dic₁₃₄	direct (N × H)
183. C₂²	Q₈×Dic₁₃₄	direct (N × H)
193. C₂	C₂×Q₈×Dic₁₃₄	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×Q₈×Dic₁₃₄	2
C₂²×#14524043	2
C₂²×#14524042	2
C₂²×Q₈×Dic₆₇	2
C₂²×C₄×Dic₁₃₄	2
C₂²×C₄×Q₈×C₆₇	2
C₂²×Q₈²	67

Subgroups — C22×Q8×Dic134 #31762898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₁₃₄ #31762898