Subgroups — C₄×#509×C₆₇ #30712703

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#509×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2177	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2175	C₂	1	2	yes	maximal, complemented
6	C₄×SD₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₆₇	—	2	1	no	—
15	C₂×C₄×D₄×C₆₇	—	2	1	no	—
16	#509×C₆₇	C₄	1	8	yes	direct product factor
17	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₄×SD₁₆×C₆₇	—	2	1	no	—
24	C₄×SD₁₆×C₆₇	—	2	1	no	—
25	C₄×D₈×C₆₇	—	2	1	no	—
26	C₄×D₈×C₆₇	—	2	1	no	—
27	C₄×D₈×C₆₇	C₂²	1	2	yes	—
28	C₄×D₈×C₆₇	C₂²	1	2	yes	—
29	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#427×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
33	#416×C₆₇	—	2	1	no	—
34	C₂×#102×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#367×C₆₇	C₂²	1	2	yes	complemented
37	#366×C₆₇	C₂²	1	2	yes	complemented
38	#358×C₆₇	C₂²	1	2	yes	complemented
39	#357×C₆₇	C₂²	1	2	yes	—
40	#357×C₆₇	C₂²	1	2	yes	—
41	#357×C₆₇	C₂²	1	2	yes	complemented
42	#346×C₆₇	C₂²	1	2	yes	complemented
43	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	—	4	1	no	—
47	#135×C₆₇	—	2	2	no	—
48	#135×C₆₇	C₂×C₄	1	4	yes	complemented
49	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×SD₁₆×C₆₇	—	2	1	no	—
51	C₂×D₈×C₆₇	—	2	1	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
54	C₂×D₈×C₆₇	C₂³	1	2	yes	—
55	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	4	1	no	—
65	C₄×D₄×C₆₇	—	4	1	no	—
66	C₄×D₄×C₆₇	—	4	1	no	—
67	C₄×D₄×C₆₇	—	4	1	no	—
68	#117×C₆₇	—	2	1	no	—
69	#117×C₆₇	—	4	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
72	C₂×C₄²×C₆₇	—	4	1	no	—
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
75	D₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
76	M₃₂×C₆₇	—	2	2	no	—
77	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
78	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
79	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
80	#107×C₆₇	—	2	1	no	—
81	#107×C₆₇	C₂³	1	1	yes	characteristic
82	#107×C₆₇	C₂³	1	1	yes	characteristic
83	#107×C₆₇	C₂³	1	2	yes	—
84	#106×C₆₇	—	2	1	no	—
85	#103×C₆₇	—	2	1	no	—
86	#103×C₆₇	—	2	1	no	—
87	#102×C₆₇	—	2	2	no	—
88	#102×C₆₇	—	2	2	no	—
89	#102×C₆₇	—	2	1	no	—
90	#102×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
92	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂⁴×C₆₇	—	4	1	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	2	1	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	4	1	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	4	1	no	—
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	8	4	no	—
112	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	Q₁₆×C₆₇	—	2	1	no	—
116	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	complemented
117	SD₁₆×C₆₇	—	2	4	no	—
118	D₈×C₆₇	—	2	4	no	—
119	D₈×C₆₇	—	2	2	no	—
120	D₈×C₆₇	—	2	1	no	—
121	D₈×C₆₇	C₂²×C₄	1	4	yes	—
122	D₈×C₆₇	C₂²×C₄	1	2	yes	complemented
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	#41	1	2	yes	—
129	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	4	1	no	—
134	C₄⋊C₄×C₆₇	—	4	1	no	—
135	C₄⋊C₄×C₆₇	—	4	1	no	—
136	C₂²⋊C₄×C₆₇	—	4	1	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	1	no	—
140	C₂²⋊C₄×C₆₇	—	4	1	no	—
141	C₄²×C₆₇	—	4	1	no	—
142	C₄²×C₆₇	—	4	1	no	—
