Subgroups — Q₈²×Dic₆₇ #31015837

Order 17152.

2324 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22563299	C₂	1	9	yes	maximal
2	Q₈×C₄⋊Dic₆₇	C₂	1	6	yes	maximal
3	C₄⋊Q₈×Dic₆₇	C₂	1	9	yes	maximal
4	C₄×Q₈×Dic₆₇	C₂	1	6	yes	maximal
5	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₆₇	C₂²	1	6	yes	—
7	C₂×Q₈×Dic₆₇	C₂²	1	6	yes	—
8	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	—
9	#16863303	C₂²	1	18	yes	—
10	#16863295	C₂²	1	36	yes	—
11	#16863266	C₂²	1	9	yes	—
12	C₄⋊C₄×Dic₆₇	C₂²	1	18	yes	—
13	C₄⋊C₄×Dic₆₇	C₂²	1	18	yes	—
14	C₄×C₄⋊Dic₆₇	C₂²	1	18	yes	—
15	C₄²×Dic₆₇	C₂²	1	9	yes	—
16	Q₈²×C₆₇	C₄	1	16	yes	complemented
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	9	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
19	Q₈×Dic₆₇	—	2	24	no	—
20	Q₈×Dic₆₇	—	4	12	no	—
21	Q₈×Dic₆₇	Q₈	1	16	yes	direct product factor
22	C₂×C₄⋊Dic₆₇	C₂³	1	18	yes	—
23	C₂×C₄⋊Dic₆₇	C₂³	1	18	yes	—
24	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
25	C₂×C₄×Dic₆₇	C₂³	1	18	yes	—
26	C₂×C₄×Dic₆₇	C₂³	1	9	yes	—
27	C₂×C₄×Dic₆₇	C₂³	1	9	yes	—
28	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
29	C₆₇⋊#95	C₂³	1	36	yes	—
30	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	72	yes	—
33	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
36	C₂²×Dic₆₇	C₂⁴	1	9	yes	—
37	C₂²×Dic₆₇	C₂⁴	1	6	yes	—
38	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
39	C₄⋊Dic₆₇	—	2	36	no	—
40	C₄⋊Dic₆₇	—	4	18	no	—
41	C₄⋊Dic₆₇	C₂×Q₈	1	24	yes	—
42	C₄×Dic₆₇	—	2	36	no	—
43	C₄×Dic₆₇	—	2	18	no	—
44	C₄×Dic₆₇	—	4	9	no	—
45	C₄×Dic₆₇	C₂×Q₈	1	24	yes	—
46	C₂×Q₈×C₆₇	—	4	6	no	—
47	C₂×Q₈×C₆₇	—	4	6	no	—
48	C₂×Q₈×C₆₇	C₂²×C₄	1	24	yes	—
49	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
50	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
51	C₂×Q₈×C₆₇	#41	1	8	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	9	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
54	C₄⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	36	yes	—
56	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
57	Q₈×C₆₇	—	4	24	no	—
58	Q₈×C₆₇	C₄×Q₈	1	32	yes	complemented
59	C₂×C₄×C₆₇	—	2	9	no	—
60	C₂×C₄×C₆₇	—	2	9	no	—
61	C₂×C₄×C₆₇	C₂³×C₄	1	18	yes	—
62	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
63	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
64	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
65	C₂×Dic₆₇	—	2	6	no	—
66	C₂×Dic₆₇	—	2	6	no	—
67	C₂×Dic₆₇	—	2	18	no	—
68	C₂×Dic₆₇	—	2	9	no	—
69	C₂×Dic₆₇	C₂²×Q₈	1	12	yes	—
70	C₂×Dic₆₇	C₂²×Q₈	1	4	yes	—
71	C₂×Dic₆₇	H(5,𝔽₂)	1	2	yes	—
72	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
74	C₂²×C₆₇	C₂²×#41	1	2	yes	—
75	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₆₇	#585	1	1	yes	characteristic
77	C₄×C₆₇	—	2	18	no	—
78	C₄×C₆₇	C₂×C₄×Q₈	1	24	yes	—
79	Dic₆₇	—	2	12	no	—
80	Dic₆₇	—	4	9	no	—
81	Dic₆₇	Q₈²	1	4	yes	direct product factor
82	C₄×Q₈²	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	C₂²×C₄×Q₈	1	2	yes	—
84	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
85	C₂×C₆₇	C₂×Q₈²	1	1	yes	characteristic
86	C₂×C₆₇	Q₈×#41	1	2	yes	—
