Subgroups — C₆₇×#10844 #30690095

Order 17152.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 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×C₄×C₈×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×C₄×C₈×C₆₇	—	2	4	no	—
11	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
13	#341×C₆₇	—	2	8	no	—
14	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
15	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₆₇	—	2	4	no	—
17	C₂×M₁₆×C₆₇	—	2	8	no	—
18	C₂×M₁₆×C₆₇	—	2	8	no	—
19	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₆₇	—	2	4	no	—
21	C₂²×C₈×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	D₄	1	4	yes	—
25	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₇	D₄	1	4	yes	—
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₄⋊C₈×C₆₇	—	2	16	no	—
32	C₄×C₈×C₆₇	—	2	4	no	—
33	C₄×C₈×C₆₇	—	2	4	no	—
34	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	8	no	—
36	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₆₇	#41	1	4	yes	—
46	M₁₆×C₆₇	—	2	16	no	—
47	M₁₆×C₆₇	—	4	8	no	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	—	2	8	no	—
52	C₂×C₈×C₆₇	—	2	8	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₂×C₈×C₆₇	—	4	4	no	—
55	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
56	C₄²×C₆₇	C₄²	1	2	yes	—
57	C₄²×C₆₇	C₄²	1	2	yes	—
58	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
59	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂³×C₆₇	—	2	4	no	—
62	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₆₇	#125	1	2	yes	—
65	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	2	16	no	—
68	C₂×C₄×C₆₇	#95	1	4	yes	—
69	C₂×C₄×C₆₇	#95	1	4	yes	—
70	C₂×C₄×C₆₇	#108	1	8	yes	—
71	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₆₇	#117	1	2	yes	—
76	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
78	C₂×C₄×C₆₇	#121	1	4	yes	—
79	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₆₇	#124	1	2	yes	—
81	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₆₇	—	2	8	no	—
84	C₈×C₆₇	—	4	8	no	—
85	C₂²×C₆₇	—	2	4	no	—
86	C₂²×C₆₇	—	2	4	no	—
87	C₂²×C₆₇	—	2	4	no	—
88	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
89	C₂²×C₆₇	#382	1	1	yes	characteristic
90	C₂²×C₆₇	#384	1	1	yes	characteristic
91	C₂²×C₆₇	#389	1	4	yes	—
92	C₂²×C₆₇	C₂×#108	1	4	yes	—
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	#341	1	8	yes	complemented
95	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
96	C₄×C₆₇	#380	1	2	yes	—
97	C₄×C₆₇	#390	1	2	yes	—
98	C₄×C₆₇	#391	1	2	yes	—
99	#10844	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₆₇	—	2	4	no	—
101	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#341	1	2	yes	—
103	C₂×C₆₇	#1837	1	2	yes	—
104	C₂×C₆₇	#1848	1	2	yes	—
105	C₂×#423	C₂×C₆₇	1	2	yes	—
106	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
107	C₂×#341	C₂×C₆₇	1	4	yes	—
108	C₆₇	#10844	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
110	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
111	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
112	#423	—	2	4	no	—
113	#423	C₄×C₆₇	1	8	yes	—
114	C₂×C₄⋊C₈	—	2	4	no	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
117	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
118	#341	—	2	8	no	—
119	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
120	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	8	no	—
123	C₂×M₁₆	—	2	8	no	—
124	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
129	C₂²×C₈	D₄×C₆₇	1	4	yes	—
130	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
131	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
134	C₂×C₄²	D₄×C₆₇	1	4	yes	—
135	C₂×C₄²	D₄×C₆₇	1	2	yes	—
136	C₄⋊C₈	—	2	16	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
142	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
143	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
147	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
149	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
150	C₂²×C₄	#41×C₆₇	1	4	yes	—
151	M₁₆	—	2	16	no	—
152	M₁₆	—	4	8	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	4	4	no	—
160	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
161	C₄²	C₄²×C₆₇	1	2	yes	—
162	C₄²	C₄²×C₆₇	1	2	yes	—
163	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
164	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
165	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
166	C₂³	—	2	4	no	—
167	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
168	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
169	C₂³	#125×C₆₇	1	2	yes	—
170	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	#95×C₆₇	1	4	yes	—
174	C₂×C₄	#95×C₆₇	1	4	yes	—
175	C₂×C₄	#108×C₆₇	1	8	yes	—
176	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
179	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
180	C₂×C₄	#117×C₆₇	1	2	yes	—
181	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
182	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
183	C₂×C₄	#121×C₆₇	1	4	yes	—
184	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
185	C₂×C₄	#124×C₆₇	1	2	yes	—
186	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
187	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
188	C₈	—	2	8	no	—
189	C₈	—	4	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
194	C₂²	#382×C₆₇	1	1	yes	characteristic
195	C₂²	#384×C₆₇	1	1	yes	characteristic
196	C₂²	#389×C₆₇	1	4	yes	—
197	C₂²	C₂×#108×C₆₇	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	#341×C₆₇	1	8	yes	complemented
200	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
201	C₄	#380×C₆₇	1	2	yes	—
202	C₄	#390×C₆₇	1	2	yes	—
203	C₄	#391×C₆₇	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
206	C₂	C₂×#341×C₆₇	1	2	yes	—
207	C₂	C₆₇×#1837	1	2	yes	—
208	C₂	C₆₇×#1848	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
94. C₄×C₆₇	#341	semidirect
99. #10844	C₆₇	direct (N × H)
108. C₆₇	#10844	direct (N × H)
199. C₄	#341×C₆₇	semidirect

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×C₄×C₈×C₆₇	2
C₂×#341×C₆₇	2
#10844	67

Subgroups — C67×#10844 #30690095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10844 #30690095