Subgroups — C₂³×#100×C₆₇ #30739322

Order 17152.

7134 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#100×C₆₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₆₇	C₄	1	2	yes	—
5	C₂³×D₄×C₆₇	—	2	14	no	—
6	C₂³×D₄×C₆₇	—	2	2	no	—
7	C₂³×D₄×C₆₇	C₄	1	14	yes	—
8	C₂³×D₄×C₆₇	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	—	2	14	no	—
11	C₂²×M₁₆×C₆₇	C₂²	1	28	yes	complemented
12	C₂³×C₈×C₆₇	—	2	2	no	—
13	C₂×#100×C₆₇	C₂²	1	112	yes	direct product factor
14	C₂⁵×C₆₇	—	2	14	no	—
15	C₂⁵×C₆₇	—	2	2	no	—
16	C₂⁵×C₆₇	—	4	16	no	—
17	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂⁵×C₆₇	C₂×C₄	1	28	yes	—
19	C₂²×D₄×C₆₇	—	2	84	no	—
20	C₂²×D₄×C₆₇	—	2	28	no	—
21	C₂²×D₄×C₆₇	—	2	28	no	—
22	C₂²×D₄×C₆₇	—	2	84	no	—
23	C₂²×D₄×C₆₇	C₂×C₄	1	84	yes	—
24	C₂²×D₄×C₆₇	C₂³	1	28	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₆₇	D₄	1	14	yes	—
27	C₂³×C₄×C₆₇	D₄	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
29	C₂×M₁₆×C₆₇	—	2	84	no	—
30	C₂×M₁₆×C₆₇	C₂³	1	56	yes	complemented
31	C₂²×C₈×C₆₇	—	2	28	no	—
32	#100×C₆₇	C₂³	1	64	yes	direct product factor
33	C₂⁴×C₆₇	—	2	8	no	—
34	C₂⁴×C₆₇	—	2	84	no	—
35	C₂⁴×C₆₇	—	2	28	no	—
36	C₂⁴×C₆₇	—	4	112	no	—
37	C₂⁴×C₆₇	—	4	112	no	—
38	C₂⁴×C₆₇	—	4	16	no	—
39	C₂⁴×C₆₇	C₂²⋊C₄	1	7	yes	—
40	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₆₇	C₂²×C₄	1	7	yes	—
42	C₂⁴×C₆₇	C₂²×C₄	1	56	yes	—
43	C₂×D₄×C₆₇	—	2	168	no	—
44	C₂×D₄×C₆₇	—	2	168	no	—
45	C₂×D₄×C₆₇	—	2	56	no	—
46	C₂×D₄×C₆₇	—	2	56	no	—
47	C₂×D₄×C₆₇	C₂²×C₄	1	56	yes	—
48	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	42	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	14	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	14	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	42	yes	—
54	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
55	M₁₆×C₆₇	—	2	56	no	—
56	M₁₆×C₆₇	C₂⁴	1	16	yes	complemented
57	C₂×C₈×C₆₇	—	2	56	no	—
58	C₂³×C₆₇	—	2	28	no	—
59	C₂³×C₆₇	—	2	28	no	—
60	C₂³×C₆₇	—	2	56	no	—
61	C₂³×C₆₇	—	2	56	no	—
62	C₂³×C₆₇	—	4	112	no	—
63	C₂³×C₆₇	—	4	336	no	—
64	C₂³×C₆₇	—	4	112	no	—
65	C₂³×C₆₇	#100	1	8	yes	direct product factor
66	C₂³×C₆₇	C₂×C₂²⋊C₄	1	21	yes	—
67	C₂³×C₆₇	C₂×C₂²⋊C₄	1	7	yes	—
68	C₂³×C₆₇	C₂³×C₄	1	7	yes	—
69	C₂³×C₆₇	C₂³×C₄	1	16	yes	—
70	D₄×C₆₇	—	2	112	no	—
71	D₄×C₆₇	—	2	16	no	—
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	42	yes	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	42	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	14	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	14	yes	—
