Subgroups — C₆₇×#21430 #30700681

Order 17152.

858 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	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1923	C₂	1	4	yes	maximal
3	C₆₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
6	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
9	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
20	#376×C₆₇	C₄	1	1	yes	characteristic
21	#376×C₆₇	C₄	1	1	yes	characteristic
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	4	yes	—
25	#336×C₆₇	C₂²	1	4	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	#117×C₆₇	—	2	4	no	—
33	#117×C₆₇	C₈	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	—	2	4	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	—	2	8	no	—
52	C₂²⋊C₈×C₆₇	—	2	8	no	—
53	C₄×C₈×C₆₇	—	2	8	no	—
54	C₄×C₈×C₆₇	—	2	4	no	—
55	C₄×C₈×C₆₇	—	2	4	no	—
56	#95×C₆₇	C₂×C₄	1	2	yes	—
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	#95×C₆₇	C₂×C₄	1	2	yes	—
60	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	#41	1	8	yes	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
86	C₂²⋊C₄×C₆₇	—	2	4	no	—
87	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	C₂×C₈	1	4	yes	—
97	C₄²×C₆₇	C₂×C₈	1	4	yes	—
98	C₄²×C₆₇	M₁₆	1	4	yes	—
99	C₄²×C₆₇	#41	1	4	yes	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
105	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	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	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	#117	1	2	yes	—
125	C₂×C₄×C₆₇	#117	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
127	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
128	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
132	C₂×C₄×C₆₇	#131	1	4	yes	—
133	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
135	C₈×C₆₇	—	2	8	no	—
136	C₈×C₆₇	—	4	4	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	—	4	1	no	—
142	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
143	C₂²×C₆₇	C₄×#41	1	2	yes	—
144	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#131	1	2	yes	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	4	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	1	no	—
154	C₄×C₆₇	#431	1	4	yes	—
155	C₄×C₆₇	#433	1	4	yes	—
156	C₄×C₆₇	#529	1	4	yes	—
157	#21430	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	4	2	no	—
159	C₂×C₆₇	#2298	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
162	C₂×C₆₇	C₈×#41	1	2	yes	—
163	C₂×C₆₇	#2965	1	2	yes	—
164	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
165	#1923	C₂×C₆₇	1	4	yes	—
166	#1916	C₂×C₆₇	1	4	yes	complemented
167	C₄×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
168	C₄×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
169	#1744	C₂×C₆₇	1	1	yes	characteristic, complemented
170	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
171	C₆₇	#21430	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
173	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
176	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
177	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
178	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
180	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
182	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
183	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
184	#376	C₄×C₆₇	1	1	yes	characteristic
185	#376	C₄×C₆₇	1	1	yes	characteristic
186	C₄³	C₂²×C₆₇	1	1	yes	characteristic
187	C₄³	C₂²×C₆₇	1	1	yes	characteristic
188	#336	C₂²×C₆₇	1	4	yes	—
189	#336	C₂²×C₆₇	1	4	yes	—
190	#336	C₂²×C₆₇	1	4	yes	—
191	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
192	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
193	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
195	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
196	#117	—	2	4	no	—
197	#117	C₈×C₆₇	1	8	yes	complemented
198	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
205	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
206	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
215	C₄⋊C₈	—	2	8	no	—
216	C₂²⋊C₈	—	2	8	no	—
217	C₄×C₈	—	2	8	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	—	2	4	no	—
220	#95	C₂×C₄×C₆₇	1	2	yes	—
221	#95	C₂×C₄×C₆₇	1	2	yes	—
222	#95	C₂×C₄×C₆₇	1	2	yes	—
223	#95	C₂×C₄×C₆₇	1	2	yes	—
224	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	#41×C₆₇	1	8	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂×C₈×C₆₇	1	4	yes	—
261	C₄²	C₂×C₈×C₆₇	1	4	yes	—
262	C₄²	M₁₆×C₆₇	1	4	yes	—
263	C₄²	#41×C₆₇	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
269	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	#117×C₆₇	1	2	yes	—
289	C₂×C₄	#117×C₆₇	1	2	yes	—
290	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
291	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
292	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
293	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
294	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
295	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
296	C₂×C₄	#131×C₆₇	1	4	yes	—
297	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
298	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
299	C₈	—	2	8	no	—
300	C₈	—	4	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
307	C₂²	C₄×#41×C₆₇	1	2	yes	—
308	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
309	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
310	C₂²	C₂×#131×C₆₇	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	#431×C₆₇	1	4	yes	—
319	C₄	#433×C₆₇	1	4	yes	—
320	C₄	#529×C₆₇	1	4	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₆₇×#2298	1	1	yes	characteristic
323	C₂	C₂×#431×C₆₇	1	1	yes	characteristic
324	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
325	C₂	C₈×#41×C₆₇	1	2	yes	—
326	C₂	C₆₇×#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1916	C₂	semidirect
4. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
6. C₆₇×#1744	C₂	semidirect
7. C₄²×C₈×C₆₇	C₂	semidirect
33. #117×C₆₇	C₈	semidirect
157. #21430	C₆₇	direct (N × H)
166. #1916	C₂×C₆₇	semidirect
167. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
169. #1744	C₂×C₆₇	semidirect
170. C₄²×C₈	C₂×C₆₇	semidirect
171. C₆₇	#21430	direct (N × H)
197. #117	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#117×C₆₇	2
C₆₇×#1923	2
C₆₇×#1916	2
C₄×C₄⋊C₈×C₆₇	2
C₄×C₂²⋊C₈×C₆₇	2
C₆₇×#1744	2
C₄²×C₈×C₆₇	2
#21430	67

Subgroups — C67×#21430 #30700681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21430 #30700681