Subgroups — C₆₇×#20806 #30700057

Order 17152.

994 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₆₇	C₂	1	4	yes	maximal
2	C₂×#423×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1849	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1848	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	#430×C₆₇	—	2	16	no	—
10	C₂×#108×C₆₇	C₂²	1	16	yes	—
11	#423×C₆₇	—	2	4	no	—
12	#423×C₆₇	C₂²	1	4	yes	—
13	#423×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
15	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
16	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₆₇	—	2	4	no	—
22	C₂×M₁₆×C₆₇	—	2	16	no	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
25	C₂×M₁₆×C₆₇	Q₈	1	8	yes	—
26	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
27	C₂²×C₈×C₆₇	—	2	4	no	—
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	4	yes	—
32	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#108×C₆₇	—	2	32	no	—
36	C₄⋊C₈×C₆₇	—	2	8	no	—
37	C₄⋊C₈×C₆₇	—	2	8	no	—
38	#97×C₆₇	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	16	no	—
54	M₁₆×C₆₇	—	4	8	no	—
55	M₁₆×C₆₇	C₄⋊C₄	1	32	yes	—
56	C₂×C₈×C₆₇	—	2	8	no	—
57	C₂×C₈×C₆₇	—	4	8	no	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
60	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
61	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
62	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₆₇	C₂×D₄	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	2	yes	—
68	C₂³×C₆₇	—	4	2	no	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
71	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	4	4	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₆₇	#127	1	4	yes	—
82	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
83	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
88	C₈×C₆₇	—	4	8	no	—
89	C₈×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
90	C₂²×C₆₇	—	2	4	no	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	—	4	4	no	—
93	C₂²×C₆₇	#384	1	4	yes	—
94	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
96	C₄×C₆₇	—	2	4	no	—
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	#384	1	4	yes	—
99	C₄×C₆₇	#430	1	4	yes	complemented
100	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
101	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
102	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	#20806	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	—	2	2	no	—
105	C₂×C₆₇	—	2	4	no	—
106	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
107	C₂×C₆₇	C₂×#430	1	2	yes	—
108	C₂×#430	C₂×C₆₇	1	4	yes	—
109	C₂×#423	C₂×C₆₇	1	2	yes	—
110	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
111	#1849	C₂×C₆₇	1	4	yes	complemented
112	#1848	C₂×C₆₇	1	4	yes	complemented
113	C₆₇	#20806	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
115	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
116	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
117	#430	—	2	16	no	—
118	C₂×#108	C₂²×C₆₇	1	16	yes	—
119	#423	—	2	4	no	—
120	#423	C₂²×C₆₇	1	4	yes	—
121	#423	C₂²×C₆₇	1	4	yes	—
122	C₂×C₄⋊C₈	—	2	4	no	—
123	C₄×M₁₆	C₄×C₆₇	1	4	yes	—
124	C₄×M₁₆	C₄×C₆₇	1	4	yes	—
125	C₂×#97	C₂²×C₆₇	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
127	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
128	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	16	no	—
131	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
132	C₂×M₁₆	D₄×C₆₇	1	8	yes	—
133	C₂×M₁₆	Q₈×C₆₇	1	8	yes	—
134	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
138	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
139	C₂×C₄²	D₄×C₆₇	1	4	yes	—
140	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
141	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
142	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
143	#108	—	2	32	no	—
144	C₄⋊C₈	—	2	8	no	—
145	C₄⋊C₈	—	2	8	no	—
146	#97	C₂×C₄×C₆₇	1	8	yes	—
147	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
148	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
149	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
152	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
155	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
156	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
157	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
158	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
160	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
161	M₁₆	—	2	16	no	—
162	M₁₆	—	4	8	no	—
163	M₁₆	C₄⋊C₄×C₆₇	1	32	yes	—
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	—	4	8	no	—
166	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
167	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
168	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
169	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
170	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
171	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
172	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
173	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
174	C₄²	C₂×D₄×C₆₇	1	2	yes	—
175	C₄²	C₂×D₄×C₆₇	1	2	yes	—
176	C₂³	—	4	2	no	—
177	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
178	C₂³	C₄⋊Q₈×C₆₇	1	4	yes	—
179	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
186	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
187	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
189	C₂×C₄	#127×C₆₇	1	4	yes	—
190	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
191	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
192	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
196	C₈	—	4	8	no	—
197	C₈	C₂×C₄⋊C₄×C₆₇	1	16	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	#384×C₆₇	1	4	yes	—
202	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
203	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	#384×C₆₇	1	4	yes	—
207	C₄	#430×C₆₇	1	4	yes	complemented
208	C₄	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
209	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
210	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	4	no	—
213	C₂	C₂×#384×C₆₇	1	1	yes	characteristic
214	C₂	C₂×#430×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1849	C₂	semidirect
5. C₆₇×#1848	C₂	semidirect
99. C₄×C₆₇	#430	semidirect
103. #20806	C₆₇	direct (N × H)
111. #1849	C₂×C₆₇	semidirect
112. #1848	C₂×C₆₇	semidirect
113. C₆₇	#20806	direct (N × H)
207. C₄	#430×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#430×C₆₇	2
C₂×#423×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#1849	2
C₆₇×#1848	2
#20806	67

Subgroups — C67×#20806 #30700057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20806 #30700057