Subgroups — C₆₇×#21184 #30700435

Order 17152.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₆₇	C₂	1	2	yes	maximal
2	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1953	C₂	1	8	yes	maximal, complemented
6	C₂×#344×C₆₇	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	#430×C₆₇	—	2	4	no	—
13	#430×C₆₇	—	2	2	no	—
14	#430×C₆₇	—	2	2	no	—
15	#430×C₆₇	C₄	1	8	yes	complemented
16	#430×C₆₇	C₂²	1	8	yes	—
17	C₂×#108×C₆₇	—	2	4	no	—
18	C₂×#108×C₆₇	C₂²	1	4	yes	—
19	C₂×#108×C₆₇	C₂²	1	4	yes	—
20	#405×C₆₇	—	2	2	no	—
21	#405×C₆₇	C₂²	1	4	yes	—
22	C₄×M₁₆×C₆₇	—	2	2	no	—
23	#344×C₆₇	C₂²	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₆₇	—	2	1	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
39	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
40	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
44	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	#117×C₆₇	—	2	1	no	—
50	#117×C₆₇	—	2	1	no	—
51	#117×C₆₇	Q₈	1	2	yes	—
52	#117×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
55	C₂×C₄²×C₆₇	—	2	1	no	—
56	#108×C₆₇	—	2	16	no	—
57	#108×C₆₇	—	2	8	no	—
58	#108×C₆₇	C₂×C₄	1	16	yes	complemented
59	#97×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	—	2	4	no	—
61	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₆₇	—	2	4	no	—
74	M₁₆×C₆₇	—	2	4	no	—
75	M₁₆×C₆₇	—	2	4	no	—
76	M₁₆×C₆₇	—	2	4	no	—
77	M₁₆×C₆₇	—	2	2	no	—
78	M₁₆×C₆₇	—	2	4	no	—
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	4	2	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
83	M₁₆×C₆₇	C₄⋊C₄	1	8	yes	—
84	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
85	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂²⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	8	1	no	—
104	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
105	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₆₇	#121	1	2	yes	—
117	C₂×C₄×C₆₇	#121	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#125	1	2	yes	—
121	C₂×C₄×C₆₇	#125	1	2	yes	—
122	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₈×C₆₇	—	2	8	no	—
130	C₈×C₆₇	—	2	4	no	—
131	C₈×C₆₇	—	4	4	no	—
132	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
133	C₂²×C₆₇	—	4	1	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#389	1	2	yes	—
139	C₂²×C₆₇	#389	1	2	yes	—
140	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	#389	1	2	yes	—
145	C₄×C₆₇	#389	1	2	yes	—
146	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
150	#21184	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
157	C₂×#430	C₂×C₆₇	1	2	yes	—
158	C₂×#430	C₂×C₆₇	1	1	yes	characteristic
159	C₂×#430	C₂×C₆₇	1	1	yes	characteristic
160	#2873	C₂×C₆₇	1	1	yes	characteristic
161	#1953	C₂×C₆₇	1	8	yes	complemented
162	C₂×#344	C₂×C₆₇	1	2	yes	—
163	C₆₇	#21184	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
166	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
168	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
169	#430	—	2	4	no	—
170	#430	—	2	2	no	—
171	#430	—	2	2	no	—
172	#430	C₄×C₆₇	1	8	yes	complemented
173	#430	C₂²×C₆₇	1	8	yes	—
174	C₂×#108	—	2	4	no	—
175	C₂×#108	C₂²×C₆₇	1	4	yes	—
176	C₂×#108	C₂²×C₆₇	1	4	yes	—
177	#405	—	2	2	no	—
178	#405	C₂²×C₆₇	1	4	yes	—
179	C₄×M₁₆	—	2	2	no	—
180	#344	C₂²×C₆₇	1	8	yes	—
181	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
193	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
194	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
195	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
196	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
197	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
198	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
199	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
200	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
201	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
206	#117	—	2	1	no	—
207	#117	—	2	1	no	—
208	#117	Q₈×C₆₇	1	2	yes	—
209	#117	C₂³×C₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	Q₈×C₆₇	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	#108	—	2	16	no	—
214	#108	—	2	8	no	—
215	#108	C₂×C₄×C₆₇	1	16	yes	complemented
216	#97	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
224	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
226	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	#41×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
240	M₁₆	C₄⋊C₄×C₆₇	1	8	yes	—
241	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
242	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
252	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
253	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₂³	—	8	1	no	—
261	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
262	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
263	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
264	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
265	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
273	C₂×C₄	#121×C₆₇	1	2	yes	—
274	C₂×C₄	#121×C₆₇	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
277	C₂×C₄	#125×C₆₇	1	2	yes	—
278	C₂×C₄	#125×C₆₇	1	2	yes	—
279	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
286	C₈	—	2	8	no	—
287	C₈	—	2	4	no	—
288	C₈	—	4	4	no	—
289	C₈	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#389×C₆₇	1	2	yes	—
296	C₂²	#389×C₆₇	1	2	yes	—
297	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
298	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
299	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
300	C₄	—	4	4	no	—
301	C₄	#389×C₆₇	1	2	yes	—
302	C₄	#389×C₆₇	1	2	yes	—
303	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
304	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
305	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
306	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₂×#389×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×#389
prime radical	C₂⁴×C₆₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1953	C₂	semidirect
15. #430×C₆₇	C₄	semidirect
58. #108×C₆₇	C₂×C₄	semidirect
150. #21184	C₆₇	direct (N × H)
161. #1953	C₂×C₆₇	semidirect
163. C₆₇	#21184	direct (N × H)
172. #430	C₄×C₆₇	semidirect
215. #108	C₂×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×#430×C₆₇	2
C₆₇×#2873	2
C₆₇×#1953	2
C₂×#344×C₆₇	2
#21184	67

Subgroups — C67×#21184 #30700435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21184 #30700435