Subgroups — C₆₇×#33970 #30713221

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3396	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2250	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2206	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#559×C₆₇	—	2	2	no	—
11	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
17	#495×C₆₇	C₂²	1	2	yes	—
18	#495×C₆₇	C₂²	1	2	yes	—
19	#493×C₆₇	C₂²	1	2	yes	—
20	#463×C₆₇	—	2	2	no	—
21	#463×C₆₇	—	2	2	no	—
22	#460×C₆₇	—	2	2	no	—
23	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
25	C₄×D₈×C₆₇	C₂²	1	2	yes	—
26	C₄×D₈×C₆₇	C₂²	1	2	yes	—
27	#366×C₆₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	#358×C₆₇	C₂²	1	2	yes	—
31	#357×C₆₇	C₂²	1	2	yes	—
32	#357×C₆₇	C₂²	1	2	yes	—
33	#346×C₆₇	C₂²	1	2	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₆₇	—	2	2	no	—
37	C₂×SD₁₆×C₆₇	—	2	2	no	—
38	C₂×D₈×C₆₇	—	2	2	no	—
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	C₂×D₈×C₆₇	C₂³	1	2	yes	—
41	#127×C₆₇	—	2	2	no	—
42	#124×C₆₇	—	2	2	no	—
43	#124×C₆₇	—	2	2	no	—
44	#124×C₆₇	—	4	2	no	—
45	#123×C₆₇	—	4	2	no	—
46	#123×C₆₇	—	4	2	no	—
47	C₄×Q₈×C₆₇	—	2	2	no	—
48	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×D₄×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	—	2	2	no	—
51	C₄×D₄×C₆₇	—	4	2	no	—
52	Q₃₂×C₆₇	—	4	1	no	—
53	SD₃₂×C₆₇	—	2	4	no	—
54	SD₃₂×C₆₇	—	4	2	no	—
55	D₁₆×C₆₇	—	2	2	no	—
56	D₁₆×C₆₇	—	4	2	no	—
57	D₁₆×C₆₇	—	4	1	no	—
58	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
59	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	C₂³	1	2	yes	—
62	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
63	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
64	#103×C₆₇	—	2	2	no	—
65	#102×C₆₇	—	2	2	no	—
66	#102×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	C₂³	1	2	yes	—
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
71	C₂×D₄×C₆₇	—	2	2	no	—
72	C₂×D₄×C₆₇	—	2	2	no	—
73	C₂×D₄×C₆₇	—	4	2	no	—
74	C₂×D₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	Q₁₆×C₆₇	—	2	2	no	—
78	Q₁₆×C₆₇	—	2	2	no	—
79	SD₁₆×C₆₇	—	8	2	no	—
80	SD₁₆×C₆₇	—	8	2	no	—
81	D₈×C₆₇	—	2	2	no	—
82	D₈×C₆₇	—	2	2	no	—
83	D₈×C₆₇	—	2	2	no	—
84	D₈×C₆₇	—	2	2	no	—
85	D₈×C₆₇	—	8	2	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	4	2	no	—
91	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
92	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
93	C₂²⋊C₄×C₆₇	—	4	2	no	—
94	C₂²⋊C₄×C₆₇	—	4	2	no	—
95	C₂²⋊C₄×C₆₇	—	4	2	no	—
96	C₂²⋊C₄×C₆₇	—	4	2	no	—
97	C₂²⋊C₄×C₆₇	—	4	2	no	—
98	C₂²⋊C₄×C₆₇	—	4	2	no	—
99	C₄²×C₆₇	—	4	2	no	—
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₁₆×C₆₇	—	2	2	no	—
103	C₁₆×C₆₇	—	2	1	no	—
104	C₁₆×C₆₇	#41	1	2	yes	—
105	C₂³×C₆₇	—	4	2	no	—
106	C₂³×C₆₇	—	4	2	no	—
107	Q₈×C₆₇	—	4	1	no	—
108	Q₈×C₆₇	—	4	2	no	—
109	Q₈×C₆₇	—	4	2	no	—
110	D₄×C₆₇	—	4	2	no	—
111	D₄×C₆₇	—	4	2	no	—
112	D₄×C₆₇	—	4	2	no	—
113	D₄×C₆₇	—	4	2	no	—
114	D₄×C₆₇	—	8	4	no	—
115	D₄×C₆₇	—	16	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	16	2	no	—
119	C₂×C₄×C₆₇	—	16	2	no	—
120	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	—	8	2	no	—
131	C₂²×C₆₇	—	8	2	no	—
132	C₂²×C₆₇	—	8	2	no	—
133	C₂²×C₆₇	—	8	2	no	—
134	C₂²×C₆₇	—	8	2	no	—
135	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	#550	1	1	yes	characteristic
143	C₄×C₆₇	#576	1	1	yes	characteristic
144	#33970	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	16	2	no	—
146	C₂×C₆₇	—	16	2	no	—
147	C₂×C₆₇	#3391	1	1	yes	characteristic
148	C₂×C₆₇	#3414	1	1	yes	characteristic
149	C₂×C₆₇	#3415	1	1	yes	characteristic
150	#3396	C₂×C₆₇	1	2	yes	complemented
151	#2250	C₂×C₆₇	1	2	yes	complemented
152	#2249	C₂×C₆₇	1	1	yes	characteristic, complemented
153	#2211	C₂×C₆₇	1	2	yes	complemented
154	#2210	C₂×C₆₇	1	2	yes	complemented
155	#2206	C₂×C₆₇	1	2	yes	complemented