143	C₄²×C₆₇	—	4	1	no	—
144	C₄²×C₆₇	D₈	1	1	yes	characteristic
145	C₄²×C₆₇	D₈	1	1	yes	characteristic
146	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₁₆×C₆₇	—	2	2	no	—
149	C₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
150	C₂³×C₆₇	—	4	1	no	—
151	C₂³×C₆₇	—	4	2	no	—
152	C₂³×C₆₇	—	4	4	no	—
153	C₂³×C₆₇	—	8	2	no	—
154	C₂³×C₆₇	—	8	2	no	—
155	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
156	Q₈×C₆₇	—	2	2	no	—
157	Q₈×C₆₇	—	4	1	no	—
158	D₄×C₆₇	—	2	4	no	—
159	D₄×C₆₇	—	2	2	no	—
160	D₄×C₆₇	—	4	4	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	4	1	no	—
163	D₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	8	2	no	—
172	C₂×C₄×C₆₇	—	8	4	no	—
173	C₂×C₄×C₆₇	—	8	2	no	—
174	C₂×C₄×C₆₇	—	8	2	no	—
175	C₂×C₄×C₆₇	—	8	4	no	—
176	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₆₇	#135	1	2	yes	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₈×C₆₇	—	2	1	no	—
185	C₈×C₆₇	—	2	1	no	—
186	C₈×C₆₇	C₄×D₄	1	2	yes	—
187	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	8	8	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	8	1	no	—
196	C₂²×C₆₇	—	8	1	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	C₄×D₈	1	2	yes	—
199	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
200	C₄×C₆₇	—	2	1	no	—
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	2	1	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	8	1	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	—	8	1	no	—
207	C₄×C₆₇	C₄×D₈	1	2	yes	—
208	C₄×C₆₇	#509	1	2	yes	direct product factor
209	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
211	C₄×#509	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	2	2	no	—
213	C₂×C₆₇	—	8	2	no	—
214	C₂×C₆₇	—	8	4	no	—
215	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
217	C₂×C₆₇	#3414	1	1	yes	characteristic
218	C₂×#509	C₂×C₆₇	1	1	yes	characteristic
219	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#2177	C₂×C₆₇	1	2	yes	complemented
222	#2175	C₂×C₆₇	1	2	yes	complemented
223	C₄×SD₃₂	C₂×C₆₇	1	2	yes	complemented
224	C₄×D₁₆	C₂×C₆₇	1	2	yes	complemented
225	#2159	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2140	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2139	C₂×C₆₇	1	1	yes	characteristic, complemented
228	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	C₄×#509	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
231	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₄×#41	—	2	1	no	—
233	C₂×C₄×D₄	—	2	1	no	—
234	#509	C₄×C₆₇	1	8	yes	direct product factor
235	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
236	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
237	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
239	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
240	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
241	C₄×SD₁₆	—	2	1	no	—
242	C₄×SD₁₆	—	2	1	no	—
243	C₄×D₈	—	2	1	no	—
244	C₄×D₈	—	2	1	no	—
245	C₄×D₈	C₂²×C₆₇	1	2	yes	—
246	C₄×D₈	C₂²×C₆₇	1	2	yes	—
247	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
248	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
249	#427	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
251	#416	—	2	1	no	—
252	C₂×#102	—	2	1	no	—
253	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	#367	C₂²×C₆₇	1	2	yes	complemented
