Subgroups — C₆₇×#33495 #30712746

Order 17152.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2169	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2113	C₂	1	4	yes	maximal, complemented
8	C₂×#131×C₆₇	C₂²	1	2	yes	—
9	#532×C₆₇	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	—
12	#436×C₆₇	C₄	1	2	yes	—
13	#436×C₆₇	C₄	1	2	yes	—
14	C₈×D₄×C₆₇	C₂²	1	2	yes	—
15	C₈×D₄×C₆₇	C₂²	1	2	yes	—
16	#423×C₆₇	C₄	1	1	yes	characteristic
17	#408×C₆₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
24	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
25	#346×C₆₇	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	#131×C₆₇	—	4	4	no	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
36	#127×C₆₇	C₈	1	1	yes	characteristic
37	#124×C₆₇	C₈	1	2	yes	—
38	#121×C₆₇	C₈	1	2	yes	—
39	#121×C₆₇	C₈	1	2	yes	—
40	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₆₇	—	4	4	no	—
44	C₂×C₁₆×C₆₇	—	4	4	no	—
45	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
47	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	#41×C₆₇	—	4	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
62	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
67	M₁₆×C₆₇	—	4	2	no	—
68	M₁₆×C₆₇	—	4	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	4	1	no	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
81	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
82	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₁₆×C₆₇	—	2	2	no	—
87	C₁₆×C₆₇	—	2	4	no	—
88	C₁₆×C₆₇	—	2	2	no	—
89	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
90	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
91	Q₈×C₆₇	—	4	2	no	—
92	D₄×C₆₇	—	2	4	no	—
93	D₄×C₆₇	—	4	2	no	—
94	D₄×C₆₇	—	4	2	no	—
95	D₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	4	1	no	—
102	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₆₇	—	2	4	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	2	1	no	—
112	C₈×C₆₇	—	2	1	no	—
113	C₈×C₆₇	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	4	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	#504	1	2	yes	—
127	C₄×C₆₇	#518	1	1	yes	characteristic
128	C₄×C₆₇	#568	1	1	yes	characteristic
129	#33495	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₆₇	—	4	1	no	—
131	C₂×C₆₇	—	4	4	no	—
132	C₂×C₆₇	#2972	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
134	C₂×C₆₇	#3407	1	1	yes	characteristic
135	#2931	C₂×C₆₇	1	1	yes	characteristic
136	#2169	C₂×C₆₇	1	2	yes	complemented
137	#2168	C₂×C₆₇	1	4	yes	complemented
138	D₄×C₁₆	C₂×C₆₇	1	2	yes	complemented
139	#2163	C₂×C₆₇	1	1	yes	characteristic, complemented
140	#2162	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#2113	C₂×C₆₇	1	4	yes	complemented
142	C₆₇	#33495	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#131	C₂²×C₆₇	1	2	yes	—
144	#532	C₄×C₆₇	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
146	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	—
147	#436	C₄×C₆₇	1	2	yes	—
148	#436	C₄×C₆₇	1	2	yes	—
149	C₈×D₄	C₂²×C₆₇	1	2	yes	—
150	C₈×D₄	C₂²×C₆₇	1	2	yes	—
151	#423	C₄×C₆₇	1	1	yes	characteristic
152	#408	C₄×C₆₇	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
155	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
156	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
157	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
158	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
159	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
160	#346	C₂²×C₆₇	1	2	yes	—
161	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
162	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
164	#131	—	4	4	no	—
165	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
166	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
167	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
170	C₄⋊Q₈	C₈×C₆₇	1	1	yes	characteristic
171	#127	C₈×C₆₇	1	1	yes	characteristic
172	#124	C₈×C₆₇	1	2	yes	—
173	#121	C₈×C₆₇	1	2	yes	—
174	#121	C₈×C₆₇	1	2	yes	—
175	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
176	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
178	M₃₂	—	4	4	no	—
179	C₂×C₁₆	—	4	4	no	—
180	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
181	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
182	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
183	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
184	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
185	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
186	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
187	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
194	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
195	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
196	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
197	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
198	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
212	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
213	C₄⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
214	C₄⋊C₄	C₂×C₈×C₆₇	1	2	yes	—
215	C₂²⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
216	C₂²⋊C₄	C₂×C₈×C₆₇	1	4	yes	—
217	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
218	C₄²	C₂×C₈×C₆₇	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
220	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	4	no	—
223	C₁₆	—	2	2	no	—
224	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
225	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
242	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	H(5,𝔽₂)×C₆₇	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#504×C₆₇	1	2	yes	—
262	C₄	#518×C₆₇	1	1	yes	characteristic
263	C₄	#568×C₆₇	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	C₆₇×#2972	1	1	yes	characteristic
267	C₂	C₂×#504×C₆₇	1	1	yes	characteristic
268	C₂	C₆₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	#532×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2169	C₂	semidirect
3. C₆₇×#2168	C₂	semidirect
4. D₄×C₁₆×C₆₇	C₂	semidirect
5. C₆₇×#2163	C₂	semidirect
6. C₆₇×#2162	C₂	semidirect
7. C₆₇×#2113	C₂	semidirect
129. #33495	C₆₇	direct (N × H)
136. #2169	C₂×C₆₇	semidirect
137. #2168	C₂×C₆₇	semidirect
138. D₄×C₁₆	C₂×C₆₇	semidirect
139. #2163	C₂×C₆₇	semidirect
140. #2162	C₂×C₆₇	semidirect
141. #2113	C₂×C₆₇	semidirect
142. C₆₇	#33495	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2931	2
C₆₇×#2169	2
C₆₇×#2168	2
D₄×C₁₆×C₆₇	2
C₆₇×#2163	2
C₆₇×#2162	2
C₆₇×#2113	2
#33495	67

Subgroups — C67×#33495 #30712746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33495 #30712746