77	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
78	C₈×C₆₇	—	2	16	no	—
79	C₂²×C₆₇	—	2	42	no	—
80	C₂²×C₆₇	—	2	14	no	—
81	C₂²×C₆₇	—	2	16	no	—
82	C₂²×C₆₇	—	4	16	no	—
83	C₂²×C₆₇	—	4	112	no	—
84	C₂²×C₆₇	—	4	112	no	—
85	C₂²×C₆₇	C₂×#100	1	28	yes	direct product factor
86	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
87	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
88	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
89	C₄×C₆₇	C₂²×C₂²⋊C₄	1	14	yes	—
90	C₄×C₆₇	C₂³×D₄	1	2	yes	—
91	C₂³×#100	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₆₇	—	2	7	no	—
93	C₂×C₆₇	—	2	1	no	—
94	C₂×C₆₇	—	4	16	no	—
95	C₂×C₆₇	C₂²×#100	1	14	yes	direct product factor
96	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×C₆₇	1	1	yes	characteristic
98	C₂³×M₁₆	C₂×C₆₇	1	2	yes	complemented
99	C₂²×#100	C₂×C₆₇	1	28	yes	direct product factor
100	C₆₇	C₂³×#100	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁶	C₄×C₆₇	1	2	yes	—
102	C₂³×D₄	—	2	2	no	—
103	C₂³×D₄	—	2	14	no	—
104	C₂³×D₄	C₄×C₆₇	1	14	yes	—
105	C₂³×D₄	C₂²×C₆₇	1	14	yes	—
106	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
107	C₂²×M₁₆	—	2	14	no	—
108	C₂²×M₁₆	C₂²×C₆₇	1	28	yes	complemented
109	C₂³×C₈	—	2	2	no	—
110	C₂×#100	C₂²×C₆₇	1	112	yes	direct product factor
111	C₂⁵	—	2	14	no	—
112	C₂⁵	—	2	2	no	—
113	C₂⁵	—	4	16	no	—
114	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
115	C₂⁵	C₂×C₄×C₆₇	1	28	yes	—
116	C₂²×D₄	—	2	84	no	—
117	C₂²×D₄	—	2	28	no	—
118	C₂²×D₄	—	2	28	no	—
119	C₂²×D₄	—	2	84	no	—
120	C₂²×D₄	C₂×C₄×C₆₇	1	84	yes	—
121	C₂²×D₄	C₂³×C₆₇	1	28	yes	—
122	C₂³×C₄	C₂×C₄×C₆₇	1	7	yes	—
123	C₂³×C₄	D₄×C₆₇	1	2	yes	—
124	C₂³×C₄	D₄×C₆₇	1	14	yes	—
125	C₂³×C₄	C₂³×C₆₇	1	7	yes	—
126	C₂×M₁₆	—	2	84	no	—
127	C₂×M₁₆	C₂³×C₆₇	1	56	yes	complemented
128	C₂²×C₈	—	2	28	no	—
129	#100	C₂³×C₆₇	1	64	yes	direct product factor
130	C₂⁴	—	2	8	no	—
131	C₂⁴	—	2	84	no	—
132	C₂⁴	—	2	28	no	—
133	C₂⁴	—	4	112	no	—
134	C₂⁴	—	4	112	no	—
135	C₂⁴	—	4	16	no	—
136	C₂⁴	C₂²⋊C₄×C₆₇	1	7	yes	—
137	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
138	C₂⁴	C₂²×C₄×C₆₇	1	7	yes	—
139	C₂⁴	C₂²×C₄×C₆₇	1	56	yes	—
140	C₂×D₄	—	2	168	no	—
141	C₂×D₄	—	2	168	no	—
142	C₂×D₄	—	2	56	no	—
143	C₂×D₄	—	2	56	no	—
144	C₂×D₄	C₂²×C₄×C₆₇	1	56	yes	—
145	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₆₇	1	42	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₆₇	1	14	yes	—
