Subgroups — C₆₇×#33133 #30712384

Order 17152.

810 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×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2625	C₂	1	2	yes	maximal
4	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1933	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1894	C₂	1	2	yes	maximal
8	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	#467×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#103×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	4	yes	—
19	#405×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#400×C₆₇	—	2	2	no	—
23	#398×C₆₇	—	2	2	no	—
24	#397×C₆₇	—	2	4	no	—
25	#391×C₆₇	C₂²	1	4	yes	—
26	#389×C₆₇	C₂²	1	4	yes	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆×C₆₇	—	2	2	no	—
33	C₂×Q₁₆×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	4	no	—
36	C₂²×C₈×C₆₇	—	2	4	no	—
37	C₂²×C₈×C₆₇	D₄	1	4	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²⋊Q₈×C₆₇	—	2	4	no	—
42	C₂²⋊Q₈×C₆₇	—	2	4	no	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	#107×C₆₇	—	2	2	no	—
56	#106×C₆₇	—	2	4	no	—
57	#103×C₆₇	—	2	4	no	—
58	#103×C₆₇	—	2	8	no	—
59	#97×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	#95×C₆₇	—	2	2	no	—
62	#95×C₆₇	—	2	4	no	—
63	#95×C₆₇	—	2	4	no	—
64	#95×C₆₇	—	2	4	no	—
65	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	4	no	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₇	—	4	4	no	—
82	M₁₆×C₆₇	—	2	8	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	8	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	4	yes	—
90	C₂×C₈×C₆₇	#41	1	4	yes	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₄⋊C₄×C₆₇	—	8	4	no	—
99	C₂²⋊C₄×C₆₇	—	2	4	no	—
100	C₂²⋊C₄×C₆₇	—	4	4	no	—
101	C₂²⋊C₄×C₆₇	—	8	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	#121	1	2	yes	—
108	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	4	2	no	—
110	Q₈×C₆₇	—	4	2	no	—
111	Q₈×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	#121	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₈×C₆₇	—	2	2	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₈×C₆₇	#123	1	4	yes	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	4	4	no	—
140	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
141	C₂²×C₆₇	#532	1	2	yes	—
142	C₂²×C₆₇	#577	1	2	yes	—
143	C₂²×C₆₇	#578	1	2	yes	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	8	2	no	—
148	C₄×C₆₇	—	8	4	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
151	C₄×C₆₇	#525	1	1	yes	characteristic
152	C₄×C₆₇	#529	1	2	yes	—
153	#33133	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₆₇	—	4	2	no	—
155	C₂×C₆₇	#2854	1	1	yes	characteristic
156	C₂×C₆₇	#3059	1	1	yes	characteristic
157	C₂×C₆₇	#3060	1	1	yes	characteristic
158	C₂×C₆₇	#3178	1	4	yes	—
159	C₂×#467	C₂×C₆₇	1	1	yes	characteristic
160	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
161	#2625	C₂×C₆₇	1	2	yes	—
162	#1971	C₂×C₆₇	1	2	yes	complemented
163	#1944	C₂×C₆₇	1	2	yes	complemented
164	#1933	C₂×C₆₇	1	4	yes	complemented
165	#1894	C₂×C₆₇	1	2	yes	—
166	#1807	C₂×C₆₇	1	1	yes	characteristic
167	C₆₇	#33133	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
170	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
172	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
173	#467	—	2	4	no	—
174	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#106	C₂²×C₆₇	1	2	yes	—
176	C₂×#103	C₂²×C₆₇	1	4	yes	—
177	C₂×#103	C₂²×C₆₇	1	2	yes	—
178	#405	—	2	4	no	—
179	C₂×#97	C₂²×C₆₇	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
181	#400	—	2	2	no	—
182	#398	—	2	2	no	—
183	#397	—	2	4	no	—
184	#391	C₂²×C₆₇	1	4	yes	—
185	#389	C₂²×C₆₇	1	4	yes	—
186	#380	C₂²×C₆₇	1	2	yes	—
187	#340	C₂²×C₆₇	1	2	yes	—
188	#340	C₂²×C₆₇	1	4	yes	—
189	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
190	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	D₄×C₆₇	1	4	yes	—
197	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
200	C₂²⋊Q₈	—	2	4	no	—
201	C₂²⋊Q₈	—	2	4	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
214	#107	—	2	2	no	—
215	#106	—	2	4	no	—
216	#103	—	2	4	no	—
217	#103	—	2	8	no	—
218	#97	—	2	4	no	—
219	C₄×C₈	—	2	4	no	—
220	#95	—	2	2	no	—
221	#95	—	2	4	no	—
222	#95	—	2	4	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂²×C₄	#41×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	Q₁₆	—	4	4	no	—
241	M₁₆	—	2	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
248	C₂×C₈	#41×C₆₇	1	4	yes	—
249	C₂×C₈	#41×C₆₇	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	8	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	#121×C₆₇	1	2	yes	—
267	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	#121×C₆₇	1	2	yes	—
288	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
290	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
291	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	4	no	—
294	C₈	#123×C₆₇	1	4	yes	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
300	C₂²	#532×C₆₇	1	2	yes	—
301	C₂²	#577×C₆₇	1	2	yes	—
302	C₂²	#578×C₆₇	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	4	no	—
309	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
310	C₄	#525×C₆₇	1	1	yes	characteristic
311	C₄	#529×C₆₇	1	2	yes	—
312	C₂	—	4	2	no	—
313	C₂	C₆₇×#2854	1	1	yes	characteristic
314	C₂	C₆₇×#3059	1	1	yes	characteristic
315	C₂	C₆₇×#3060	1	1	yes	characteristic
316	C₂	C₆₇×#3178	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1971	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1933	C₂	semidirect
153. #33133	C₆₇	direct (N × H)
162. #1971	C₂×C₆₇	semidirect
163. #1944	C₂×C₆₇	semidirect
164. #1933	C₂×C₆₇	semidirect
167. C₆₇	#33133	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#467×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2625	2
C₆₇×#1971	2
C₆₇×#1944	2
C₆₇×#1933	2
C₆₇×#1894	2
C₆₇×#1807	2
#33133	67

Subgroups — C67×#33133 #30712384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33133 #30712384