Subgroups — C₂²×#446×C₆₇ #30739437

Order 17152.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₆₇	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₄	1	12	yes	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
17	#446×C₆₇	C₂²	1	64	yes	direct product factor
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	12	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	24	yes	—
20	C₂×#97×C₆₇	C₂²	1	24	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	—
22	C₂²×Q₈×C₆₇	—	2	8	no	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	12	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	4	no	—
30	C₂²×C₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
35	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	—
36	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
43	C₄⋊C₈×C₆₇	C₂³	1	16	yes	—
44	C₄⋊C₈×C₆₇	C₂³	1	32	yes	—
45	#97×C₆₇	C₂³	1	32	yes	—
46	C₄×C₈×C₆₇	C₂³	1	16	yes	—
47	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	48	no	—
49	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
50	C₂²×C₄×C₆₇	—	2	8	no	—
51	C₂²×C₄×C₆₇	—	2	4	no	—
52	C₂²×C₄×C₆₇	—	2	8	no	—
53	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₆₇	#41	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
62	C₂×C₈×C₆₇	—	2	12	no	—
63	C₂×C₈×C₆₇	—	2	24	no	—
64	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	complemented
65	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
67	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
68	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
69	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
71	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
72	C₄²×C₆₇	C₂⁴	1	4	yes	—
73	C₄²×C₆₇	C₂⁴	1	8	yes	—
74	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
75	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
76	C₂³×C₆₇	#131	1	4	yes	—
77	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
78	Q₈×C₆₇	—	2	32	no	—
79	C₂×C₄×C₆₇	—	2	24	no	—
80	C₂×C₄×C₆₇	—	2	12	no	—
81	C₂×C₄×C₆₇	—	2	24	no	—
82	C₂×C₄×C₆₇	C₄×Q₈	1	12	yes	—
83	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
84	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
90	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
91	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
92	C₈×C₆₇	—	2	4	no	—
93	C₈×C₆₇	—	2	8	no	—
94	C₈×C₆₇	C₂²×Q₈	1	8	yes	complemented
95	C₂²×C₆₇	#446	1	16	yes	direct product factor
96	C₂²×C₆₇	C₂×C₄×Q₈	1	6	yes	—
97	C₂²×C₆₇	C₂²×M₁₆	1	6	yes	—
98	C₂²×C₆₇	C₂×#131	1	6	yes	—
99	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₆₇	—	2	8	no	—
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	2	8	no	—
103	C₄×C₆₇	C₂×C₄×Q₈	1	6	yes	—
104	C₄×C₆₇	C₂²×M₁₆	1	8	yes	—
105	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
106	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
107	C₂²×#446	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₆₇	C₂×#446	1	12	yes	direct product factor
109	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
111	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
112	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
113	C₂×#446	C₂×C₆₇	1	24	yes	direct product factor
114	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	—
115	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
116	C₂²×#97	C₂×C₆₇	1	2	yes	—
117	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
118	C₆₇	C₂²×#446	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×Q₈	C₄×C₆₇	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
122	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₄×C₆₇	1	12	yes	—
124	C₂×C₄×Q₈	C₂²×C₆₇	1	12	yes	—
125	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
126	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
127	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
128	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
129	#446	C₂²×C₆₇	1	64	yes	direct product factor
130	C₂×C₄⋊C₈	C₂²×C₆₇	1	12	yes	—
131	C₂×C₄⋊C₈	C₂²×C₆₇	1	24	yes	—
132	C₂×#97	C₂²×C₆₇	1	24	yes	—
133	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	—
134	C₂²×Q₈	—	2	8	no	—
135	C₂²×Q₈	C₂×C₄×C₆₇	1	12	yes	—
136	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
137	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
138	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
139	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
140	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	8	no	—
143	C₂²×C₈	Q₈×C₆₇	1	8	yes	complemented
144	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
145	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
146	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
147	C₄×Q₈	C₂×C₄×C₆₇	1	48	yes	—
148	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	24	yes	—
150	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	12	yes	—
151	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
152	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
153	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
154	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
155	C₄⋊C₈	C₂³×C₆₇	1	16	yes	—
156	C₄⋊C₈	C₂³×C₆₇	1	32	yes	—
157	#97	C₂³×C₆₇	1	32	yes	—
158	C₄×C₈	C₂³×C₆₇	1	16	yes	—
159	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
160	C₂×Q₈	—	2	48	no	—
161	C₂×Q₈	C₂²×C₄×C₆₇	1	16	yes	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
166	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
167	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
168	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
169	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
171	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
172	C₂²×C₄	#41×C₆₇	1	4	yes	—
173	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
174	C₂×C₈	—	2	12	no	—
175	C₂×C₈	—	2	24	no	—
176	C₂×C₈	C₂×Q₈×C₆₇	1	24	yes	complemented
177	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
178	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
179	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
180	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
181	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
182	C₄²	C₂²×C₄×C₆₇	1	24	yes	—
183	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
184	C₄²	C₂⁴×C₆₇	1	4	yes	—
185	C₄²	C₂⁴×C₆₇	1	8	yes	—
186	C₂³	C₄×Q₈×C₆₇	1	4	yes	—
187	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
188	C₂³	#131×C₆₇	1	4	yes	—
189	C₂³	C₂³×C₄×C₆₇	1	3	yes	—
190	Q₈	—	2	32	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	24	no	—
194	C₂×C₄	C₄×Q₈×C₆₇	1	12	yes	—
195	C₂×C₄	C₂×M₁₆×C₆₇	1	24	yes	—
196	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
197	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
198	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
199	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
200	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
201	C₂×C₄	C₂²×Q₈×C₆₇	1	6	yes	—
202	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
203	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	2	8	no	—
206	C₈	C₂²×Q₈×C₆₇	1	8	yes	complemented
207	C₂²	#446×C₆₇	1	16	yes	direct product factor
208	C₂²	C₂×C₄×Q₈×C₆₇	1	6	yes	—
209	C₂²	C₂²×M₁₆×C₆₇	1	6	yes	—
210	C₂²	C₂×#131×C₆₇	1	6	yes	—
211	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
212	C₄	—	2	8	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×Q₈×C₆₇	1	6	yes	—
216	C₄	C₂²×M₁₆×C₆₇	1	8	yes	—
217	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
218	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
219	C₂	C₂×#446×C₆₇	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Q₈×C₆₇	1	1	yes	characteristic
221	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic
222	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#446×C₆₇	C₂	direct (N × H)
17. #446×C₆₇	C₂²	direct (N × H)
31. C₂²×C₈×C₆₇	Q₈	semidirect
64. C₂×C₈×C₆₇	C₂×Q₈	semidirect
94. C₈×C₆₇	C₂²×Q₈	semidirect
95. C₂²×C₆₇	#446	direct (N × H)
107. C₂²×#446	C₆₇	direct (N × H)
108. C₂×C₆₇	C₂×#446	direct (N × H)
113. C₂×#446	C₂×C₆₇	direct (N × H)
118. C₆₇	C₂²×#446	direct (N × H)
129. #446	C₂²×C₆₇	direct (N × H)
143. C₂²×C₈	Q₈×C₆₇	semidirect
176. C₂×C₈	C₂×Q₈×C₆₇	semidirect
206. C₈	C₂²×Q₈×C₆₇	semidirect
207. C₂²	#446×C₆₇	direct (N × H)
219. C₂	C₂×#446×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#446×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#446	67

Subgroups — C22×#446×C67 #30739437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#446×C₆₇ #30739437