Subgroups — C₂²×#433×C₆₇ #30739367

Order 17152.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#433×C₆₇	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#117×C₆₇	C₄	1	12	yes	—
11	C₂×#117×C₆₇	C₂²	1	12	yes	—
12	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#433×C₆₇	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	24	yes	complemented
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	24	yes	—
19	C₂×#97×C₆₇	C₂²	1	12	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
21	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	—	2	8	no	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	8	no	—
30	C₂²×C₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
33	#117×C₆₇	C₂×C₄	1	48	yes	—
34	#117×C₆₇	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
41	C₄⋊C₈×C₆₇	C₂³	1	32	yes	complemented
42	C₂²⋊C₈×C₆₇	C₂³	1	32	yes	—
43	#97×C₆₇	C₂³	1	16	yes	complemented
44	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
45	C₂⁴×C₆₇	—	2	4	no	—
46	C₂⁴×C₆₇	—	2	8	no	—
47	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
48	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	2	8	no	—
51	C₂²×C₄×C₆₇	—	2	8	no	—
52	C₂²×C₄×C₆₇	—	2	24	no	—
53	C₂²×C₄×C₆₇	—	2	24	no	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₆₇	#41	1	8	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
62	C₂×C₈×C₆₇	—	2	24	no	—
63	C₂×C₈×C₆₇	—	2	24	no	—
64	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
65	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
66	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
67	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
68	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
70	C₄²×C₆₇	C₂⁴	1	4	yes	—
71	C₄²×C₆₇	C₂⁴	1	4	yes	—
72	C₂³×C₆₇	—	2	12	no	—
73	C₂³×C₆₇	—	2	24	no	—
74	C₂³×C₆₇	—	4	16	no	—
75	C₂³×C₆₇	#117	1	4	yes	—
76	C₂³×C₆₇	#131	1	8	yes	—
77	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
78	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
79	C₂×C₄×C₆₇	—	2	12	no	—
80	C₂×C₄×C₆₇	—	2	24	no	—
81	C₂×C₄×C₆₇	—	2	24	no	—
82	C₂×C₄×C₆₇	—	2	24	no	—
83	C₂×C₄×C₆₇	—	2	8	no	—
84	C₂×C₄×C₆₇	#117	1	12	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	4	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
90	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
91	C₈×C₆₇	—	2	8	no	—
92	C₈×C₆₇	—	2	8	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	—	2	8	no	—
95	C₂²×C₆₇	—	4	24	no	—
96	C₂²×C₆₇	#433	1	16	yes	direct product factor
97	C₂²×C₆₇	C₂×#117	1	6	yes	—
98	C₂²×C₆₇	C₂×#131	1	12	yes	—
99	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	—	2	8	no	—
102	C₄×C₆₇	—	2	8	no	—
103	C₄×C₆₇	C₂×#117	1	6	yes	—
104	C₄×C₆₇	C₂²×#41	1	2	yes	—
105	C₂²×#433	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂×C₆₇	—	4	4	no	—
107	C₂×C₆₇	C₂×#433	1	12	yes	direct product factor
108	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
109	C₂×C₆₇	C₂²×#131	1	2	yes	—
110	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic
111	C₂×#433	C₂×C₆₇	1	24	yes	direct product factor
112	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
113	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
114	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
115	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
116	C₆₇	C₂²×#433	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
119	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
120	C₂×#117	C₄×C₆₇	1	12	yes	—
121	C₂×#117	C₂²×C₆₇	1	12	yes	—
122	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
123	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
124	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
125	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
126	#433	C₂²×C₆₇	1	64	yes	direct product factor
127	C₂×C₄⋊C₈	C₂²×C₆₇	1	24	yes	complemented
128	C₂×C₂²⋊C₈	C₂²×C₆₇	1	24	yes	—
129	C₂×#97	C₂²×C₆₇	1	12	yes	complemented
130	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
131	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
132	C₂³×C₄	—	2	8	no	—
133	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
134	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
