Subgroups — C₆₇×#33190 #30712441

Order 17152.

1082 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3038	C₂	1	2	yes	maximal, complemented
4	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1978	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1965	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
9	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
10	C₂×#131×C₆₇	C₂²	1	2	yes	—
11	C₂×#127×C₆₇	C₂²	1	2	yes	—
12	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
13	#493×C₆₇	—	2	1	no	—
14	#493×C₆₇	—	2	1	no	—
15	#493×C₆₇	—	2	2	no	—
16	#472×C₆₇	—	2	16	no	—
17	#460×C₆₇	—	2	4	no	—
18	#459×C₆₇	—	2	4	no	—
19	C₂×#108×C₆₇	C₂²	1	4	yes	—
20	#423×C₆₇	C₂²	1	2	yes	—
21	#418×C₆₇	C₂²	1	4	yes	—
22	C₂×#100×C₆₇	C₂²	1	8	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	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₆₇	—	2	4	no	—
31	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
32	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
33	C₂×SD₁₆×C₆₇	—	2	4	no	—
34	C₂×D₈×C₆₇	—	2	4	no	—
35	C₂×D₈×C₆₇	—	2	8	no	—
36	C₂×D₈×C₆₇	—	2	4	no	—
37	C₂×D₈×C₆₇	—	2	4	no	—
38	C₂×D₈×C₆₇	—	2	8	no	—
39	#131×C₆₇	—	2	4	no	—
40	#131×C₆₇	D₄	1	8	yes	complemented
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	#127×C₆₇	—	2	4	no	—
47	#127×C₆₇	—	2	2	no	—
48	#127×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	D₄	1	2	yes	—
50	C₄×D₄×C₆₇	—	2	2	no	—
51	C₄×D₄×C₆₇	D₄	1	2	yes	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	—	2	1	no	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#108×C₆₇	—	2	8	no	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	#102×C₆₇	—	2	8	no	—
59	#100×C₆₇	—	2	16	no	—
60	#97×C₆₇	—	2	2	no	—
61	#97×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
66	C₄×C₈×C₆₇	D₄	1	2	yes	—
67	C₂⁴×C₆₇	—	2	4	no	—
68	#41×C₆₇	—	2	2	no	—
69	#41×C₆₇	C₂×D₄	1	4	yes	—
70	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×D₄×C₆₇	—	2	1	no	—
72	C₂×D₄×C₆₇	—	2	8	no	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	4	no	—
75	C₂×D₄×C₆₇	—	2	8	no	—
76	C₂×D₄×C₆₇	—	4	4	no	—
77	C₂×D₄×C₆₇	—	4	4	no	—
78	C₂×D₄×C₆₇	—	8	8	no	—
79	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₆₇	—	4	8	no	—
85	D₈×C₆₇	—	4	8	no	—
86	D₈×C₆₇	—	4	8	no	—
87	D₈×C₆₇	—	4	8	no	—
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	2	8	no	—
90	M₁₆×C₆₇	—	2	8	no	—
91	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	8	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	#41	1	4	yes	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²⋊C₄×C₆₇	—	2	1	no	—
102	C₂²⋊C₄×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	1	no	—
105	C₄²×C₆₇	C₂×D₄	1	2	yes	—
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	4	4	no	—
110	C₂³×C₆₇	—	4	8	no	—
111	C₂³×C₆₇	—	8	8	no	—
112	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	1	no	—
114	Q₈×C₆₇	#127	1	2	yes	—
115	D₄×C₆₇	—	2	4	no	—
116	D₄×C₆₇	—	2	1	no	—
117	D₄×C₆₇	—	4	8	no	—
118	D₄×C₆₇	—	4	4	no	—
119	D₄×C₆₇	—	4	4	no	—
120	D₄×C₆₇	—	8	8	no	—
121	D₄×C₆₇	—	8	8	no	—
122	D₄×C₆₇	#127	1	2	yes	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	#121	1	2	yes	—
129	C₂×C₄×C₆₇	#121	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
134	C₈×C₆₇	—	2	4	no	—
135	C₈×C₆₇	—	4	4	no	—
136	C₈×C₆₇	#121	1	4	yes	—
137	C₈×C₆₇	#121	1	4	yes	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	—	8	4	no	—
142	C₂²×C₆₇	—	8	8	no	—
143	C₂²×C₆₇	—	16	8	no	—
144	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
145	C₂²×C₆₇	#532	1	2	yes	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	#472	1	4	yes	complemented
150	C₄×C₆₇	C₂×#121	1	2	yes	—
151	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
152	C₄×C₆₇	#532	1	1	yes	characteristic
153	#33190	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	4	2	no	—
156	C₂×C₆₇	—	16	4	no	—
157	C₂×C₆₇	#2856	1	1	yes	characteristic
158	C₂×C₆₇	C₂×#472	1	2	yes	—
159	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₂×#472	C₂×C₆₇	1	4	yes	complemented
161	#3038	C₂×C₆₇	1	2	yes	complemented
162	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
