Subgroups — C₆₇×#22738 #30701989

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1948	C₂	1	6	yes	maximal
4	C₆₇×#1834	C₂	1	3	yes	maximal
5	C₄×#106×C₆₇	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₆₇	C₄	1	8	yes	complemented
9	C₂×#106×C₆₇	C₂²	1	6	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
12	#389×C₆₇	—	2	6	no	—
13	#384×C₆₇	C₂²	1	6	yes	—
14	C₄×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
15	#340×C₆₇	C₂²	1	1	yes	characteristic
16	#340×C₆₇	C₂²	1	3	yes	—
17	#340×C₆₇	C₂²	1	6	yes	—
18	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
21	C₄×Q₈×C₆₇	—	4	8	no	—
22	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
24	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
26	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
29	C₂×C₄²×C₆₇	—	2	6	no	—
30	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
31	#106×C₆₇	—	2	12	no	—
32	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
33	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
34	#95×C₆₇	—	2	6	no	—
35	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
36	C₂×Q₈×C₆₇	SD₁₆	1	8	yes	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	2	6	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₆₇	—	2	6	no	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
46	C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₄⋊C₄×C₆₇	—	2	6	no	—
48	C₄⋊C₄×C₆₇	—	2	6	no	—
49	C₄⋊C₄×C₆₇	—	4	12	no	—
50	C₄⋊C₄×C₆₇	—	8	6	no	—
51	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
52	C₄⋊C₄×C₆₇	#41	1	4	yes	—
53	C₄²×C₆₇	—	4	12	no	—
54	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
55	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
56	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
57	Q₈×C₆₇	#106	1	16	yes	complemented
58	C₂×C₄×C₆₇	—	2	3	no	—
59	C₂×C₄×C₆₇	—	2	3	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	—	4	2	no	—
63	C₂×C₄×C₆₇	—	4	6	no	—
64	C₂×C₄×C₆₇	—	4	6	no	—
65	C₂×C₄×C₆₇	—	4	6	no	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	12	yes	—
69	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
72	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
73	C₈×C₆₇	—	2	6	no	—
74	C₈×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₆₇	#548	1	2	yes	—
77	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
78	C₂²×C₆₇	#576	1	1	yes	characteristic
79	C₂²×C₆₇	#578	1	1	yes	characteristic
80	C₄×C₆₇	—	2	6	no	—
81	C₄×C₆₇	—	4	4	no	—
82	C₄×C₆₇	—	8	6	no	—
83	C₄×C₆₇	C₂×#106	1	12	yes	—
84	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
85	C₄×C₆₇	#519	1	1	yes	characteristic
86	C₄×C₆₇	#557	1	2	yes	—
87	#22738	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₆₇	#2351	1	1	yes	characteristic
89	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
90	C₂×C₆₇	#2913	1	1	yes	characteristic
91	C₂×C₆₇	#3359	1	2	yes	—
92	C₂×C₆₇	#3362	1	2	yes	—
93	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
94	Q₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
95	#1948	C₂×C₆₇	1	6	yes	—
96	#1834	C₂×C₆₇	1	3	yes	—
97	C₄×#106	C₂×C₆₇	1	3	yes	—
98	C₆₇	#22738	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₄×Q₈	—	2	2	no	—
100	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
101	C₈×Q₈	C₄×C₆₇	1	8	yes	complemented
102	C₂×#106	C₂²×C₆₇	1	6	yes	—
103	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
104	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
105	#389	—	2	6	no	—
106	#384	C₂²×C₆₇	1	6	yes	—
107	C₄×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
108	#340	C₂²×C₆₇	1	3	yes	—
109	#340	C₂²×C₆₇	1	6	yes	—
110	#340	C₂²×C₆₇	1	1	yes	characteristic
111	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
114	C₄×Q₈	—	4	8	no	—
115	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
116	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
117	C₂×C₄⋊C₄	—	2	6	no	—
118	C₂×C₄⋊C₄	—	2	6	no	—
119	C₂×C₄⋊C₄	D₄×C₆₇	1	3	yes	—
120	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
121	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
122	C₂×C₄²	—	2	6	no	—
123	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
124	#106	—	2	12	no	—
125	C₄⋊C₈	C₂×C₄×C₆₇	1	12	yes	—
126	C₄×C₈	C₂×C₄×C₆₇	1	12	yes	—
127	#95	—	2	6	no	—
128	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
129	C₂×Q₈	SD₁₆×C₆₇	1	8	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	6	no	—
132	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
133	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	6	no	—
141	C₄⋊C₄	—	2	6	no	—
142	C₄⋊C₄	—	4	12	no	—
143	C₄⋊C₄	—	8	6	no	—
144	C₄⋊C₄	C₄⋊C₄×C₆₇	1	12	yes	—
145	C₄⋊C₄	#41×C₆₇	1	4	yes	—
146	C₄²	—	4	12	no	—
147	C₄²	C₂²×C₄×C₆₇	1	6	yes	—
148	C₄²	C₂×Q₈×C₆₇	1	6	yes	—
149	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
150	Q₈	#106×C₆₇	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	6	no	—
157	C₂×C₄	—	4	6	no	—
158	C₂×C₄	—	4	6	no	—
159	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
160	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
161	C₂×C₄	C₂×SD₁₆×C₆₇	1	12	yes	—
162	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
164	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
165	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
166	C₈	—	2	6	no	—
167	C₈	—	2	2	no	—
168	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
169	C₂²	#548×C₆₇	1	2	yes	—
170	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
171	C₂²	#576×C₆₇	1	1	yes	characteristic
172	C₂²	#578×C₆₇	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	6	no	—
176	C₄	C₂×#106×C₆₇	1	12	yes	—
177	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
178	C₄	#519×C₆₇	1	1	yes	characteristic
179	C₄	#557×C₆₇	1	2	yes	—
180	C₂	C₆₇×#2351	1	1	yes	characteristic
181	C₂	C₂²×#106×C₆₇	1	1	yes	characteristic
182	C₂	C₆₇×#2913	1	1	yes	characteristic
183	C₂	C₆₇×#3359	1	2	yes	—
184	C₂	C₆₇×#3362	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₆₇	C₄	semidirect
57. Q₈×C₆₇	#106	semidirect
87. #22738	C₆₇	direct (N × H)
98. C₆₇	#22738	direct (N × H)
101. C₈×Q₈	C₄×C₆₇	semidirect
150. Q₈	#106×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₆₇	2
Q₈×C₄⋊C₄×C₆₇	2
C₆₇×#1948	2
C₆₇×#1834	2
C₄×#106×C₆₇	2
#22738	67

Subgroups — C67×#22738 #30701989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22738 #30701989