148	C₂²×C₄	C₂²×C₄×C₆₇	1	21	yes	—
149	C₂²×C₄	C₂×D₄×C₆₇	1	14	yes	—
150	C₂²×C₄	C₂×D₄×C₆₇	1	42	yes	—
151	C₂²×C₄	C₂⁴×C₆₇	1	7	yes	—
152	M₁₆	—	2	56	no	—
153	M₁₆	C₂⁴×C₆₇	1	16	yes	complemented
154	C₂×C₈	—	2	56	no	—
155	C₂³	—	2	28	no	—
156	C₂³	—	2	28	no	—
157	C₂³	—	2	56	no	—
158	C₂³	—	2	56	no	—
159	C₂³	—	4	112	no	—
160	C₂³	—	4	336	no	—
161	C₂³	—	4	112	no	—
162	C₂³	#100×C₆₇	1	8	yes	direct product factor
163	C₂³	C₂×C₂²⋊C₄×C₆₇	1	21	yes	—
164	C₂³	C₂×C₂²⋊C₄×C₆₇	1	7	yes	—
165	C₂³	C₂³×C₄×C₆₇	1	7	yes	—
166	C₂³	C₂³×C₄×C₆₇	1	16	yes	—
167	D₄	—	2	112	no	—
168	D₄	—	2	16	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	42	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	42	yes	—
171	C₂×C₄	C₂³×C₄×C₆₇	1	7	yes	—
172	C₂×C₄	C₂²×D₄×C₆₇	1	14	yes	—
173	C₂×C₄	C₂²×D₄×C₆₇	1	14	yes	—
174	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
175	C₈	—	2	16	no	—
176	C₂²	—	2	42	no	—
177	C₂²	—	2	14	no	—
178	C₂²	—	2	16	no	—
179	C₂²	—	4	16	no	—
180	C₂²	—	4	112	no	—
181	C₂²	—	4	112	no	—
182	C₂²	C₂×#100×C₆₇	1	28	yes	direct product factor
183	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
184	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
185	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
186	C₄	C₂²×C₂²⋊C₄×C₆₇	1	14	yes	—
187	C₄	C₂³×D₄×C₆₇	1	2	yes	—
188	C₂	—	2	7	no	—
189	C₂	—	2	1	no	—
190	C₂	—	4	16	no	—
191	C₂	C₂²×#100×C₆₇	1	14	yes	direct product factor
192	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₆₇	C₂	semidirect
3. C₂²×#100×C₆₇	C₂	direct (N × H)
11. C₂²×M₁₆×C₆₇	C₂²	semidirect
13. C₂×#100×C₆₇	C₂²	direct (N × H)
30. C₂×M₁₆×C₆₇	C₂³	semidirect
32. #100×C₆₇	C₂³	direct (N × H)
56. M₁₆×C₆₇	C₂⁴	semidirect
65. C₂³×C₆₇	#100	direct (N × H)
85. C₂²×C₆₇	C₂×#100	direct (N × H)
91. C₂³×#100	C₆₇	direct (N × H)
95. C₂×C₆₇	C₂²×#100	direct (N × H)
98. C₂³×M₁₆	C₂×C₆₇	semidirect
99. C₂²×#100	C₂×C₆₇	direct (N × H)
100. C₆₇	C₂³×#100	direct (N × H)
108. C₂²×M₁₆	C₂²×C₆₇	semidirect
110. C₂×#100	C₂²×C₆₇	direct (N × H)
127. C₂×M₁₆	C₂³×C₆₇	semidirect
129. #100	C₂³×C₆₇	direct (N × H)
153. M₁₆	C₂⁴×C₆₇	semidirect
162. C₂³	#100×C₆₇	direct (N × H)
182. C₂²	C₂×#100×C₆₇	direct (N × H)
191. C₂	C₂²×#100×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
C₂⁴×D₄×C₆₇	2
C₂³×M₁₆×C₆₇	2
C₂²×#100×C₆₇	2
C₂³×#100	67

Subgroups — C23×#100×C67 #30739322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#100×C₆₇ #30739322