87	C₂×C₆₇	#3483	1	1	yes	characteristic
88	C₂×Q₈²	D₆₇	1	1	yes	characteristic
89	#2369	—	67	9	no	—
90	Q₈×C₄⋊C₄	—	67	6	no	—
91	C₄×C₄⋊Q₈	—	67	9	no	—
92	C₄²×Q₈	—	67	6	no	—
93	C₆₇	C₄×Q₈²	1	1	yes	characteristic, complemented, Sylow
94	Q₈²	Dic₆₇	1	16	yes	direct product factor
95	C₂×C₄⋊Q₈	C₂×D₆₇	1	9	yes	—
96	C₂×C₄×Q₈	—	67	6	no	—
97	C₂×C₄×Q₈	—	67	2	no	—
98	C₂×C₄×Q₈	—	67	6	no	—
99	C₂×C₄×Q₈	C₂×D₆₇	1	6	yes	—
100	#391	—	67	18	no	—
101	#389	—	67	36	no	—
102	#384	—	67	9	no	—
103	C₄×C₄⋊C₄	—	67	18	no	—
104	C₄×C₄⋊C₄	—	67	18	no	—
105	C₄×C₄⋊C₄	—	67	18	no	—
106	C₄³	—	67	9	no	—
107	C₂²×Q₈	C₂²×D₆₇	1	6	yes	—
108	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
109	C₄⋊Q₈	C₂×Dic₆₇	1	72	yes	—
110	C₄×Q₈	—	268	12	no	—
111	C₄×Q₈	—	67	16	no	—
112	C₄×Q₈	—	134	24	no	—
113	C₄×Q₈	C₂×Dic₆₇	1	48	yes	—
114	C₂×C₄⋊C₄	—	67	18	no	—
115	C₂×C₄⋊C₄	—	67	6	no	—
116	C₂×C₄⋊C₄	—	67	18	no	—
117	C₂×C₄⋊C₄	C₂²×D₆₇	1	18	yes	—
118	C₂×C₄²	—	67	9	no	—
119	C₂×C₄²	—	67	6	no	—
120	C₂×C₄²	—	67	9	no	—
121	C₂×C₄²	—	67	18	no	—
122	C₂×C₄²	C₂²×D₆₇	1	9	yes	—
123	#95	—	67	36	no	—
124	C₂×Q₈	—	4	6	no	—
125	C₂×Q₈	—	4	6	no	—
126	C₂×Q₈	C₂²×Dic₆₇	1	24	yes	—
127	C₂×Q₈	C₂²×Dic₆₇	1	8	yes	—
128	C₂×Q₈	Q₈×D₆₇	1	8	yes	—
129	C₂×Q₈	#11761629	1	8	yes	—
130	C₂²×C₄	—	67	1	no	—
131	C₂²×C₄	—	67	6	no	—
132	C₂²×C₄	—	67	9	no	—
133	C₂²×C₄	C₂³×D₆₇	1	9	yes	—
134	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
135	C₄⋊C₄	—	268	18	no	—
136	C₄⋊C₄	—	67	24	no	—
137	C₄⋊C₄	—	134	36	no	—
138	C₄⋊C₄	C₂²×Dic₆₇	1	72	yes	—
139	C₄²	—	268	9	no	—
140	C₄²	—	67	24	no	—
141	C₄²	—	134	18	no	—
142	C₄²	—	134	36	no	—
143	C₄²	C₂²×Dic₆₇	1	36	yes	—
144	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
145	Q₈	—	4	24	no	—
146	Q₈	Q₈×Dic₆₇	1	32	yes	direct product factor
147	C₂×C₄	—	2	9	no	—
148	C₂×C₄	—	2	9	no	—
149	C₂×C₄	—	67	4	no	—
150	C₂×C₄	—	67	2	no	—
151	C₂×C₄	—	67	12	no	—
152	C₂×C₄	—	134	18	no	—
153	C₂×C₄	—	134	9	no	—
154	C₂×C₄	—	134	6	no	—
155	C₂×C₄	—	134	6	no	—
156	C₂×C₄	C₂³×Dic₆₇	1	18	yes	—
157	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
158	C₂×C₄	C₂×Q₈×D₆₇	1	12	yes	—
159	C₂×C₄	C₂×#11761629	1	12	yes	—
160	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
161	C₂²	C₂²×Q₈×D₆₇	1	2	yes	—
162	C₂²	C₂²×#11761629	1	2	yes	—
163	C₂²	H(5,𝔽₂)×D₆₇	1	1	yes	characteristic
164	C₂²	#16864231	1	1	yes	characteristic
165	C₄	—	2	18	no	—
166	C₄	—	268	9	no	—
167	C₄	—	67	4	no	—
168	C₄	—	134	12	no	—
169	C₄	C₂×Q₈×Dic₆₇	1	24	yes	—
170	C₂	C₂²×Q₈×Dic₆₇	1	2	yes	—
171	C₂	H(5,𝔽₂)×Dic₆₇	1	1	yes	characteristic
172	C₂	Q₈²×D₆₇	1	1	yes	characteristic
173	C₂	Q₈×#11761629	1	2	yes	—
174	C₂	#22573836	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
16. Q₈²×C₆₇	C₄	semidirect
21. Q₈×Dic₆₇	Q₈	direct (N × H)
58. Q₈×C₆₇	C₄×Q₈	semidirect
81. Dic₆₇	Q₈²	direct (N × H)
93. C₆₇	C₄×Q₈²	semidirect
94. Q₈²	Dic₆₇	direct (N × H)
146. Q₈	Q₈×Dic₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#22563299	2
Q₈×C₄⋊Dic₆₇	2
C₄⋊Q₈×Dic₆₇	2
C₄×Q₈×Dic₆₇	2
C₂×Q₈²×C₆₇	2
C₄×Q₈²	67

Subgroups — Q82×Dic67 #31015837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×Dic₆₇ #31015837