Subgroups — C₆₇×#33078 #30712329

Order 17152.

906 conjugacy classes of subgroups.

The subgroup lattice has 262 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, complemented
2	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1925	C₂	1	3	yes	maximal, complemented
4	C₆₇×#1918	C₂	1	3	yes	maximal, complemented
5	C₆₇×#1879	C₂	1	3	yes	maximal
6	C₆₇×#1862	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#126×C₆₇	—	2	1	no	—
10	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₆₇	—	2	1	no	—
12	C₂×#107×C₆₇	C₂²	1	3	yes	—
13	C₂×#102×C₆₇	C₂²	1	3	yes	—
14	C₂×#102×C₆₇	C₂²	1	3	yes	—
15	#390×C₆₇	C₂²	1	3	yes	—
16	#389×C₆₇	C₂²	1	3	yes	—
17	#388×C₆₇	—	2	3	no	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₆₇	—	2	3	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×D₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	—	2	3	no	—
28	C₂×D₈×C₆₇	—	2	6	no	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
31	#126×C₆₇	—	4	4	no	—
32	C₄×D₄×C₆₇	—	2	6	no	—
33	C₄×D₄×C₆₇	—	2	6	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×C₂²⋊C₄×C₆₇	—	2	3	no	—
40	C₂×C₄²×C₆₇	—	2	3	no	—
41	C₂×C₄²×C₆₇	—	4	4	no	—
42	C₂×C₄²×C₆₇	D₄	1	3	yes	—
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
45	#107×C₆₇	—	2	6	no	—
46	#102×C₆₇	—	2	6	no	—
47	#102×C₆₇	—	2	6	no	—
48	#95×C₆₇	—	2	3	no	—
49	#95×C₆₇	—	2	3	no	—
50	C₂⁴×C₆₇	—	2	1	no	—
51	C₂⁴×C₆₇	—	2	1	no	—
52	C₂×D₄×C₆₇	—	2	3	no	—
53	C₂×D₄×C₆₇	—	2	3	no	—
54	C₂×D₄×C₆₇	—	2	3	no	—
55	C₂×D₄×C₆₇	#41	1	6	yes	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	2	3	no	—
58	C₂²×C₄×C₆₇	—	2	3	no	—
59	C₂²×C₄×C₆₇	—	4	6	no	—
60	C₂²×C₄×C₆₇	—	4	6	no	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	D₈×C₆₇	—	4	4	no	—
65	C₂×C₈×C₆₇	—	2	3	no	—
66	C₂×C₈×C₆₇	—	2	3	no	—
67	C₂×C₈×C₆₇	—	2	3	no	—
68	C₂×C₈×C₆₇	—	2	3	no	—
69	C₄⋊C₄×C₆₇	—	2	3	no	—
70	C₄⋊C₄×C₆₇	—	2	3	no	—
71	C₄⋊C₄×C₆₇	—	2	3	no	—
72	C₄⋊C₄×C₆₇	—	4	6	no	—
73	C₄⋊C₄×C₆₇	#41	1	6	yes	—
74	C₂²⋊C₄×C₆₇	—	2	12	no	—
75	C₂²⋊C₄×C₆₇	—	4	6	no	—
76	C₂²⋊C₄×C₆₇	—	4	6	no	—
77	C₄²×C₆₇	—	2	3	no	—
78	C₄²×C₆₇	—	2	3	no	—
79	C₄²×C₆₇	—	4	3	no	—
80	C₄²×C₆₇	—	4	6	no	—
81	C₄²×C₆₇	D₈	1	4	yes	complemented
82	C₂³×C₆₇	—	2	6	no	—
83	C₂³×C₆₇	—	4	1	no	—
84	C₂³×C₆₇	—	4	3	no	—
85	C₂³×C₆₇	—	4	3	no	—
86	C₂³×C₆₇	—	4	1	no	—
87	C₂³×C₆₇	—	4	3	no	—
88	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	D₄×C₆₇	—	2	6	no	—
90	D₄×C₆₇	—	4	1	no	—
91	D₄×C₆₇	—	4	3	no	—
92	D₄×C₆₇	#126	1	4	yes	complemented
93	C₂×C₄×C₆₇	—	2	3	no	—
94	C₂×C₄×C₆₇	—	2	3	no	—
95	C₂×C₄×C₆₇	—	2	3	no	—
96	C₂×C₄×C₆₇	—	2	6	no	—
97	C₂×C₄×C₆₇	—	2	3	no	—
98	C₂×C₄×C₆₇	—	2	3	no	—
99	C₂×C₄×C₆₇	—	4	24	no	—
100	C₂×C₄×C₆₇	—	4	3	no	—
101	C₂×C₄×C₆₇	—	4	3	no	—
102	C₂×C₄×C₆₇	—	4	3	no	—
103	C₂×C₄×C₆₇	—	4	3	no	—
104	C₂×C₄×C₆₇	—	4	3	no	—