156	#2183	C₂×C₆₇	1	1	yes	characteristic, complemented
157	#2175	C₂×C₆₇	1	2	yes	complemented
158	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
159	C₆₇	#33970	1	1	yes	characteristic, direct product factor, Sylow
160	#559	—	2	2	no	—
161	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
162	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
163	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
164	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
167	#495	C₂²×C₆₇	1	2	yes	—
168	#495	C₂²×C₆₇	1	2	yes	—
169	#493	C₂²×C₆₇	1	2	yes	—
170	#463	—	2	2	no	—
171	#463	—	2	2	no	—
172	#460	—	2	2	no	—
173	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
175	C₄×D₈	C₂²×C₆₇	1	2	yes	—
176	C₄×D₈	C₂²×C₆₇	1	2	yes	—
177	#366	C₂²×C₆₇	1	1	yes	characteristic
178	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
179	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
180	#358	C₂²×C₆₇	1	2	yes	—
181	#357	C₂²×C₆₇	1	2	yes	—
182	#357	C₂²×C₆₇	1	2	yes	—
183	#346	C₂²×C₆₇	1	2	yes	—
184	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×SD₁₆	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	C₂³×C₆₇	1	2	yes	—
190	C₂×D₈	C₂³×C₆₇	1	2	yes	—
191	#127	—	2	2	no	—
192	#124	—	2	2	no	—
193	#124	—	2	2	no	—
194	#124	—	4	2	no	—
195	#123	—	4	2	no	—
196	#123	—	4	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	4	2	no	—
202	Q₃₂	—	4	1	no	—
203	SD₃₂	—	2	4	no	—
204	SD₃₂	—	4	2	no	—
205	D₁₆	—	2	2	no	—
206	D₁₆	—	4	2	no	—
207	D₁₆	—	4	1	no	—
208	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
210	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
211	#107	C₂³×C₆₇	1	2	yes	—
212	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
213	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
214	#103	—	2	2	no	—
215	#102	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₆₇	1	2	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	8	2	no	—
230	SD₁₆	—	8	2	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	D₈	—	2	2	no	—
234	D₈	—	2	2	no	—
235	D₈	—	8	2	no	—
236	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
238	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
242	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₆₇	1	2	yes	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	2	1	no	—
254	C₁₆	#41×C₆₇	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	D₄	—	8	4	no	—
265	D₄	—	16	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	16	2	no	—
269	C₂×C₄	—	16	2	no	—
270	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
274	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	2	1	no	—
277	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
278	C₈	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	#550×C₆₇	1	1	yes	characteristic
293	C₄	#576×C₆₇	1	1	yes	characteristic
294	C₂	—	16	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₆₇×#3391	1	1	yes	characteristic
297	C₂	C₆₇×#3414	1	1	yes	characteristic
298	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3396	C₂	semidirect
2. C₆₇×#2250	C₂	semidirect
3. C₆₇×#2249	C₂	semidirect
4. C₆₇×#2211	C₂	semidirect
5. C₆₇×#2210	C₂	semidirect
6. C₆₇×#2206	C₂	semidirect
7. C₆₇×#2183	C₂	semidirect
8. C₆₇×#2175	C₂	semidirect
9. C₄×D₁₆×C₆₇	C₂	semidirect
144. #33970	C₆₇	direct (N × H)
150. #3396	C₂×C₆₇	semidirect
151. #2250	C₂×C₆₇	semidirect
152. #2249	C₂×C₆₇	semidirect
153. #2211	C₂×C₆₇	semidirect
154. #2210	C₂×C₆₇	semidirect
155. #2206	C₂×C₆₇	semidirect
156. #2183	C₂×C₆₇	semidirect
157. #2175	C₂×C₆₇	semidirect
158. C₄×D₁₆	C₂×C₆₇	semidirect
159. C₆₇	#33970	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×#3396	2
C₆₇×#2250	2
C₆₇×#2249	2
C₆₇×#2211	2
C₆₇×#2210	2
C₆₇×#2206	2
C₆₇×#2183	2
C₆₇×#2175	2
C₄×D₁₆×C₆₇	2
#33970	67

Subgroups — C67×#33970 #30713221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33970 #30713221