Subgroups — C₆₇×#22689 #30701940

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2952	C₂	1	2	yes	maximal
3	C₆₇×#2912	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1951	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1859	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1815	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	C₂²	1	2	yes	—
11	#520×C₆₇	C₂²	1	4	yes	—
12	#443×C₆₇	C₄	1	8	yes	complemented
13	#442×C₆₇	—	2	2	no	—
14	#441×C₆₇	—	2	2	no	—
15	C₄×SD₁₆×C₆₇	—	2	4	no	—
16	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
18	#428×C₆₇	—	2	4	no	—
19	#428×C₆₇	C₂²	1	4	yes	—
20	#427×C₆₇	—	2	2	no	—
21	#427×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₆₇	—	2	2	no	—
23	C₂×#106×C₆₇	C₂²	1	2	yes	—
24	C₂×#106×C₆₇	C₂²	1	2	yes	—
25	C₂×#104×C₆₇	C₂²	1	2	yes	—
26	#417×C₆₇	C₂²	1	4	yes	—
27	#405×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#384×C₆₇	C₂²	1	2	yes	—
30	#343×C₆₇	C₂²	1	4	yes	—
31	#339×C₆₇	C₂²	1	2	yes	—
32	C₂×#41×C₆₇	C₂³	1	2	yes	—
33	#135×C₆₇	C₂×C₄	1	4	yes	—
34	#135×C₆₇	Q₈	1	4	yes	—
35	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
36	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
37	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
38	#131×C₆₇	—	2	4	no	—
39	#131×C₆₇	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	—	2	4	no	—
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	—	4	4	no	—
49	#117×C₆₇	—	2	4	no	—
50	#117×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	2	2	no	—
56	#108×C₆₇	C₂×C₄	1	4	yes	complemented
57	#107×C₆₇	—	2	4	no	—
58	#107×C₆₇	D₄	1	2	yes	—
59	#106×C₆₇	—	2	2	no	—
60	#106×C₆₇	—	2	2	no	—
61	#106×C₆₇	—	2	2	no	—
62	#106×C₆₇	—	2	2	no	—
63	#106×C₆₇	—	2	4	no	—
64	#106×C₆₇	D₄	1	2	yes	—
65	#104×C₆₇	C₂×C₄	1	8	yes	complemented
66	#103×C₆₇	—	2	4	no	—
67	#102×C₆₇	—	2	4	no	—
68	#97×C₆₇	—	2	2	no	—
69	C₄×C₈×C₆₇	—	2	2	no	—
70	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	—	2	2	no	—
73	#41×C₆₇	C₂²×C₄	1	4	yes	—
74	#41×C₆₇	C₂×Q₈	1	4	yes	—
75	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	—	2	1	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	4	4	no	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	Q₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
84	SD₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
85	D₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	2	8	no	—
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	8	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₆₇	#41	1	2	yes	—
102	C₄⋊C₄×C₆₇	—	2	4	no	—
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	4	4	no	—
105	C₄⋊C₄×C₆₇	—	8	4	no	—
106	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂²⋊C₄×C₆₇	—	4	4	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	1	no	—
111	C₄²×C₆₇	—	4	4	no	—
112	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	D₄×C₆₇	—	2	4	no	—
118	D₄×C₆₇	—	2	4	no	—
119	D₄×C₆₇	—	2	2	no	—
120	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	8	4	no	—
129	C₂×C₄×C₆₇	—	16	4	no	—
130	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₆₇	—	2	4	no	—
137	C₈×C₆₇	—	4	2	no	—
138	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₆₇	C₄×D₄	1	2	yes	—
141	C₈×C₆₇	C₄×D₄	1	2	yes	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	—	4	4	no	—
145	C₂²×C₆₇	#520	1	1	yes	characteristic
146	C₂²×C₆₇	D₄²	1	1	yes	characteristic
147	C₂²×C₆₇	#554	1	1	yes	characteristic
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	4	no	—
151	C₄×C₆₇	—	8	4	no	—
152	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₆₇	#547	1	1	yes	characteristic
155	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
156	#22689	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	4	4	no	—
160	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₆₇	#3291	1	2	yes	—
162	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
163	#2952	C₂×C₆₇	1	2	yes	—
164	#2912	C₂×C₆₇	1	2	yes	complemented
165	#1951	C₂×C₆₇	1	2	yes	complemented
166	#1859	C₂×C₆₇	1	4	yes	complemented
167	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
168	#1815	C₂×C₆₇	1	2	yes	complemented
169	#1772	C₂×C₆₇	1	1	yes	characteristic, complemented