105	C₂×C₄×C₆₇	C₂×D₈	1	6	yes	—
106	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
107	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
108	C₈×C₆₇	—	4	1	no	—
109	C₈×C₆₇	—	4	3	no	—
110	C₂²×C₆₇	—	2	4	no	—
111	C₂²×C₆₇	—	4	6	no	—
112	C₂²×C₆₇	—	4	6	no	—
113	C₂²×C₆₇	—	8	3	no	—
114	C₂²×C₆₇	—	8	1	no	—
115	C₂²×C₆₇	—	8	3	no	—
116	C₂²×C₆₇	#525	1	3	yes	—
117	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
118	C₂²×C₆₇	#575	1	3	yes	—
119	C₄×C₆₇	—	2	6	no	—
120	C₄×C₆₇	—	4	6	no	—
121	C₄×C₆₇	—	4	3	no	—
122	C₄×C₆₇	—	8	3	no	—
123	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
124	C₄×C₆₇	#529	1	3	yes	—
125	#33078	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	4	4	no	—
127	C₂×C₆₇	—	16	1	no	—
128	C₂×C₆₇	#2852	1	1	yes	characteristic
129	C₂×C₆₇	#3100	1	3	yes	—
130	C₂×C₆₇	#3111	1	3	yes	—
131	#2860	C₂×C₆₇	1	1	yes	characteristic, complemented
132	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
133	#1925	C₂×C₆₇	1	3	yes	complemented
134	#1918	C₂×C₆₇	1	3	yes	complemented
135	#1879	C₂×C₆₇	1	3	yes	—
136	#1862	C₂×C₆₇	1	3	yes	complemented
137	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
138	C₆₇	#33078	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
140	C₂×#126	—	2	1	no	—
141	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
142	C₂²×C₄²	—	2	1	no	—
143	C₂×#107	C₂²×C₆₇	1	3	yes	—
144	C₂×#102	C₂²×C₆₇	1	3	yes	—
145	C₂×#102	C₂²×C₆₇	1	3	yes	—
146	#390	C₂²×C₆₇	1	3	yes	—
147	#389	C₂²×C₆₇	1	3	yes	—
148	#388	—	2	3	no	—
149	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
150	C₄×C₂²⋊C₄	—	2	3	no	—
151	C₄³	C₂²×C₆₇	1	1	yes	characteristic
152	#340	C₂²×C₆₇	1	3	yes	—
153	#340	C₂²×C₆₇	1	3	yes	—
154	#336	C₂²×C₆₇	1	3	yes	—
155	#336	C₂²×C₆₇	1	3	yes	—
156	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
157	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
158	C₂³×C₄	—	2	3	no	—
159	C₂×D₈	—	2	6	no	—
160	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
162	#126	—	4	4	no	—
163	C₄×D₄	—	2	6	no	—
164	C₄×D₄	—	2	6	no	—
165	C₂×C₄⋊C₄	—	2	3	no	—
166	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
167	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
168	C₂×C₂²⋊C₄	—	2	3	no	—
169	C₂×C₂²⋊C₄	—	2	3	no	—
170	C₂×C₂²⋊C₄	—	2	3	no	—
171	C₂×C₄²	—	2	3	no	—
172	C₂×C₄²	—	4	4	no	—
173	C₂×C₄²	D₄×C₆₇	1	3	yes	—
174	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
176	#107	—	2	6	no	—
177	#102	—	2	6	no	—
178	#102	—	2	6	no	—
179	#95	—	2	3	no	—
180	#95	—	2	3	no	—
181	C₂⁴	—	2	1	no	—
182	C₂⁴	—	2	1	no	—
183	C₂×D₄	—	2	3	no	—
184	C₂×D₄	—	2	3	no	—
185	C₂×D₄	—	2	3	no	—
186	C₂×D₄	#41×C₆₇	1	6	yes	—
187	C₂²×C₄	—	2	12	no	—
188	C₂²×C₄	—	2	3	no	—
189	C₂²×C₄	—	2	3	no	—
190	C₂²×C₄	—	4	6	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