136	C₂³×C₄	C₂×C₄×C₆₇	1	6	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
142	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
143	#117	C₂×C₄×C₆₇	1	48	yes	—
144	#117	C₂³×C₆₇	1	16	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	24	yes	—
146	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	24	yes	—
147	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
148	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
149	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
150	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
151	C₄⋊C₈	C₂³×C₆₇	1	32	yes	complemented
152	C₂²⋊C₈	C₂³×C₆₇	1	32	yes	—
153	#97	C₂³×C₆₇	1	16	yes	complemented
154	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	C₂²×C₄×C₆₇	1	6	yes	—
158	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	24	no	—
163	C₂²×C₄	—	2	24	no	—
164	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
165	C₂²×C₄	C₂²×C₄×C₆₇	1	12	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	3	yes	—
169	C₂²×C₄	#41×C₆₇	1	8	yes	—
170	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
172	C₂×C₈	—	2	24	no	—
173	C₂×C₈	—	2	24	no	—
174	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
175	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
176	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
177	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
178	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
179	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
180	C₄²	C₂⁴×C₆₇	1	4	yes	—
181	C₄²	C₂⁴×C₆₇	1	4	yes	—
182	C₂³	—	2	12	no	—
183	C₂³	—	2	24	no	—
184	C₂³	—	4	16	no	—
185	C₂³	#117×C₆₇	1	4	yes	—
186	C₂³	#131×C₆₇	1	8	yes	—
187	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
188	C₂³	C₂³×C₄×C₆₇	1	3	yes	—
189	C₂×C₄	—	2	12	no	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	#117×C₆₇	1	12	yes	—
195	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
196	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
197	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
198	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
199	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
200	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
201	C₈	—	2	8	no	—
202	C₈	—	2	8	no	—
203	C₂²	—	2	8	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	4	24	no	—
206	C₂²	#433×C₆₇	1	16	yes	direct product factor
207	C₂²	C₂×#117×C₆₇	1	6	yes	—
208	C₂²	C₂×#131×C₆₇	1	12	yes	—
209	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	—	2	8	no	—
212	C₄	—	2	4	no	—
213	C₄	C₂×#117×C₆₇	1	6	yes	—
214	C₄	C₂²×#41×C₆₇	1	2	yes	—
215	C₂	—	4	4	no	—
216	C₂	C₂×#433×C₆₇	1	12	yes	direct product factor
217	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
218	C₂	C₂²×#131×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×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×#433×C₆₇	C₂	direct (N × H)
3. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
5. C₂²×#97×C₆₇	C₂	semidirect
6. C₂²×C₄×C₈×C₆₇	C₂	semidirect
16. #433×C₆₇	C₂²	direct (N × H)
17. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
19. C₂×#97×C₆₇	C₂²	semidirect
20. C₂×C₄×C₈×C₆₇	C₂²	semidirect
41. C₄⋊C₈×C₆₇	C₂³	semidirect
43. #97×C₆₇	C₂³	semidirect
44. C₄×C₈×C₆₇	C₂³	semidirect
96. C₂²×C₆₇	#433	direct (N × H)
105. C₂²×#433	C₆₇	direct (N × H)
107. C₂×C₆₇	C₂×#433	direct (N × H)
111. C₂×#433	C₂×C₆₇	direct (N × H)
112. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
114. C₂²×#97	C₂×C₆₇	semidirect
115. C₂²×C₄×C₈	C₂×C₆₇	semidirect
116. C₆₇	C₂²×#433	direct (N × H)
126. #433	C₂²×C₆₇	direct (N × H)
127. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
129. C₂×#97	C₂²×C₆₇	semidirect
130. C₂×C₄×C₈	C₂²×C₆₇	semidirect
151. C₄⋊C₈	C₂³×C₆₇	semidirect
153. #97	C₂³×C₆₇	semidirect
154. C₄×C₈	C₂³×C₆₇	semidirect
206. C₂²	#433×C₆₇	direct (N × H)
216. C₂	C₂×#433×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×#117×C₆₇	2
C₂×#433×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#433	67

Subgroups — C22×#433×C67 #30739367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#433×C₆₇ #30739367