170	C₆₇	#22689	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×#131	C₂²×C₆₇	1	2	yes	—
173	#520	C₂²×C₆₇	1	4	yes	—
174	#443	C₄×C₆₇	1	8	yes	complemented
175	#442	—	2	2	no	—
176	#441	—	2	2	no	—
177	C₄×SD₁₆	—	2	4	no	—
178	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
179	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
180	#428	—	2	4	no	—
181	#428	C₂²×C₆₇	1	4	yes	—
182	#427	—	2	2	no	—
183	#427	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×#106	—	2	2	no	—
185	C₂×#106	C₂²×C₆₇	1	2	yes	—
186	C₂×#106	C₂²×C₆₇	1	2	yes	—
187	C₂×#104	C₂²×C₆₇	1	2	yes	—
188	#417	C₂²×C₆₇	1	4	yes	—
189	#405	C₂²×C₆₇	1	2	yes	—
190	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
191	#384	C₂²×C₆₇	1	2	yes	—
192	#343	C₂²×C₆₇	1	4	yes	—
193	#339	C₂²×C₆₇	1	2	yes	—
194	C₂×#41	C₂³×C₆₇	1	2	yes	—
195	#135	C₂×C₄×C₆₇	1	4	yes	—
196	#135	Q₈×C₆₇	1	4	yes	—
197	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
198	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
199	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
200	#131	—	2	4	no	—
201	#131	C₂×C₄×C₆₇	1	8	yes	complemented
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
204	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
208	C₄×Q₈	—	2	4	no	—
209	C₄×D₄	—	2	4	no	—
210	C₄×D₄	—	4	4	no	—
211	#117	—	2	4	no	—
212	#117	C₂³×C₆₇	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	4	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	C₂×C₄×C₆₇	1	4	yes	complemented
219	#107	—	2	4	no	—
220	#107	D₄×C₆₇	1	2	yes	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	4	no	—
226	#106	D₄×C₆₇	1	2	yes	—
227	#104	C₂×C₄×C₆₇	1	8	yes	complemented
228	#103	—	2	4	no	—
229	#102	—	2	4	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	C₂²×C₄×C₆₇	1	4	yes	—
236	#41	C₂×Q₈×C₆₇	1	4	yes	—
237	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
245	Q₁₆	C₄⋊C₄×C₆₇	1	4	yes	complemented
246	SD₁₆	C₄⋊C₄×C₆₇	1	8	yes	complemented
247	D₈	C₄⋊C₄×C₆₇	1	4	yes	complemented
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
257	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
263	C₂×C₈	#41×C₆₇	1	2	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	8	4	no	—
268	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	4	4	no	—
274	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	16	4	no	—
292	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
297	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
298	C₈	—	2	4	no	—
299	C₈	—	4	2	no	—
300	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
301	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
302	C₈	C₄×D₄×C₆₇	1	2	yes	—
303	C₈	C₄×D₄×C₆₇	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	#520×C₆₇	1	1	yes	characteristic
308	C₂²	D₄²×C₆₇	1	1	yes	characteristic
309	C₂²	#554×C₆₇	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
315	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
316	C₄	#547×C₆₇	1	1	yes	characteristic
317	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
322	C₂	C₆₇×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2912	C₂	semidirect
4. C₆₇×#1951	C₂	semidirect
5. C₆₇×#1859	C₂	semidirect
6. C₆₇×#1844	C₂	semidirect
7. C₆₇×#1815	C₂	semidirect
8. C₆₇×#1772	C₂	semidirect
12. #443×C₆₇	C₄	semidirect
39. #131×C₆₇	C₂×C₄	semidirect
56. #108×C₆₇	C₂×C₄	semidirect
65. #104×C₆₇	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
83. Q₁₆×C₆₇	C₄⋊C₄	semidirect
84. SD₁₆×C₆₇	C₄⋊C₄	semidirect
85. D₈×C₆₇	C₄⋊C₄	semidirect
156. #22689	C₆₇	direct (N × H)
164. #2912	C₂×C₆₇	semidirect
165. #1951	C₂×C₆₇	semidirect
166. #1859	C₂×C₆₇	semidirect
167. #1844	C₂×C₆₇	semidirect
168. #1815	C₂×C₆₇	semidirect
169. #1772	C₂×C₆₇	semidirect
170. C₆₇	#22689	direct (N × H)
174. #443	C₄×C₆₇	semidirect
201. #131	C₂×C₄×C₆₇	semidirect
218. #108	C₂×C₄×C₆₇	semidirect
227. #104	C₂×C₄×C₆₇	semidirect
232. C₄×C₈	C₂×C₄×C₆₇	semidirect
245. Q₁₆	C₄⋊C₄×C₆₇	semidirect
246. SD₁₆	C₄⋊C₄×C₆₇	semidirect
247. D₈	C₄⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#443×C₆₇	2
C₆₇×#2952	2
C₆₇×#2912	2
C₆₇×#1951	2
C₆₇×#1859	2
C₆₇×#1844	2
C₆₇×#1815	2
C₆₇×#1772	2
#22689	67

Subgroups — C67×#22689 #30701940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22689 #30701940