Subgroups — C₆₇×#33083 #30712334

Order 17152.

732 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	C₆₇×#2863	C₂	1	1	yes	maximal, characteristic
2	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1928	C₂	1	3	yes	maximal
4	C₆₇×#1921	C₂	1	3	yes	maximal
5	C₆₇×#1878	C₂	1	3	yes	maximal
6	C₆₇×#1863	C₂	1	3	yes	maximal
7	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
10	C₂×#107×C₆₇	C₂²	1	3	yes	—
11	C₂×#103×C₆₇	C₂²	1	3	yes	—
12	C₂×#103×C₆₇	C₂²	1	3	yes	—
13	#391×C₆₇	C₂²	1	3	yes	—
14	#389×C₆₇	C₂²	1	3	yes	—
15	#387×C₆₇	—	2	3	no	—
16	#383×C₆₇	—	2	1	no	—
17	C₄×C₄⋊C₄×C₆₇	—	2	3	no	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
19	C₄³×C₆₇	—	2	1	no	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#340×C₆₇	C₂²	1	3	yes	—
22	#340×C₆₇	C₂²	1	3	yes	—
23	#336×C₆₇	C₂²	1	3	yes	—
24	#336×C₆₇	C₂²	1	3	yes	—
25	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₆₇	—	2	6	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
30	C₄×Q₈×C₆₇	—	2	6	no	—
31	C₄×Q₈×C₆₇	—	2	6	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
36	C₂×C₄²×C₆₇	—	2	3	no	—
37	C₂×C₄²×C₆₇	—	2	3	no	—
38	C₂×C₄²×C₆₇	D₄	1	3	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
41	#107×C₆₇	—	2	6	no	—
42	#103×C₆₇	—	2	6	no	—
43	#103×C₆₇	—	2	6	no	—
44	#95×C₆₇	—	2	3	no	—
45	#95×C₆₇	—	2	3	no	—
46	#95×C₆₇	—	2	3	no	—
47	#95×C₆₇	—	2	3	no	—
48	C₂×Q₈×C₆₇	—	2	3	no	—
49	C₂×Q₈×C₆₇	—	2	3	no	—
50	C₂×Q₈×C₆₇	—	2	3	no	—
51	C₂×Q₈×C₆₇	#41	1	6	yes	—
52	C₂²×C₄×C₆₇	—	2	1	no	—
53	C₂²×C₄×C₆₇	—	2	3	no	—
54	C₂²×C₄×C₆₇	—	2	3	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₆₇	—	4	4	no	—
60	C₂×C₈×C₆₇	—	2	3	no	—
61	C₂×C₈×C₆₇	—	2	3	no	—
62	C₂×C₈×C₆₇	—	2	3	no	—
63	C₂×C₈×C₆₇	—	2	3	no	—
64	C₄⋊C₄×C₆₇	—	2	12	no	—
65	C₄⋊C₄×C₆₇	—	2	3	no	—
66	C₄⋊C₄×C₆₇	—	2	3	no	—
67	C₄⋊C₄×C₆₇	—	2	3	no	—
68	C₄⋊C₄×C₆₇	—	4	6	no	—
69	C₄⋊C₄×C₆₇	#41	1	6	yes	—
70	C₄²×C₆₇	—	2	12	no	—
71	C₄²×C₆₇	—	2	3	no	—
72	C₄²×C₆₇	—	2	3	no	—
73	C₄²×C₆₇	—	4	3	no	—
74	C₄²×C₆₇	—	4	6	no	—
75	C₄²×C₆₇	Q₁₆	1	4	yes	complemented
76	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₆₇	—	2	6	no	—
78	Q₈×C₆₇	—	4	1	no	—
79	Q₈×C₆₇	—	4	3	no	—
80	Q₈×C₆₇	#126	1	4	yes	—
81	C₂×C₄×C₆₇	—	2	3	no	—
82	C₂×C₄×C₆₇	—	2	6	no	—
83	C₂×C₄×C₆₇	—	2	3	no	—
84	C₂×C₄×C₆₇	—	2	3	no	—
85	C₂×C₄×C₆₇	—	2	6	no	—
86	C₂×C₄×C₆₇	—	2	3	no	—
87	C₂×C₄×C₆₇	—	2	3	no	—
88	C₂×C₄×C₆₇	—	4	6	no	—
89	C₂×C₄×C₆₇	—	4	3	no	—
90	C₂×C₄×C₆₇	—	4	3	no	—
91	C₂×C₄×C₆₇	—	4	6	no	—
92	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
93	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
94	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
95	C₈×C₆₇	—	4	1	no	—
96	C₈×C₆₇	—	4	3	no	—
97	C₂²×C₆₇	#525	1	3	yes	—
98	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
99	C₂²×C₆₇	#575	1	3	yes	—
100	C₄×C₆₇	—	2	6	no	—
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	4	3	no	—
103	C₄×C₆₇	—	4	6	no	—
104	C₄×C₆₇	—	8	1	no	—
105	C₄×C₆₇	—	8	3	no	—
106	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
107	C₄×C₆₇	#529	1	3	yes	—
108	#33083	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₆₇	#2852	1	1	yes	characteristic
110	C₂×C₆₇	#3104	1	3	yes	—
111	C₂×C₆₇	#3111	1	3	yes	—
112	#2863	C₂×C₆₇	1	1	yes	characteristic
