Subgroups — C₆₇×#33220 #30712471

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3079	C₂	1	2	yes	maximal
3	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1997	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1951	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1815	C₂	1	4	yes	maximal
8	C₂×#131×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
11	#498×C₆₇	—	2	1	no	—
12	#498×C₆₇	—	2	1	no	—
13	#498×C₆₇	—	2	2	no	—
14	#477×C₆₇	—	2	2	no	—
15	#476×C₆₇	—	2	2	no	—
16	#475×C₆₇	—	2	2	no	—
17	#474×C₆₇	—	2	2	no	—
18	C₂×#108×C₆₇	C₂²	1	4	yes	—
19	#428×C₆₇	C₂²	1	4	yes	—
20	C₂×#106×C₆₇	C₂²	1	4	yes	—
21	#423×C₆₇	C₂²	1	2	yes	—
22	#417×C₆₇	C₂²	1	4	yes	—
23	#405×C₆₇	—	2	2	no	—
24	#405×C₆₇	C₂²	1	2	yes	—
25	C₄×M₁₆×C₆₇	—	2	1	no	—
26	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	—	2	1	no	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#343×C₆₇	C₂²	1	8	yes	—
30	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	#131×C₆₇	—	2	4	no	—
33	#131×C₆₇	Q₈	1	8	yes	complemented
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
36	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₆₇	—	2	2	no	—
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄×Q₈×C₆₇	D₄	1	2	yes	—
44	C₄×D₄×C₆₇	—	2	2	no	—
45	C₄×D₄×C₆₇	D₄	1	2	yes	—
46	#117×C₆₇	—	2	1	no	—
47	#117×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#108×C₆₇	—	2	8	no	—
53	#107×C₆₇	—	2	4	no	—
54	#106×C₆₇	—	2	4	no	—
55	#106×C₆₇	—	2	4	no	—
56	#106×C₆₇	—	2	4	no	—
57	C₄⋊C₈×C₆₇	—	2	2	no	—
58	C₄⋊C₈×C₆₇	—	2	2	no	—
59	#103×C₆₇	—	2	4	no	—
60	#102×C₆₇	—	2	4	no	—
61	#97×C₆₇	—	2	2	no	—
62	#97×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	—	2	2	no	—
64	C₄×C₈×C₆₇	—	2	1	no	—
65	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
66	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	D₄	1	2	yes	—
68	#41×C₆₇	—	2	2	no	—
69	#41×C₆₇	C₂×Q₈	1	4	yes	—
70	C₂×Q₈×C₆₇	—	4	2	no	—
71	C₂×Q₈×C₆₇	—	4	2	no	—
72	C₂×Q₈×C₆₇	—	8	4	no	—
73	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₆₇	—	2	1	no	—
75	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₇	—	2	2	no	—
82	M₁₆×C₆₇	—	2	8	no	—
83	M₁₆×C₆₇	—	2	8	no	—
84	M₁₆×C₆₇	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	8	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
91	C₂×C₈×C₆₇	#41	1	4	yes	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	4	2	no	—
96	C₄⋊C₄×C₆₇	—	4	2	no	—
97	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	C₂×D₄	1	2	yes	—
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
107	Q₈×C₆₇	—	2	1	no	—
108	Q₈×C₆₇	—	8	4	no	—
109	Q₈×C₆₇	—	8	4	no	—
110	Q₈×C₆₇	C₄⋊Q₈	1	2	yes	—
111	D₄×C₆₇	—	2	4	no	—
112	D₄×C₆₇	—	2	1	no	—
113	D₄×C₆₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	8	4	no	—
121	C₂×C₄×C₆₇	#121	1	2	yes	—
122	C₂×C₄×C₆₇	#121	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
127	C₈×C₆₇	—	2	4	no	—
128	C₈×C₆₇	—	4	4	no	—
129	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
130	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
135	C₂²×C₆₇	#533	1	2	yes	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	4	2	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#471	1	4	yes	—
141	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
142	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₆₇	#532	1	1	yes	characteristic
144	#33220	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	4	2	no	—
147	C₂×C₆₇	#2857	1	1	yes	characteristic
148	C₂×C₆₇	C₂×#471	1	2	yes	—
149	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
150	#3079	C₂×C₆₇	1	2	yes	—
151	C₄×#131	C₂×C₆₇	1	1	yes	characteristic
152	#1997	C₂×C₆₇	1	2	yes	complemented
153	#1951	C₂×C₆₇	1	4	yes	complemented