255	#366	C₂²×C₆₇	1	2	yes	complemented
256	#358	C₂²×C₆₇	1	2	yes	complemented
257	#357	C₂²×C₆₇	1	2	yes	—
258	#357	C₂²×C₆₇	1	2	yes	—
259	#357	C₂²×C₆₇	1	2	yes	complemented
260	#346	C₂²×C₆₇	1	2	yes	complemented
261	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
262	C₂×#41	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂³×C₄	—	4	1	no	—
265	#135	—	2	2	no	—
266	#135	C₂×C₄×C₆₇	1	4	yes	complemented
267	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
268	C₂×SD₁₆	—	2	1	no	—
269	C₂×D₈	—	2	1	no	—
270	C₂×D₈	—	2	2	no	—
271	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
272	C₂×D₈	C₂³×C₆₇	1	2	yes	—
273	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
274	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
276	C₄×Q₈	—	2	1	no	—
277	C₄×Q₈	—	2	1	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	—	4	1	no	—
283	C₄×D₄	—	4	1	no	—
284	C₄×D₄	—	4	1	no	—
285	C₄×D₄	—	4	1	no	—
286	#117	—	2	1	no	—
287	#117	—	4	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	4	1	no	—
290	C₂×C₄²	—	4	1	no	—
291	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
292	SD₃₂	C₂×C₄×C₆₇	1	8	yes	complemented
293	D₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
294	M₃₂	—	2	2	no	—
295	M₃₂	C₂×C₄×C₆₇	1	4	yes	complemented
296	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
297	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
298	#107	—	2	1	no	—
299	#107	C₂³×C₆₇	1	1	yes	characteristic
300	#107	C₂³×C₆₇	1	1	yes	characteristic
301	#107	C₂³×C₆₇	1	2	yes	—
302	#106	—	2	1	no	—
303	#103	—	2	1	no	—
304	#103	—	2	1	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	2	1	no	—
308	#102	—	2	1	no	—
309	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
310	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
313	C₂⁴	—	4	1	no	—
314	#41	—	2	4	no	—
315	#41	—	4	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	1	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	—	4	1	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	—	8	4	no	—
330	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	Q₁₆	—	2	1	no	—
334	Q₁₆	C₂²×C₄×C₆₇	1	2	yes	complemented
335	SD₁₆	—	2	4	no	—
336	D₈	—	2	4	no	—
337	D₈	—	2	2	no	—
338	D₈	—	2	1	no	—
339	D₈	C₂²×C₄×C₆₇	1	4	yes	—
340	D₈	C₂²×C₄×C₆₇	1	2	yes	complemented
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
343	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂×C₈	#41×C₆₇	1	2	yes	—
347	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	4	2	no	—
357	C₂²⋊C₄	—	4	1	no	—
358	C₂²⋊C₄	—	4	1	no	—
359	C₄²	—	4	1	no	—
360	C₄²	—	4	1	no	—
361	C₄²	—	4	1	no	—
362	C₄²	D₈×C₆₇	1	1	yes	characteristic
363	C₄²	D₈×C₆₇	1	1	yes	characteristic
364	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₁₆	—	2	2	no	—
367	C₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
368	C₂³	—	4	1	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	4	no	—
371	C₂³	—	8	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	4	1	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	2	no	—
378	D₄	—	4	4	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	1	no	—
381	D₄	—	4	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	8	4	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	8	4	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
