Subgroups — C₆₇×#33108 #30712359

Order 17152.

964 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#483×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1948	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1927	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1872	C₂	1	2	yes	maximal
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	—	2	1	no	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#483×C₆₇	—	2	8	no	—
16	C₂×#106×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#103×C₆₇	C₂²	1	4	yes	—
19	C₂×#103×C₆₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	#391×C₆₇	C₂²	1	2	yes	—
22	#389×C₆₇	—	2	1	no	—
23	#389×C₆₇	—	2	1	no	—
24	#384×C₆₇	C₂²	1	1	yes	characteristic
25	#380×C₆₇	—	2	1	no	—
26	#380×C₆₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂×#95×C₆₇	—	2	1	no	—
32	#340×C₆₇	C₂²	1	4	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	—	2	1	no	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₆₇	—	2	4	no	—
42	C₂²⋊Q₈×C₆₇	—	2	4	no	—
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	4	no	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	4	4	no	—
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#106×C₆₇	—	2	8	no	—
67	C₄⋊C₈×C₆₇	—	2	4	no	—
68	#103×C₆₇	—	2	8	no	—
69	#103×C₆₇	—	2	8	no	—
70	C₂²⋊C₈×C₆₇	—	2	4	no	—
71	#95×C₆₇	—	2	2	no	—
72	#95×C₆₇	—	2	2	no	—
73	#95×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	2	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₆₇	—	2	4	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	—	4	4	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
121	C₄⋊C₄×C₆₇	#41	1	4	yes	—
122	C₂²⋊C₄×C₆₇	—	4	4	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	4	4	no	—
127	C₄²×C₆₇	—	4	4	no	—
128	C₄²×C₆₇	#41	1	4	yes	—
129	C₂³×C₆₇	—	2	1	no	—
130	C₂³×C₆₇	—	2	1	no	—
131	C₂³×C₆₇	—	2	1	no	—
132	C₂³×C₆₇	—	2	1	no	—
133	C₂³×C₆₇	#121	1	2	yes	—
134	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
135	C₂³×C₆₇	#137	1	2	yes	—
136	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₆₇	—	4	4	no	—
138	Q₈×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	8	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	8	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	8	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	#121	1	2	yes	—
167	C₂×C₄×C₆₇	#123	1	2	yes	—
168	C₂×C₄×C₆₇	#123	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
171	C₂×C₄×C₆₇	#137	1	2	yes	—
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
175	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
176	C₈×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	#461	1	4	yes	—
184	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
185	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
186	C₂²×C₆₇	#525	1	1	yes	characteristic
187	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
190	C₂²×C₆₇	#575	1	1	yes	characteristic
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	2	2	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	4	no	—
196	C₄×C₆₇	—	4	4	no	—
197	C₄×C₆₇	—	8	2	no	—
198	C₄×C₆₇	—	8	2	no	—
199	C₄×C₆₇	#483	1	4	yes	complemented
200	C₄×C₆₇	#529	1	2	yes	—
201	C₄×C₆₇	#532	1	2	yes	—
202	#33108	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	4	no	—
204	C₂×C₆₇	#2854	1	1	yes	characteristic
205	C₂×C₆₇	C₂×#461	1	1	yes	characteristic
206	C₂×C₆₇	#3037	1	1	yes	characteristic
207	C₂×C₆₇	C₂×#483	1	1	yes	characteristic
208	C₂×C₆₇	#3095	1	1	yes	characteristic
209	C₂×C₆₇	#3101	1	1	yes	characteristic
210	C₂×C₆₇	#3113	1	1	yes	characteristic
211	C₂×#483	C₂×C₆₇	1	2	yes	—
212	#2857	C₂×C₆₇	1	1	yes	characteristic
213	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
214	#1981	C₂×C₆₇	1	2	yes	complemented
215	#1948	C₂×C₆₇	1	2	yes	complemented
216	#1927	C₂×C₆₇	1	4	yes	complemented
217	#1872	C₂×C₆₇	1	2	yes	—
218	#1797	C₂×C₆₇	1	1	yes	characteristic
219	C₆₇	#33108	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
222	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
223	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂²×C₄²	—	2	1	no	—
225	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
