Subgroups — C₆₇×#22316 #30701567

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2399	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1985	C₂	1	2	yes	maximal
3	C₆₇×#1984	C₂	1	2	yes	maximal
4	C₆₇×#1939	C₂	1	2	yes	maximal
5	C₆₇×#1917	C₂	1	2	yes	maximal
6	C₆₇×#1916	C₂	1	2	yes	maximal
7	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1771	C₂	1	2	yes	maximal
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	#398×C₆₇	C₄	1	2	yes	—
20	#398×C₆₇	C₄	1	2	yes	—
21	#395×C₆₇	C₄	1	2	yes	—
22	#395×C₆₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	#376×C₆₇	C₂²	1	1	yes	characteristic
26	#336×C₆₇	C₂²	1	2	yes	—
27	#336×C₆₇	C₂²	1	2	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	#336×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	2	yes	—
32	#336×C₆₇	C₂²	1	1	yes	characteristic
33	#336×C₆₇	C₂²	1	1	yes	characteristic
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₄×Q₈×C₆₇	—	4	2	no	—
42	C₄×Q₈×C₆₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	—	2	4	no	—
57	C₄⋊C₈×C₆₇	—	4	2	no	—
58	C₄⋊C₈×C₆₇	—	4	2	no	—
59	#97×C₆₇	—	4	2	no	—
60	#97×C₆₇	Q₈	1	4	yes	complemented
61	C₄×C₈×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	—	4	2	no	—
63	#95×C₆₇	C₂×C₄	1	2	yes	—
64	#95×C₆₇	C₂×C₄	1	2	yes	—
65	#95×C₆₇	C₂×C₄	1	2	yes	—
66	#95×C₆₇	C₂×C₄	1	2	yes	—
67	#95×C₆₇	C₂×C₄	1	2	yes	—
68	#95×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×Q₈×C₆₇	—	2	4	no	—
70	C₂×Q₈×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	4	1	no	—
72	C₂×Q₈×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	2	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	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	4	2	no	—
97	C₄⋊C₄×C₆₇	—	4	2	no	—
98	C₄⋊C₄×C₆₇	—	4	2	no	—
99	C₄⋊C₄×C₆₇	—	4	1	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
109	C₄²×C₆₇	#41	1	2	yes	—
110	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	4	no	—
112	Q₈×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
134	C₂×C₄×C₆₇	#131	1	4	yes	—
135	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
139	C₈×C₆₇	—	2	4	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂²×C₆₇	#518	1	1	yes	characteristic
145	C₂²×C₆₇	#519	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#131	1	2	yes	—
147	C₂²×C₆₇	#568	1	2	yes	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	4	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	1	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	#542	1	2	yes	—
157	C₄×C₆₇	#544	1	2	yes	—
158	#22316	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₆₇	#2331	1	1	yes	characteristic
160	C₂×C₆₇	#2962	1	2	yes	—
161	C₂×C₆₇	#2977	1	1	yes	characteristic
162	C₂×C₆₇	#2978	1	1	yes	characteristic
163	C₂×C₆₇	#2985	1	2	yes	—
164	#2399	C₂×C₆₇	1	1	yes	characteristic
165	#1985	C₂×C₆₇	1	2	yes	—
166	#1984	C₂×C₆₇	1	2	yes	—
167	#1939	C₂×C₆₇	1	2	yes	—
168	#1917	C₂×C₆₇	1	2	yes	—
169	#1916	C₂×C₆₇	1	2	yes	—
170	#1832	C₂×C₆₇	1	1	yes	characteristic
171	#1771	C₂×C₆₇	1	2	yes	—
172	#1746	C₂×C₆₇	1	1	yes	characteristic
173	C₆₇	#22316	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
178	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
179	C₂×#97	C₂²×C₆₇	1	2	yes	—
180	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
183	#398	C₄×C₆₇	1	2	yes	—
184	#398	C₄×C₆₇	1	2	yes	—
185	#395	C₄×C₆₇	1	2	yes	—
186	#395	C₄×C₆₇	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
189	#376	C₂²×C₆₇	1	1	yes	characteristic
190	#336	C₂²×C₆₇	1	1	yes	characteristic
191	#336	C₂²×C₆₇	1	1	yes	characteristic
192	#336	C₂²×C₆₇	1	2	yes	—
193	#336	C₂²×C₆₇	1	2	yes	—
194	#336	C₂²×C₆₇	1	2	yes	—
195	#336	C₂²×C₆₇	1	2	yes	—
196	#336	C₂²×C₆₇	1	2	yes	—
197	#336	C₂²×C₆₇	1	2	yes	—
198	#336	C₂²×C₆₇	1	2	yes	—
199	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
200	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
204	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
205	C₄×Q₈	—	4	2	no	—
206	C₄×Q₈	—	4	2	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
217	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
218	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
220	C₄⋊C₈	—	2	4	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	2	no	—
223	#97	—	4	2	no	—
224	#97	Q₈×C₆₇	1	4	yes	complemented
225	C₄×C₈	—	2	2	no	—
226	C₄×C₈	—	4	2	no	—
227	#95	C₂×C₄×C₆₇	1	2	yes	—
228	#95	C₂×C₄×C₆₇	1	2	yes	—
229	#95	C₂×C₄×C₆₇	1	2	yes	—
230	#95	C₂×C₄×C₆₇	1	2	yes	—
231	#95	C₂×C₄×C₆₇	1	2	yes	—
232	#95	C₂×C₄×C₆₇	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
273	C₄²	#41×C₆₇	1	2	yes	—
274	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
275	Q₈	—	4	4	no	—
276	Q₈	—	4	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
298	C₂×C₄	#131×C₆₇	1	4	yes	—
299	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
301	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
302	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
308	C₂²	#518×C₆₇	1	1	yes	characteristic
309	C₂²	#519×C₆₇	1	1	yes	characteristic
310	C₂²	C₂×#131×C₆₇	1	2	yes	—
311	C₂²	#568×C₆₇	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	#542×C₆₇	1	2	yes	—
321	C₄	#544×C₆₇	1	2	yes	—
322	C₂	C₆₇×#2331	1	1	yes	characteristic
323	C₂	C₆₇×#2962	1	2	yes	—
324	C₂	C₆₇×#2977	1	1	yes	characteristic
325	C₂	C₆₇×#2978	1	1	yes	characteristic
326	C₂	C₆₇×#2985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
60. #97×C₆₇	Q₈	semidirect
158. #22316	C₆₇	direct (N × H)
173. C₆₇	#22316	direct (N × H)
224. #97	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2399	2
C₆₇×#1985	2
C₆₇×#1984	2
C₆₇×#1939	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1832	2
C₆₇×#1771	2
C₆₇×#1746	2
#22316	67

Subgroups — C67×#22316 #30701567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22316 #30701567