154	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
155	#1815	C₂×C₆₇	1	4	yes	—
156	C₆₇	#33220	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#131	C₂²×C₆₇	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
159	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
160	#498	—	2	1	no	—
161	#498	—	2	1	no	—
162	#498	—	2	2	no	—
163	#477	—	2	2	no	—
164	#476	—	2	2	no	—
165	#475	—	2	2	no	—
166	#474	—	2	2	no	—
167	C₂×#108	C₂²×C₆₇	1	4	yes	—
168	#428	C₂²×C₆₇	1	4	yes	—
169	C₂×#106	C₂²×C₆₇	1	4	yes	—
170	#423	C₂²×C₆₇	1	2	yes	—
171	#417	C₂²×C₆₇	1	4	yes	—
172	#405	—	2	2	no	—
173	#405	C₂²×C₆₇	1	2	yes	—
174	C₄×M₁₆	—	2	1	no	—
175	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
176	C₂×C₄×C₈	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
178	#343	C₂²×C₆₇	1	8	yes	—
179	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
180	C₂²×Q₈	—	2	2	no	—
181	#131	—	2	4	no	—
182	#131	Q₈×C₆₇	1	8	yes	complemented
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
185	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄×Q₈	D₄×C₆₇	1	2	yes	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	D₄×C₆₇	1	2	yes	—
195	#117	—	2	1	no	—
196	#117	C₂³×C₆₇	1	1	yes	characteristic
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
201	#108	—	2	8	no	—
202	#107	—	2	4	no	—
203	#106	—	2	4	no	—
204	#106	—	2	4	no	—
205	#106	—	2	4	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	#103	—	2	4	no	—
209	#102	—	2	4	no	—
210	#97	—	2	2	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
215	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
216	C₄×C₈	D₄×C₆₇	1	2	yes	—
217	#41	—	2	2	no	—
218	#41	C₂×Q₈×C₆₇	1	4	yes	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	4	no	—
222	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	#41×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	8	no	—
232	M₁₆	—	2	8	no	—
233	M₁₆	C₂×Q₈×C₆₇	1	4	yes	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	8	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
239	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
240	C₂×C₈	#41×C₆₇	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₆₇	1	2	yes	—
254	C₂³	—	2	1	no	—
255	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
256	Q₈	—	2	1	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	8	4	no	—
259	Q₈	C₄⋊Q₈×C₆₇	1	2	yes	—
260	D₄	—	2	4	no	—
261	D₄	—	2	1	no	—
262	D₄	C₄⋊Q₈×C₆₇	1	2	yes	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	#121×C₆₇	1	2	yes	—
271	C₂×C₄	#121×C₆₇	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
275	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
279	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
284	C₂²	#533×C₆₇	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	8	4	no	—
289	C₄	#471×C₆₇	1	4	yes	—
290	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
291	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
292	C₄	#532×C₆₇	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₆₇×#2857	1	1	yes	characteristic
296	C₂	C₂×#471×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#498×C₆₇	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
5. C₆₇×#1951	C₂	semidirect
6. C₆₇×#1848	C₂	semidirect
33. #131×C₆₇	Q₈	semidirect
144. #33220	C₆₇	direct (N × H)
149. C₂×#498	C₂×C₆₇	semidirect
152. #1997	C₂×C₆₇	semidirect
153. #1951	C₂×C₆₇	semidirect
154. #1848	C₂×C₆₇	semidirect
156. C₆₇	#33220	direct (N × H)
182. #131	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498×C₆₇	2
C₆₇×#3079	2
C₄×#131×C₆₇	2
C₆₇×#1997	2
C₆₇×#1951	2
C₆₇×#1848	2
C₆₇×#1815	2
#33220	67

Subgroups — C67×#33220 #30712471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33220 #30712471