Subgroups — C₂²×#441×C₆₇ #30739404

Order 17152.

2790 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	2	yes	maximal
3	C₂×#441×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#103×C₆₇	C₂	1	2	yes	maximal
6	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₆₇	C₄	1	16	yes	complemented
9	C₂²×Q₁₆×C₆₇	C₂²	1	12	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	—	2	12	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	24	yes	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	—	2	2	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#441×C₆₇	C₂²	1	64	yes	direct product factor
19	C₂×#106×C₆₇	C₂²	1	12	yes	—
20	C₂×#103×C₆₇	C₂²	1	24	yes	—
21	C₂×#97×C₆₇	C₂²	1	12	yes	—
22	C₂²×Q₈×C₆₇	—	2	6	no	—
23	C₂²×Q₈×C₆₇	—	2	8	no	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	16	yes	—
25	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
26	C₂³×C₄×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	—	2	2	no	—
28	C₂³×C₄×C₆₇	D₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×Q₁₆×C₆₇	C₂×C₄	1	96	yes	complemented
31	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	—
32	C₂²×C₈×C₆₇	—	2	4	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
36	C₄×Q₈×C₆₇	—	2	48	no	—
37	C₄×Q₈×C₆₇	C₂³	1	32	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
41	C₂×C₄²×C₆₇	—	2	24	no	—
42	C₂×C₄²×C₆₇	D₄	1	6	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
44	#106×C₆₇	C₂³	1	16	yes	—
45	#103×C₆₇	C₂³	1	32	yes	—
46	#97×C₆₇	C₂³	1	16	yes	—
47	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	48	no	—
49	C₂×Q₈×C₆₇	—	2	12	no	—
50	C₂×Q₈×C₆₇	—	2	24	no	—
51	C₂×Q₈×C₆₇	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₆₇	—	2	16	no	—
54	C₂²×C₄×C₆₇	—	2	8	no	—
55	C₂²×C₄×C₆₇	—	2	12	no	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	4	8	no	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₆₇	#41	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
63	Q₁₆×C₆₇	C₂²×C₄	1	64	yes	complemented
64	C₂×C₈×C₆₇	—	2	12	no	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
66	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
67	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
68	C₄⋊C₄×C₆₇	—	2	32	no	—
69	C₄⋊C₄×C₆₇	—	2	12	no	—
70	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
71	C₄²×C₆₇	—	2	32	no	—
72	C₄²×C₆₇	C₂×D₄	1	12	yes	—
73	C₄²×C₆₇	C₂⁴	1	4	yes	—
74	C₂³×C₆₇	C₄×D₄	1	4	yes	—
75	C₂³×C₆₇	#137	1	8	yes	—
76	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
77	Q₈×C₆₇	—	2	48	no	—
78	Q₈×C₆₇	—	2	8	no	—
79	Q₈×C₆₇	C₂³×C₄	1	16	yes	—
80	C₂×C₄×C₆₇	—	2	48	no	—
81	C₂×C₄×C₆₇	—	2	24	no	—
82	C₂×C₄×C₆₇	—	2	8	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	—	4	24	no	—
85	C₂×C₄×C₆₇	C₄×D₄	1	12	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
90	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
91	C₈×C₆₇	—	2	4	no	—
92	C₈×C₆₇	C₂³×C₄	1	8	yes	—
93	C₂²×C₆₇	#441	1	16	yes	direct product factor
94	C₂²×C₆₇	C₂×C₄×D₄	1	6	yes	—
95	C₂²×C₆₇	C₂×#137	1	12	yes	—
96	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
97	C₄×C₆₇	—	2	16	no	—
98	C₄×C₆₇	—	2	8	no	—
99	C₄×C₆₇	—	4	8	no	—
100	C₄×C₆₇	C₂×C₄×D₄	1	6	yes	—
101	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
102	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
103	C₂²×#441	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	C₂×#441	1	12	yes	direct product factor
105	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×#137	1	2	yes	—
107	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
108	C₂²×C₄×Q₈	C₂×C₆₇	1	2	yes	—
109	C₂×#441	C₂×C₆₇	1	24	yes	direct product factor
110	C₂²×#106	C₂×C₆₇	1	1	yes	characteristic
111	C₂²×#103	C₂×C₆₇	1	2	yes	—
112	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
113	C₆₇	C₂²×#441	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
115	C₂²×Q₁₆	C₄×C₆₇	1	16	yes	complemented
116	C₂²×Q₁₆	C₂²×C₆₇	1	12	yes	—
117	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
119	C₂×C₄×Q₈	—	2	12	no	—
120	C₂×C₄×Q₈	C₂²×C₆₇	1	24	yes	—
121	C₂²×C₄⋊C₄	—	2	2	no	—
122	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
