Subgroups — C₆₇×#33079 #30712330

Order 17152.

786 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2860	C₂	1	1	yes	maximal, characteristic
2	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1926	C₂	1	3	yes	maximal
4	C₆₇×#1919	C₂	1	3	yes	maximal, complemented
5	C₆₇×#1880	C₂	1	3	yes	maximal
6	C₆₇×#1865	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#126×C₆₇	—	2	1	no	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
11	C₂×#106×C₆₇	C₂²	1	3	yes	—
12	C₂×#103×C₆₇	C₂²	1	3	yes	—
13	C₂×#102×C₆₇	C₂²	1	3	yes	—
14	#390×C₆₇	C₂²	1	3	yes	—
15	#389×C₆₇	C₂²	1	3	yes	—
16	#388×C₆₇	—	2	3	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×D₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×SD₁₆×C₆₇	—	2	6	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
30	#126×C₆₇	—	4	4	no	—
31	C₄×Q₈×C₆₇	—	2	6	no	—
32	C₄×Q₈×C₆₇	—	2	6	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
37	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
38	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
39	C₂×C₄²×C₆₇	—	2	3	no	—
40	C₂×C₄²×C₆₇	—	2	3	no	—
41	C₂×C₄²×C₆₇	D₄	1	3	yes	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
44	#106×C₆₇	—	2	6	no	—
45	#103×C₆₇	—	2	6	no	—
46	#102×C₆₇	—	2	6	no	—
47	#95×C₆₇	—	2	3	no	—
48	#95×C₆₇	—	2	3	no	—
49	C₂⁴×C₆₇	—	2	1	no	—
50	C₂×Q₈×C₆₇	—	2	3	no	—
51	C₂×Q₈×C₆₇	#41	1	6	yes	—
52	C₂×D₄×C₆₇	—	2	3	no	—
53	C₂×D₄×C₆₇	—	2	3	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	3	no	—
56	C₂²×C₄×C₆₇	—	2	3	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
60	SD₁₆×C₆₇	—	4	4	no	—
61	C₂×C₈×C₆₇	—	2	3	no	—
62	C₂×C₈×C₆₇	—	2	3	no	—
63	C₂×C₈×C₆₇	—	2	3	no	—
64	C₂×C₈×C₆₇	—	2	3	no	—
65	C₄⋊C₄×C₆₇	—	2	12	no	—
66	C₄⋊C₄×C₆₇	—	2	3	no	—
67	C₄⋊C₄×C₆₇	—	2	3	no	—
68	C₄⋊C₄×C₆₇	—	2	3	no	—
69	C₄⋊C₄×C₆₇	—	4	6	no	—
70	C₄⋊C₄×C₆₇	#41	1	6	yes	—
71	C₂²⋊C₄×C₆₇	—	4	6	no	—
72	C₂²⋊C₄×C₆₇	—	4	6	no	—
73	C₄²×C₆₇	—	2	12	no	—
74	C₄²×C₆₇	—	2	3	no	—
75	C₄²×C₆₇	—	2	3	no	—
76	C₄²×C₆₇	—	4	3	no	—
77	C₄²×C₆₇	—	4	6	no	—
78	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
79	C₂³×C₆₇	—	4	3	no	—
80	C₂³×C₆₇	—	4	1	no	—
81	C₂³×C₆₇	—	4	3	no	—
82	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₆₇	—	2	6	no	—
84	Q₈×C₆₇	#126	1	4	yes	complemented
85	D₄×C₆₇	—	4	1	no	—
86	D₄×C₆₇	—	4	3	no	—
87	C₂×C₄×C₆₇	—	2	3	no	—
88	C₂×C₄×C₆₇	—	2	6	no	—
89	C₂×C₄×C₆₇	—	2	3	no	—
90	C₂×C₄×C₆₇	—	2	3	no	—
91	C₂×C₄×C₆₇	—	2	6	no	—
92	C₂×C₄×C₆₇	—	2	3	no	—
93	C₂×C₄×C₆₇	—	2	3	no	—
94	C₂×C₄×C₆₇	—	4	6	no	—
95	C₂×C₄×C₆₇	—	4	3	no	—
96	C₂×C₄×C₆₇	—	4	3	no	—
97	C₂×C₄×C₆₇	—	4	3	no	—
98	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
99	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
100	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
101	C₈×C₆₇	—	4	1	no	—
102	C₈×C₆₇	—	4	3	no	—
103	C₂²×C₆₇	—	8	3	no	—
104	C₂²×C₆₇	—	8	1	no	—
105	C₂²×C₆₇	—	8	3	no	—
106	C₂²×C₆₇	#525	1	3	yes	—
107	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
108	C₂²×C₆₇	#575	1	3	yes	—
109	C₄×C₆₇	—	2	6	no	—
110	C₄×C₆₇	—	2	4	no	—
111	C₄×C₆₇	—	4	3	no	—
112	C₄×C₆₇	—	4	6	no	—
113	C₄×C₆₇	—	8	3	no	—
114	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
115	C₄×C₆₇	#529	1	3	yes	—
116	#33079	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₆₇	—	16	1	no	—
118	C₂×C₆₇	#2852	1	1	yes	characteristic
119	C₂×C₆₇	#3103	1	3	yes	—
120	C₂×C₆₇	#3111	1	3	yes	—
121	#2860	C₂×C₆₇	1	1	yes	characteristic
122	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
123	#1926	C₂×C₆₇	1	3	yes	—
124	#1919	C₂×C₆₇	1	3	yes	complemented
125	#1880	C₂×C₆₇	1	3	yes	—
126	#1865	C₂×C₆₇	1	3	yes	complemented
127	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
128	C₆₇	#33079	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×#126	—	2	1	no	—
131	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
132	C₂×#106	C₂²×C₆₇	1	3	yes	—