193	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
194	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
195	D₈	—	4	4	no	—
196	C₂×C₈	—	2	3	no	—
197	C₂×C₈	—	2	3	no	—
198	C₂×C₈	—	2	3	no	—
199	C₂×C₈	—	2	3	no	—
200	C₄⋊C₄	—	2	3	no	—
201	C₄⋊C₄	—	2	3	no	—
202	C₄⋊C₄	—	2	3	no	—
203	C₄⋊C₄	—	4	6	no	—
204	C₄⋊C₄	#41×C₆₇	1	6	yes	—
205	C₂²⋊C₄	—	2	12	no	—
206	C₂²⋊C₄	—	4	6	no	—
207	C₂²⋊C₄	—	4	6	no	—
208	C₄²	—	2	3	no	—
209	C₄²	—	2	3	no	—
210	C₄²	—	4	3	no	—
211	C₄²	—	4	6	no	—
212	C₄²	D₈×C₆₇	1	4	yes	complemented
213	C₂³	—	2	6	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	3	no	—
216	C₂³	—	4	3	no	—
217	C₂³	—	4	1	no	—
218	C₂³	—	4	3	no	—
219	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
220	D₄	—	2	6	no	—
221	D₄	—	4	1	no	—
222	D₄	—	4	3	no	—
223	D₄	#126×C₆₇	1	4	yes	complemented
224	C₂×C₄	—	2	3	no	—
225	C₂×C₄	—	2	3	no	—
226	C₂×C₄	—	2	3	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	3	no	—
229	C₂×C₄	—	2	3	no	—
230	C₂×C₄	—	4	24	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	—	4	3	no	—
233	C₂×C₄	—	4	3	no	—
234	C₂×C₄	—	4	3	no	—
235	C₂×C₄	—	4	3	no	—
236	C₂×C₄	C₂×D₈×C₆₇	1	6	yes	—
237	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
238	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
239	C₈	—	4	1	no	—
240	C₈	—	4	3	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	6	no	—
243	C₂²	—	4	6	no	—
244	C₂²	—	8	3	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	8	3	no	—
247	C₂²	#525×C₆₇	1	3	yes	—
248	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
249	C₂²	#575×C₆₇	1	3	yes	—
250	C₄	—	2	6	no	—
251	C₄	—	4	6	no	—
252	C₄	—	4	3	no	—
253	C₄	—	8	3	no	—
254	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
255	C₄	#529×C₆₇	1	3	yes	—
256	C₂	—	4	4	no	—
257	C₂	—	16	1	no	—
258	C₂	C₆₇×#2852	1	1	yes	characteristic
259	C₂	C₆₇×#3100	1	3	yes	—
260	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×D₈×C₆₇	C₂²
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
1. C₆₇×#2860	C₂	semidirect
2. C₄²×D₄×C₆₇	C₂	semidirect
3. C₆₇×#1925	C₂	semidirect
4. C₆₇×#1918	C₂	semidirect
6. C₆₇×#1862	C₂	semidirect
7. C₆₇×#1839	C₂	semidirect
81. C₄²×C₆₇	D₈	semidirect
92. D₄×C₆₇	#126	semidirect
125. #33078	C₆₇	direct (N × H)
131. #2860	C₂×C₆₇	semidirect
132. C₄²×D₄	C₂×C₆₇	semidirect
133. #1925	C₂×C₆₇	semidirect
134. #1918	C₂×C₆₇	semidirect
136. #1862	C₂×C₆₇	semidirect
137. #1839	C₂×C₆₇	semidirect
138. C₆₇	#33078	direct (N × H)
212. C₄²	D₈×C₆₇	semidirect
223. D₄	#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×D₄×C₆₇	2
C₆₇×#1925	2
C₆₇×#1918	2
C₆₇×#1879	2
C₆₇×#1862	2
C₆₇×#1839	2
#33078	67

Subgroups — C67×#33078 #30712329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33078 #30712329