Subgroups — C₆₇×#33222 #30712473

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×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×#1952	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1816	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	#500×C₆₇	—	2	1	no	—
12	#500×C₆₇	—	2	1	no	—
13	#500×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×#107×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	1	yes	characteristic
44	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×D₄×C₆₇	—	2	1	no	—
46	C₄×D₄×C₆₇	—	2	1	no	—
47	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
48	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
49	#117×C₆₇	—	2	1	no	—
50	#117×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	2	8	no	—
56	#107×C₆₇	—	2	4	no	—
57	#107×C₆₇	—	2	4	no	—
58	#107×C₆₇	—	2	4	no	—
59	#106×C₆₇	—	2	4	no	—
60	C₄⋊C₈×C₆₇	—	2	2	no	—
61	C₄⋊C₈×C₆₇	—	2	2	no	—
62	#103×C₆₇	—	2	4	no	—
63	#102×C₆₇	—	2	4	no	—
64	#97×C₆₇	—	2	2	no	—
65	#97×C₆₇	—	2	1	no	—
66	C₄×C₈×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	D₄	1	2	yes	—
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂×Q₈×C₆₇	—	4	2	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	8	4	no	—
76	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×D₄×C₆₇	—	2	1	no	—
78	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	—	2	1	no	—
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	2	8	no	—
88	M₁₆×C₆₇	—	2	8	no	—
89	M₁₆×C₆₇	C₂×Q₈	1	4	yes	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	8	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
96	C₂×C₈×C₆₇	#41	1	4	yes	—
97	C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₆₇	—	2	1	no	—
106	C₂²⋊C₄×C₆₇	—	2	1	no	—
107	C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	—	2	1	no	—
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	—	2	1	no	—
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₆₇	—	2	1	no	—
116	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
117	Q₈×C₆₇	—	2	1	no	—
118	Q₈×C₆₇	—	8	4	no	—
119	Q₈×C₆₇	—	8	4	no	—
120	Q₈×C₆₇	C₄⋊Q₈	1	2	yes	—
121	D₄×C₆₇	—	2	4	no	—
122	D₄×C₆₇	—	2	1	no	—
123	D₄×C₆₇	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	8	4	no	—
134	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
142	C₈×C₆₇	—	2	4	no	—
143	C₈×C₆₇	—	4	4	no	—
144	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
145	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	4	1	no	—
150	C₂²×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₆₇	#533	1	2	yes	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	4	1	no	—
157	C₄×C₆₇	—	4	1	no	—
158	C₄×C₆₇	—	8	4	no	—
159	C₄×C₆₇	#472	1	2	yes	—
160	C₄×C₆₇	#473	1	2	yes	—
161	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
162	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₄×C₆₇	#532	1	1	yes	characteristic
164	#33222	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	—	4	2	no	—
167	C₂×C₆₇	#2857	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
170	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#3079	C₂×C₆₇	1	2	yes	—
172	C₄×#131	C₂×C₆₇	1	1	yes	characteristic
173	#1997	C₂×C₆₇	1	2	yes	complemented
174	#1952	C₂×C₆₇	1	4	yes	complemented
175	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#1816	C₂×C₆₇	1	4	yes	—
177	C₆₇	#33222	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#131	C₂²×C₆₇	1	2	yes	—
179	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
180	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
181	#500	—	2	1	no	—
182	#500	—	2	1	no	—
183	#500	—	2	2	no	—
184	#477	—	2	2	no	—
185	#476	—	2	2	no	—
186	#475	—	2	2	no	—
187	#474	—	2	2	no	—
188	C₂×#108	C₂²×C₆₇	1	4	yes	—
189	#428	C₂²×C₆₇	1	4	yes	—
190	C₂×#107	C₂²×C₆₇	1	4	yes	—
191	#423	C₂²×C₆₇	1	2	yes	—
192	#417	C₂²×C₆₇	1	4	yes	—
193	#405	—	2	2	no	—
194	#405	C₂²×C₆₇	1	2	yes	—
195	C₄×M₁₆	—	2	1	no	—
196	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×C₄×C₈	—	2	1	no	—
198	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
199	#343	C₂²×C₆₇	1	8	yes	—
200	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
201	C₂²×Q₈	—	2	2	no	—
202	#131	—	2	4	no	—
203	#131	Q₈×C₆₇	1	8	yes	complemented
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
206	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
214	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
218	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
219	#117	—	2	1	no	—
220	#117	C₂³×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
225	#108	—	2	8	no	—
226	#107	—	2	4	no	—
227	#107	—	2	4	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	#97	—	2	2	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
239	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
240	C₄×C₈	D₄×C₆₇	1	2	yes	—
241	#41	—	2	2	no	—
242	#41	C₂×Q₈×C₆₇	1	4	yes	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	8	4	no	—
246	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
254	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	8	no	—
258	M₁₆	—	2	8	no	—
259	M₁₆	C₂×Q₈×C₆₇	1	4	yes	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
265	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
266	C₂×C₈	#41×C₆₇	1	4	yes	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	8	4	no	—
289	Q₈	—	8	4	no	—
290	Q₈	C₄⋊Q₈×C₆₇	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	C₄⋊Q₈×C₆₇	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
305	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
306	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
307	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
311	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
312	C₈	—	2	4	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
315	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
322	C₂²	#533×C₆₇	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	4	no	—
329	C₄	#472×C₆₇	1	2	yes	—
330	C₄	#473×C₆₇	1	2	yes	—
331	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
332	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
333	C₄	#532×C₆₇	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	C₆₇×#2857	1	1	yes	characteristic
337	C₂	C₂×#472×C₆₇	1	1	yes	characteristic
338	C₂	C₂×#473×C₆₇	1	1	yes	characteristic

	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×#500×C₆₇	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
5. C₆₇×#1952	C₂	semidirect
6. C₆₇×#1848	C₂	semidirect
33. #131×C₆₇	Q₈	semidirect
164. #33222	C₆₇	direct (N × H)
170. C₂×#500	C₂×C₆₇	semidirect
173. #1997	C₂×C₆₇	semidirect
174. #1952	C₂×C₆₇	semidirect
175. #1848	C₂×C₆₇	semidirect
177. C₆₇	#33222	direct (N × H)
203. #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×#500×C₆₇	2
C₆₇×#3079	2
C₄×#131×C₆₇	2
C₆₇×#1997	2
C₆₇×#1952	2
C₆₇×#1848	2
C₆₇×#1816	2
#33222	67

Subgroups — C67×#33222 #30712473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33222 #30712473