226	#483	—	2	8	no	—
227	C₂×#106	C₂²×C₆₇	1	4	yes	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×#103	C₂²×C₆₇	1	4	yes	—
230	C₂×#103	C₂²×C₆₇	1	4	yes	—
231	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
232	#391	C₂²×C₆₇	1	2	yes	—
233	#389	—	2	1	no	—
234	#389	—	2	1	no	—
235	#384	C₂²×C₆₇	1	1	yes	characteristic
236	#380	—	2	1	no	—
237	#380	—	2	1	no	—
238	C₄×C₄⋊C₄	—	2	1	no	—
239	C₄×C₄⋊C₄	—	2	1	no	—
240	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
241	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₂×#95	—	2	1	no	—
243	#340	C₂²×C₆₇	1	4	yes	—
244	#336	C₂²×C₆₇	1	2	yes	—
245	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
246	C₂³×C₄	—	2	1	no	—
247	C₂³×C₄	—	2	1	no	—
248	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
249	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
252	C₄⋊Q₈	—	2	4	no	—
253	C₂²⋊Q₈	—	2	4	no	—
254	C₂²⋊Q₈	—	2	4	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	4	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
265	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₄²	—	2	4	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	4	4	no	—
273	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
277	#106	—	2	8	no	—
278	C₄⋊C₈	—	2	4	no	—
279	#103	—	2	8	no	—
280	#103	—	2	8	no	—
281	C₂²⋊C₈	—	2	4	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	4	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	4	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	#41×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	4	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
332	C₄⋊C₄	#41×C₆₇	1	4	yes	—
333	C₂²⋊C₄	—	4	4	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	4	4	no	—
338	C₄²	—	4	4	no	—
339	C₄²	#41×C₆₇	1	4	yes	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	#121×C₆₇	1	2	yes	—
345	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
346	C₂³	#137×C₆₇	1	2	yes	—
347	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
348	Q₈	—	4	4	no	—
349	Q₈	—	4	4	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	8	no	—
366	C₂×C₄	—	4	8	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	#121×C₆₇	1	2	yes	—
378	C₂×C₄	#123×C₆₇	1	2	yes	—
379	C₂×C₄	#123×C₆₇	1	2	yes	—
380	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
381	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
382	C₂×C₄	#137×C₆₇	1	2	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
386	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
387	C₈	—	4	4	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	#461×C₆₇	1	4	yes	—
395	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
396	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
397	C₂²	#525×C₆₇	1	1	yes	characteristic
398	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
399	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
400	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
401	C₂²	#575×C₆₇	1	1	yes	characteristic
402	C₄	—	2	2	no	—
403	C₄	—	2	2	no	—
404	C₄	—	2	2	no	—
405	C₄	—	4	4	no	—
406	C₄	—	4	1	no	—
407	C₄	—	4	4	no	—
408	C₄	—	8	2	no	—
409	C₄	—	8	2	no	—
410	C₄	#483×C₆₇	1	4	yes	complemented
411	C₄	#529×C₆₇	1	2	yes	—
412	C₄	#532×C₆₇	1	2	yes	—
413	C₂	—	2	4	no	—
414	C₂	C₆₇×#2854	1	1	yes	characteristic
415	C₂	C₂×#461×C₆₇	1	1	yes	characteristic
416	C₂	C₆₇×#3037	1	1	yes	characteristic
417	C₂	C₂×#483×C₆₇	1	1	yes	characteristic
418	C₂	C₆₇×#3095	1	1	yes	characteristic
419	C₂	C₆₇×#3101	1	1	yes	characteristic
420	C₂	C₆₇×#3113	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1981	C₂	semidirect
5. C₆₇×#1948	C₂	semidirect
6. C₆₇×#1927	C₂	semidirect
199. C₄×C₆₇	#483	semidirect
202. #33108	C₆₇	direct (N × H)
214. #1981	C₂×C₆₇	semidirect
215. #1948	C₂×C₆₇	semidirect
216. #1927	C₂×C₆₇	semidirect
219. C₆₇	#33108	direct (N × H)
410. C₄	#483×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#483×C₆₇	2
C₆₇×#2857	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1981	2
C₆₇×#1948	2
C₆₇×#1927	2
C₆₇×#1872	2
C₆₇×#1797	2
#33108	67

Subgroups — C67×#33108 #30712359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33108 #30712359