Subgroups — C₆₇×#10845 #30690096

Order 17152.

774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	2	yes	maximal
2	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₆₇	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
7	#423×C₆₇	—	2	4	no	—
8	#423×C₆₇	C₄	1	8	yes	—
9	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
10	C₂×#97×C₆₇	—	2	4	no	—
11	C₂×#97×C₆₇	C₄	1	4	yes	—
12	#341×C₆₇	—	2	8	no	—
13	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
14	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
15	C₂×M₁₆×C₆₇	—	2	4	no	—
16	C₂×M₁₆×C₆₇	—	2	8	no	—
17	C₂×M₁₆×C₆₇	—	2	8	no	—
18	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₆₇	—	2	4	no	—
20	C₂²×C₈×C₆₇	—	2	4	no	—
21	C₂²×C₈×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₇	D₄	1	4	yes	—
24	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₆₇	D₄	1	4	yes	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₄⋊C₈×C₆₇	—	2	16	no	—
30	#97×C₆₇	—	2	8	no	—
31	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	—	4	4	no	—
33	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
41	C₂²×C₄×C₆₇	#41	1	4	yes	—
42	M₁₆×C₆₇	—	2	16	no	—
43	M₁₆×C₆₇	—	4	8	no	—
44	C₂×C₈×C₆₇	—	2	4	no	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	—	2	8	no	—
47	C₂×C₈×C₆₇	—	2	8	no	—
48	C₂×C₈×C₆₇	—	2	8	no	—
49	C₂×C₈×C₆₇	—	4	4	no	—
50	C₂×C₈×C₆₇	—	4	4	no	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
52	C₄²×C₆₇	C₄²	1	4	yes	—
53	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
54	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₂³×C₆₇	—	2	4	no	—
56	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₆₇	#125	1	2	yes	—
59	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
60	C₂×C₄×C₆₇	—	2	8	no	—
61	C₂×C₄×C₆₇	—	4	8	no	—
62	C₂×C₄×C₆₇	#95	1	8	yes	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₆₇	#117	1	2	yes	—
68	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
70	C₂×C₄×C₆₇	#121	1	4	yes	—
71	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
72	C₂×C₄×C₆₇	#124	1	2	yes	—
73	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₆₇	—	2	8	no	—
76	C₈×C₆₇	—	4	8	no	—
77	C₂²×C₆₇	—	2	4	no	—
78	C₂²×C₆₇	—	2	4	no	—
79	C₂²×C₆₇	—	2	4	no	—
80	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
81	C₂²×C₆₇	#382	1	1	yes	characteristic
82	C₂²×C₆₇	#384	1	1	yes	characteristic
83	C₂²×C₆₇	#389	1	4	yes	—
84	C₂²×C₆₇	#430	1	4	yes	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
87	C₄×C₆₇	#380	1	2	yes	—
88	C₄×C₆₇	#390	1	2	yes	—
89	C₄×C₆₇	#391	1	2	yes	—
90	#10845	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₆₇	—	2	4	no	—
92	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
93	C₂×C₆₇	#1738	1	2	yes	—
94	C₂×C₆₇	#1838	1	2	yes	—
95	C₂×C₆₇	#1849	1	2	yes	—
96	C₂×#423	C₂×C₆₇	1	2	yes	—
97	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
98	C₂×#341	C₂×C₆₇	1	4	yes	—
99	C₆₇	#10845	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
101	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
102	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
103	#423	—	2	4	no	—
104	#423	C₄×C₆₇	1	8	yes	—
105	C₂×C₄⋊C₈	—	2	4	no	—
106	C₂×#97	—	2	4	no	—
107	C₂×#97	C₄×C₆₇	1	4	yes	—
108	#341	—	2	8	no	—
109	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
110	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	—	2	8	no	—
113	C₂×M₁₆	—	2	8	no	—
114	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
115	C₂²×C₈	—	2	4	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
119	C₂²×C₈	D₄×C₆₇	1	4	yes	—
120	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
121	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
123	C₂×C₄²	D₄×C₆₇	1	4	yes	—
124	C₂×C₄²	D₄×C₆₇	1	2	yes	—
125	C₄⋊C₈	—	2	16	no	—
126	#97	—	2	8	no	—
127	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
130	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
131	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
133	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
136	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
137	C₂²×C₄	#41×C₆₇	1	4	yes	—
138	M₁₆	—	2	16	no	—
139	M₁₆	—	4	8	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	8	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	4	4	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
148	C₄²	C₄²×C₆₇	1	4	yes	—
149	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
150	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
151	C₂³	—	2	4	no	—
152	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
153	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
154	C₂³	#125×C₆₇	1	2	yes	—
155	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	#95×C₆₇	1	8	yes	—
159	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
163	C₂×C₄	#117×C₆₇	1	2	yes	—
164	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
165	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
166	C₂×C₄	#121×C₆₇	1	4	yes	—
167	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
168	C₂×C₄	#124×C₆₇	1	2	yes	—
169	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
170	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
171	C₈	—	2	8	no	—
172	C₈	—	4	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
177	C₂²	#382×C₆₇	1	1	yes	characteristic
178	C₂²	#384×C₆₇	1	1	yes	characteristic
179	C₂²	#389×C₆₇	1	4	yes	—
180	C₂²	#430×C₆₇	1	4	yes	—
181	C₄	—	2	8	no	—
182	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
183	C₄	#380×C₆₇	1	2	yes	—
184	C₄	#390×C₆₇	1	2	yes	—
185	C₄	#391×C₆₇	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
188	C₂	C₆₇×#1738	1	2	yes	—
189	C₂	C₆₇×#1838	1	2	yes	—
190	C₂	C₆₇×#1849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
90. #10845	C₆₇	direct (N × H)
99. C₆₇	#10845	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×#423×C₆₇	2
C₂²×#97×C₆₇	2
C₂×#341×C₆₇	2
#10845	67

Subgroups — C67×#10845 #30690096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10845 #30690096