163	#1978	C₂×C₆₇	1	4	yes	complemented
164	#1965	C₂×C₆₇	1	2	yes	complemented
165	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
166	C₆₇	#33190	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
168	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
169	C₂×#131	C₂²×C₆₇	1	2	yes	—
170	C₂×#127	C₂²×C₆₇	1	2	yes	—
171	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
172	#493	—	2	1	no	—
173	#493	—	2	1	no	—
174	#493	—	2	2	no	—
175	#472	—	2	16	no	—
176	#460	—	2	4	no	—
177	#459	—	2	4	no	—
178	C₂×#108	C₂²×C₆₇	1	4	yes	—
179	#423	C₂²×C₆₇	1	2	yes	—
180	#418	C₂²×C₆₇	1	4	yes	—
181	C₂×#100	C₂²×C₆₇	1	8	yes	—
182	#405	—	2	2	no	—
183	#405	C₂²×C₆₇	1	2	yes	—
184	C₄×M₁₆	—	2	1	no	—
185	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×C₄×C₈	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
191	AGL(1,ℤ/8)	—	2	16	no	—
192	C₂×SD₁₆	—	2	4	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	8	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	4	no	—
197	C₂×D₈	—	2	8	no	—
198	#131	—	2	4	no	—
199	#131	D₄×C₆₇	1	8	yes	complemented
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
202	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
205	#127	—	2	4	no	—
206	#127	—	2	2	no	—
207	#127	—	2	2	no	—
208	C₄×Q₈	D₄×C₆₇	1	2	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	D₄×C₆₇	1	2	yes	—
211	#117	—	2	1	no	—
212	#117	C₂³×C₆₇	1	1	yes	characteristic
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
215	#108	—	2	8	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#102	—	2	8	no	—
218	#100	—	2	16	no	—
219	#97	—	2	2	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
224	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
225	C₄×C₈	D₄×C₆₇	1	2	yes	—
226	C₂⁴	—	2	4	no	—
227	#41	—	2	2	no	—
228	#41	C₂×D₄×C₆₇	1	4	yes	—
229	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	8	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	8	8	no	—
238	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	#41×C₆₇	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
243	SD₁₆	—	4	8	no	—
244	D₈	—	4	8	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
257	C₂×C₈	#41×C₆₇	1	4	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₆₇	1	2	yes	—
267	C₂³	—	2	1	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	8	8	no	—
271	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#127×C₆₇	1	2	yes	—
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	8	8	no	—
280	D₄	—	8	8	no	—
281	D₄	#127×C₆₇	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	#121×C₆₇	1	2	yes	—
288	C₂×C₄	#121×C₆₇	1	2	yes	—
289	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	4	4	no	—
295	C₈	#121×C₆₇	1	4	yes	—
296	C₈	#121×C₆₇	1	4	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
304	C₂²	#532×C₆₇	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	#472×C₆₇	1	4	yes	complemented
309	C₄	C₂×#121×C₆₇	1	2	yes	—
310	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
311	C₄	#532×C₆₇	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	—	16	4	no	—
315	C₂	C₆₇×#2856	1	1	yes	characteristic
316	C₂	C₂×#472×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
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×#493×C₆₇	C₂	semidirect
2. C₂×#472×C₆₇	C₂	semidirect
3. C₆₇×#3038	C₂	semidirect
4. C₄×#131×C₆₇	C₂	semidirect
5. C₆₇×#1978	C₂	semidirect
6. C₆₇×#1965	C₂	semidirect
7. C₆₇×#1848	C₂	semidirect
40. #131×C₆₇	D₄	semidirect
149. C₄×C₆₇	#472	semidirect
153. #33190	C₆₇	direct (N × H)
159. C₂×#493	C₂×C₆₇	semidirect
160. C₂×#472	C₂×C₆₇	semidirect
161. #3038	C₂×C₆₇	semidirect
162. C₄×#131	C₂×C₆₇	semidirect
163. #1978	C₂×C₆₇	semidirect
164. #1965	C₂×C₆₇	semidirect
165. #1848	C₂×C₆₇	semidirect
166. C₆₇	#33190	direct (N × H)
199. #131	D₄×C₆₇	semidirect
308. C₄	#472×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₆₇	2
C₂×#472×C₆₇	2
C₆₇×#3038	2
C₄×#131×C₆₇	2
C₆₇×#1978	2
C₆₇×#1965	2
C₆₇×#1848	2
#33190	67

Subgroups — C67×#33190 #30712441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33190 #30712441