395	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
397	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
400	C₂×C₄	#135×C₆₇	1	2	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	C₄×D₄×C₆₇	1	2	yes	—
405	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
406	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	8	8	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	2	no	—
416	C₂²	C₄×D₈×C₆₇	1	2	yes	—
417	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	2	no	—
423	C₄	—	8	1	no	—
424	C₄	—	8	1	no	—
425	C₄	C₄×D₈×C₆₇	1	2	yes	—
426	C₄	#509×C₆₇	1	2	yes	direct product factor
427	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
428	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
429	C₂	—	2	2	no	—
430	C₂	—	8	4	no	—
431	C₂	—	8	2	no	—
432	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
433	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
434	C₂	C₆₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#509×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#135×C₆₇	C₂	semidirect
3. C₂×C₄×D₈×C₆₇	C₂	semidirect
4. C₆₇×#2177	C₂	semidirect
5. C₆₇×#2175	C₂	semidirect
6. C₄×SD₃₂×C₆₇	C₂	semidirect
7. C₄×D₁₆×C₆₇	C₂	semidirect
8. C₆₇×#2159	C₂	semidirect
9. C₆₇×#2140	C₂	semidirect
10. C₆₇×#2139	C₂	semidirect
11. C₄×M₃₂×C₆₇	C₂	semidirect
16. #509×C₆₇	C₄	direct (N × H)
21. C₄×Q₁₆×C₆₇	C₂²	semidirect
22. C₄×Q₁₆×C₆₇	C₂²	semidirect
29. C₄×D₈×C₆₇	C₂²	semidirect
30. C₄×D₈×C₆₇	C₂²	semidirect
36. #367×C₆₇	C₂²	semidirect
37. #366×C₆₇	C₂²	semidirect
38. #358×C₆₇	C₂²	semidirect
41. #357×C₆₇	C₂²	semidirect
42. #346×C₆₇	C₂²	semidirect
43. C₄×C₁₆×C₆₇	C₂²	semidirect
48. #135×C₆₇	C₂×C₄	semidirect
53. C₂×D₈×C₆₇	C₂×C₄	semidirect
74. SD₃₂×C₆₇	C₂×C₄	semidirect
75. D₁₆×C₆₇	C₂×C₄	semidirect
77. M₃₂×C₆₇	C₂×C₄	semidirect
116. Q₁₆×C₆₇	C₂²×C₄	semidirect
122. D₈×C₆₇	C₂²×C₄	semidirect
149. C₁₆×C₆₇	C₂²×C₄	semidirect
208. C₄×C₆₇	#509	direct (N × H)
211. C₄×#509	C₆₇	direct (N × H)
219. C₄×#135	C₂×C₆₇	semidirect
220. C₂×C₄×D₈	C₂×C₆₇	semidirect
221. #2177	C₂×C₆₇	semidirect
222. #2175	C₂×C₆₇	semidirect
223. C₄×SD₃₂	C₂×C₆₇	semidirect
224. C₄×D₁₆	C₂×C₆₇	semidirect
225. #2159	C₂×C₆₇	semidirect
226. #2140	C₂×C₆₇	semidirect
227. #2139	C₂×C₆₇	semidirect
228. C₄×M₃₂	C₂×C₆₇	semidirect
229. C₆₇	C₄×#509	direct (N × H)
234. #509	C₄×C₆₇	direct (N × H)
239. C₄×Q₁₆	C₂²×C₆₇	semidirect
240. C₄×Q₁₆	C₂²×C₆₇	semidirect
247. C₄×D₈	C₂²×C₆₇	semidirect
248. C₄×D₈	C₂²×C₆₇	semidirect
254. #367	C₂²×C₆₇	semidirect
255. #366	C₂²×C₆₇	semidirect
256. #358	C₂²×C₆₇	semidirect
259. #357	C₂²×C₆₇	semidirect
260. #346	C₂²×C₆₇	semidirect
261. C₄×C₁₆	C₂²×C₆₇	semidirect
266. #135	C₂×C₄×C₆₇	semidirect
271. C₂×D₈	C₂×C₄×C₆₇	semidirect
292. SD₃₂	C₂×C₄×C₆₇	semidirect
293. D₁₆	C₂×C₄×C₆₇	semidirect
295. M₃₂	C₂×C₄×C₆₇	semidirect
334. Q₁₆	C₂²×C₄×C₆₇	semidirect
340. D₈	C₂²×C₄×C₆₇	semidirect
367. C₁₆	C₂²×C₄×C₆₇	semidirect
426. C₄	#509×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#509×C₆₇	2
C₄×#135×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2177	2
C₆₇×#2175	2
C₄×SD₃₂×C₆₇	2
C₄×D₁₆×C₆₇	2
C₆₇×#2159	2
C₆₇×#2140	2
C₆₇×#2139	2
C₄×M₃₂×C₆₇	2
C₄×#509	67

Subgroups — C4×#509×C67 #30712703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#509×C₆₇ #30712703