133	C₂×#103	C₂²×C₆₇	1	3	yes	—
134	C₂×#102	C₂²×C₆₇	1	3	yes	—
135	#390	C₂²×C₆₇	1	3	yes	—
136	#389	C₂²×C₆₇	1	3	yes	—
137	#388	—	2	3	no	—
138	C₄×C₄⋊C₄	—	2	3	no	—
139	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
140	C₄³	—	2	1	no	—
141	C₄³	C₂²×C₆₇	1	1	yes	characteristic
142	#340	C₂²×C₆₇	1	3	yes	—
143	#340	C₂²×C₆₇	1	3	yes	—
144	#336	C₂²×C₆₇	1	3	yes	—
145	#336	C₂²×C₆₇	1	3	yes	—
146	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
147	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
148	C₂×SD₁₆	—	2	6	no	—
149	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
151	#126	—	4	4	no	—
152	C₄×Q₈	—	2	6	no	—
153	C₄×Q₈	—	2	6	no	—
154	C₂×C₄⋊C₄	—	2	3	no	—
155	C₂×C₄⋊C₄	—	2	3	no	—
156	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
157	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
158	C₂×C₂²⋊C₄	—	2	3	no	—
159	C₂×C₂²⋊C₄	—	2	3	no	—
160	C₂×C₄²	—	2	3	no	—
161	C₂×C₄²	—	2	3	no	—
162	C₂×C₄²	D₄×C₆₇	1	3	yes	—
163	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
165	#106	—	2	6	no	—
166	#103	—	2	6	no	—
167	#102	—	2	6	no	—
168	#95	—	2	3	no	—
169	#95	—	2	3	no	—
170	C₂⁴	—	2	1	no	—
171	C₂×Q₈	—	2	3	no	—
172	C₂×Q₈	#41×C₆₇	1	6	yes	—
173	C₂×D₄	—	2	3	no	—
174	C₂×D₄	—	2	3	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	3	no	—
177	C₂²×C₄	—	2	3	no	—
178	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
180	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
181	SD₁₆	—	4	4	no	—
182	C₂×C₈	—	2	3	no	—
183	C₂×C₈	—	2	3	no	—
184	C₂×C₈	—	2	3	no	—
185	C₂×C₈	—	2	3	no	—
186	C₄⋊C₄	—	2	12	no	—
187	C₄⋊C₄	—	2	3	no	—
188	C₄⋊C₄	—	2	3	no	—
189	C₄⋊C₄	—	2	3	no	—
190	C₄⋊C₄	—	4	6	no	—
191	C₄⋊C₄	#41×C₆₇	1	6	yes	—
192	C₂²⋊C₄	—	4	6	no	—
193	C₂²⋊C₄	—	4	6	no	—
194	C₄²	—	2	12	no	—
195	C₄²	—	2	3	no	—
196	C₄²	—	2	3	no	—
197	C₄²	—	4	3	no	—
198	C₄²	—	4	6	no	—
199	C₄²	SD₁₆×C₆₇	1	4	yes	complemented
200	C₂³	—	4	3	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	3	no	—
203	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
204	Q₈	—	2	6	no	—
205	Q₈	#126×C₆₇	1	4	yes	complemented
206	D₄	—	4	1	no	—
207	D₄	—	4	3	no	—
208	C₂×C₄	—	2	3	no	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	2	3	no	—
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	2	3	no	—
214	C₂×C₄	—	2	3	no	—
215	C₂×C₄	—	4	6	no	—
216	C₂×C₄	—	4	3	no	—
217	C₂×C₄	—	4	3	no	—
218	C₂×C₄	—	4	3	no	—
219	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
220	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
221	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
222	C₈	—	4	1	no	—
223	C₈	—	4	3	no	—
224	C₂²	—	8	3	no	—
225	C₂²	—	8	1	no	—
226	C₂²	—	8	3	no	—
227	C₂²	#525×C₆₇	1	3	yes	—
228	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
229	C₂²	#575×C₆₇	1	3	yes	—
230	C₄	—	2	6	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	3	no	—
233	C₄	—	4	6	no	—
234	C₄	—	8	3	no	—
235	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
236	C₄	#529×C₆₇	1	3	yes	—
237	C₂	—	16	1	no	—
238	C₂	C₆₇×#2852	1	1	yes	characteristic
239	C₂	C₆₇×#3103	1	3	yes	—
240	C₂	C₆₇×#3111	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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
2. C₄²×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#1919	C₂	semidirect
6. C₆₇×#1865	C₂	semidirect
7. C₆₇×#1839	C₂	semidirect
78. C₄²×C₆₇	SD₁₆	semidirect
84. Q₈×C₆₇	#126	semidirect
116. #33079	C₆₇	direct (N × H)
122. C₄²×Q₈	C₂×C₆₇	semidirect
124. #1919	C₂×C₆₇	semidirect
126. #1865	C₂×C₆₇	semidirect
127. #1839	C₂×C₆₇	semidirect
128. C₆₇	#33079	direct (N × H)
199. C₄²	SD₁₆×C₆₇	semidirect
205. Q₈	#126×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2860	2
C₄²×Q₈×C₆₇	2
C₆₇×#1926	2
C₆₇×#1919	2
C₆₇×#1880	2
C₆₇×#1865	2
C₆₇×#1839	2
#33079	67

Subgroups — C67×#33079 #30712330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33079 #30712330