123	C₂²×C₄²	—	2	2	no	—
124	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
125	#441	C₂²×C₆₇	1	64	yes	direct product factor
126	C₂×#106	C₂²×C₆₇	1	12	yes	—
127	C₂×#103	C₂²×C₆₇	1	24	yes	—
128	C₂×#97	C₂²×C₆₇	1	12	yes	—
129	C₂²×Q₈	—	2	6	no	—
130	C₂²×Q₈	—	2	8	no	—
131	C₂²×Q₈	C₂×C₄×C₆₇	1	16	yes	—
132	C₂²×Q₈	C₂³×C₆₇	1	12	yes	—
133	C₂³×C₄	—	2	2	no	—
134	C₂³×C₄	—	2	2	no	—
135	C₂³×C₄	D₄×C₆₇	1	2	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
137	C₂×Q₁₆	C₂×C₄×C₆₇	1	96	yes	complemented
138	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
141	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
142	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
143	C₄×Q₈	—	2	48	no	—
144	C₄×Q₈	C₂³×C₆₇	1	32	yes	—
145	C₂×C₄⋊C₄	—	2	24	no	—
146	C₂×C₄⋊C₄	—	2	6	no	—
147	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
148	C₂×C₄²	—	2	24	no	—
149	C₂×C₄²	D₄×C₆₇	1	6	yes	—
150	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
151	#106	C₂³×C₆₇	1	16	yes	—
152	#103	C₂³×C₆₇	1	32	yes	—
153	#97	C₂³×C₆₇	1	16	yes	—
154	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
155	C₂×Q₈	—	2	48	no	—
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	24	no	—
158	C₂×Q₈	C₂²×C₄×C₆₇	1	48	yes	—
159	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
160	C₂²×C₄	—	2	16	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
166	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
167	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
168	C₂²×C₄	#41×C₆₇	1	4	yes	—
169	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
170	Q₁₆	C₂²×C₄×C₆₇	1	64	yes	complemented
171	C₂×C₈	—	2	12	no	—
172	C₂×C₈	C₂²×C₄×C₆₇	1	24	yes	—
173	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
174	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
175	C₄⋊C₄	—	2	32	no	—
176	C₄⋊C₄	—	2	12	no	—
177	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
178	C₄²	—	2	32	no	—
179	C₄²	C₂×D₄×C₆₇	1	12	yes	—
180	C₄²	C₂⁴×C₆₇	1	4	yes	—
181	C₂³	C₄×D₄×C₆₇	1	4	yes	—
182	C₂³	#137×C₆₇	1	8	yes	—
183	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
184	Q₈	—	2	48	no	—
185	Q₈	—	2	8	no	—
186	Q₈	C₂³×C₄×C₆₇	1	16	yes	—
187	C₂×C₄	—	2	48	no	—
188	C₂×C₄	—	2	24	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	4	24	no	—
192	C₂×C₄	C₄×D₄×C₆₇	1	12	yes	—
193	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
194	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
195	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
196	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
197	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₂³×C₄×C₆₇	1	8	yes	—
200	C₂²	#441×C₆₇	1	16	yes	direct product factor
201	C₂²	C₂×C₄×D₄×C₆₇	1	6	yes	—
202	C₂²	C₂×#137×C₆₇	1	12	yes	—
203	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	—	2	16	no	—
206	C₄	—	4	8	no	—
207	C₄	C₂×C₄×D₄×C₆₇	1	6	yes	—
208	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
209	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
210	C₂	C₂×#441×C₆₇	1	12	yes	direct product factor
211	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
212	C₂	C₂²×#137×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#441×C₆₇	C₂	direct (N × H)
8. C₂²×Q₁₆×C₆₇	C₄	semidirect
18. #441×C₆₇	C₂²	direct (N × H)
30. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
63. Q₁₆×C₆₇	C₂²×C₄	semidirect
93. C₂²×C₆₇	#441	direct (N × H)
103. C₂²×#441	C₆₇	direct (N × H)
104. C₂×C₆₇	C₂×#441	direct (N × H)
109. C₂×#441	C₂×C₆₇	direct (N × H)
113. C₆₇	C₂²×#441	direct (N × H)
115. C₂²×Q₁₆	C₄×C₆₇	semidirect
125. #441	C₂²×C₆₇	direct (N × H)
137. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
170. Q₁₆	C₂²×C₄×C₆₇	semidirect
200. C₂²	#441×C₆₇	direct (N × H)
210. C₂	C₂×#441×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
C₂³×Q₁₆×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₂×#441×C₆₇	2
C₂²×#106×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×#441	67

Subgroups — C22×#441×C67 #30739404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#441×C₆₇ #30739404