113	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
114	#1928	C₂×C₆₇	1	3	yes	—
115	#1921	C₂×C₆₇	1	3	yes	—
116	#1878	C₂×C₆₇	1	3	yes	—
117	#1863	C₂×C₆₇	1	3	yes	—
118	#1839	C₂×C₆₇	1	1	yes	characteristic
119	C₆₇	#33083	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
122	C₂×#107	C₂²×C₆₇	1	3	yes	—
123	C₂×#103	C₂²×C₆₇	1	3	yes	—
124	C₂×#103	C₂²×C₆₇	1	3	yes	—
125	#391	C₂²×C₆₇	1	3	yes	—
126	#389	C₂²×C₆₇	1	3	yes	—
127	#387	—	2	3	no	—
128	#383	—	2	1	no	—
129	C₄×C₄⋊C₄	—	2	3	no	—
130	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
131	C₄³	—	2	1	no	—
132	C₄³	C₂²×C₆₇	1	1	yes	characteristic
133	#340	C₂²×C₆₇	1	3	yes	—
134	#340	C₂²×C₆₇	1	3	yes	—
135	#336	C₂²×C₆₇	1	3	yes	—
136	#336	C₂²×C₆₇	1	3	yes	—
137	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
138	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
139	C₂×Q₁₆	—	2	6	no	—
140	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
142	C₄×Q₈	—	2	6	no	—
143	C₄×Q₈	—	2	6	no	—
144	C₂×C₄⋊C₄	—	2	3	no	—
145	C₂×C₄⋊C₄	—	2	3	no	—
146	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
147	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
148	C₂×C₄²	—	2	3	no	—
149	C₂×C₄²	—	2	3	no	—
150	C₂×C₄²	D₄×C₆₇	1	3	yes	—
151	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
153	#107	—	2	6	no	—
154	#103	—	2	6	no	—
155	#103	—	2	6	no	—
156	#95	—	2	3	no	—
157	#95	—	2	3	no	—
158	#95	—	2	3	no	—
159	#95	—	2	3	no	—
160	C₂×Q₈	—	2	3	no	—
161	C₂×Q₈	—	2	3	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	#41×C₆₇	1	6	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
171	Q₁₆	—	4	4	no	—
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₄⋊C₄	—	2	12	no	—
177	C₄⋊C₄	—	2	3	no	—
178	C₄⋊C₄	—	2	3	no	—
179	C₄⋊C₄	—	2	3	no	—
180	C₄⋊C₄	—	4	6	no	—
181	C₄⋊C₄	#41×C₆₇	1	6	yes	—
182	C₄²	—	2	12	no	—
183	C₄²	—	2	3	no	—
184	C₄²	—	2	3	no	—
185	C₄²	—	4	3	no	—
186	C₄²	—	4	6	no	—
187	C₄²	Q₁₆×C₆₇	1	4	yes	complemented
188	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
189	Q₈	—	2	6	no	—
190	Q₈	—	4	1	no	—
191	Q₈	—	4	3	no	—
192	Q₈	#126×C₆₇	1	4	yes	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	4	6	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	3	no	—
203	C₂×C₄	—	4	6	no	—
204	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
205	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
206	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
207	C₈	—	4	1	no	—
208	C₈	—	4	3	no	—
209	C₂²	#525×C₆₇	1	3	yes	—
210	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
211	C₂²	#575×C₆₇	1	3	yes	—
212	C₄	—	2	6	no	—
213	C₄	—	2	4	no	—
214	C₄	—	4	3	no	—
215	C₄	—	4	6	no	—
216	C₄	—	8	1	no	—
217	C₄	—	8	3	no	—
218	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
219	C₄	#529×C₆₇	1	3	yes	—
220	C₂	C₆₇×#2852	1	1	yes	characteristic
221	C₂	C₆₇×#3104	1	3	yes	—
222	C₂	C₆₇×#3111	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
75. C₄²×C₆₇	Q₁₆	semidirect
108. #33083	C₆₇	direct (N × H)
119. C₆₇	#33083	direct (N × H)
187. C₄²	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2863	2
C₄²×Q₈×C₆₇	2
C₆₇×#1928	2
C₆₇×#1921	2
C₆₇×#1878	2
C₆₇×#1863	2
C₆₇×#1839	2
#33083	67

Subgroups — C67×#33083 #30712334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33083 #30712334