Subgroups — C₆₇×#33203 #30712454

Order 17152.

864 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#478×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2660	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2275	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1989	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1949	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1947	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1804	C₂	1	2	yes	maximal
11	C₆₇×#1802	C₂	1	2	yes	maximal
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
16	#478×C₆₇	—	2	4	no	—
17	#474×C₆₇	—	2	4	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	#398×C₆₇	—	2	2	no	—
29	#395×C₆₇	—	2	1	no	—
30	#395×C₆₇	—	2	1	no	—
31	#384×C₆₇	C₂²	1	2	yes	—
32	#382×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
37	#376×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#95×C₆₇	—	2	1	no	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	#125×C₆₇	—	2	2	no	—
53	#125×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	—	2	4	no	—
56	C₄×D₄×C₆₇	D₄	1	4	yes	—
57	C₄×D₄×C₆₇	Q₈	1	8	yes	complemented
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	—	2	4	no	—
75	#107×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	1	no	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	D₄	1	1	yes	characteristic
89	#95×C₆₇	D₄	1	1	yes	characteristic
90	#95×C₆₇	D₄	1	2	yes	—
91	C₂⁴×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	4	2	no	—
93	C₂×Q₈×C₆₇	—	8	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	8	2	no	—
106	C₂²×C₄×C₆₇	—	8	2	no	—
107	C₂²×C₄×C₆₇	—	8	2	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	—	4	2	no	—
131	C₄⋊C₄×C₆₇	—	8	2	no	—
132	C₄⋊C₄×C₆₇	—	8	2	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	8	no	—
137	C₄²×C₆₇	—	8	2	no	—
138	C₄²×C₆₇	—	8	1	no	—
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
141	C₄²×C₆₇	#41	1	2	yes	—
142	C₄²×C₆₇	#41	1	4	yes	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	4	1	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	8	4	no	—
150	D₄×C₆₇	—	2	2	no	—
151	D₄×C₆₇	—	2	4	no	—
152	D₄×C₆₇	C₄⋊Q₈	1	4	yes	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	4	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	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	8	8	no	—
174	C₂×C₄×C₆₇	—	8	4	no	—
175	C₂×C₄×C₆₇	#121	1	2	yes	—
176	C₂×C₄×C₆₇	#121	1	2	yes	—
177	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
178	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₈×C₆₇	—	4	1	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
192	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
193	C₂²×C₆₇	#576	1	2	yes	—
194	C₂²×C₆₇	#577	1	1	yes	characteristic
195	C₂²×C₆₇	#578	1	1	yes	characteristic
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	8	1	no	—
203	C₄×C₆₇	—	8	2	no	—
204	C₄×C₆₇	—	8	1	no	—
205	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
206	C₄×C₆₇	#532	1	1	yes	characteristic
207	C₄×C₆₇	#533	1	2	yes	—
208	#33203	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	4	4	no	—
210	C₂×C₆₇	#2857	1	1	yes	characteristic
211	C₂×C₆₇	#3041	1	1	yes	characteristic
212	C₂×C₆₇	#3046	1	1	yes	characteristic
213	C₂×C₆₇	#3081	1	2	yes	—
214	C₂×C₆₇	#3083	1	2	yes	—
215	C₂×#478	C₂×C₆₇	1	1	yes	characteristic
216	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
217	#2660	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#2275	C₂×C₆₇	1	1	yes	characteristic
219	#1989	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1949	C₂×C₆₇	1	2	yes	complemented
222	#1947	C₂×C₆₇	1	2	yes	complemented
223	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1804	C₂×C₆₇	1	2	yes	—
225	#1802	C₂×C₆₇	1	2	yes	—
226	C₆₇	#33203	1	1	yes	characteristic, direct product factor, Sylow
227	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
231	#478	—	2	4	no	—
232	#474	—	2	4	no	—
233	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#107	C₂²×C₆₇	1	2	yes	—
235	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#102	C₂²×C₆₇	1	2	yes	—
243	#398	—	2	2	no	—
244	#395	—	2	1	no	—
245	#395	—	2	1	no	—
246	#384	C₂²×C₆₇	1	2	yes	—
247	#382	C₂²×C₆₇	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
250	C₄×C₂²⋊C₄	—	2	1	no	—
251	C₄×C₂²⋊C₄	—	2	2	no	—
252	#376	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#95	—	2	1	no	—
254	#340	C₂²×C₆₇	1	2	yes	—
255	#340	C₂²×C₆₇	1	2	yes	—
256	#340	C₂²×C₆₇	1	2	yes	—
257	#340	C₂²×C₆₇	1	2	yes	—
258	C₂²×Q₈	—	2	1	no	—
259	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
260	C₂³×C₄	—	2	1	no	—
261	C₂³×C₄	—	2	2	no	—
262	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
264	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	2	2	no	—
266	C₄⋊Q₈	—	2	2	no	—
267	#125	—	2	2	no	—
268	#125	—	2	2	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	D₄×C₆₇	1	4	yes	—
272	C₄×D₄	Q₈×C₆₇	1	8	yes	complemented
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
283	C₂×C₂²⋊C₄	—	2	1	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
289	#107	—	2	4	no	—
290	#107	—	2	2	no	—
291	#106	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	4	no	—
295	C₄⋊C₈	—	2	4	no	—
296	#102	—	2	4	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#95	—	2	1	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	D₄×C₆₇	1	1	yes	characteristic
304	#95	D₄×C₆₇	1	1	yes	characteristic
305	#95	D₄×C₆₇	1	2	yes	—
306	C₂⁴	—	2	1	no	—
307	C₂×Q₈	—	4	2	no	—
308	C₂×Q₈	—	8	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	8	2	no	—
321	C₂²×C₄	—	8	2	no	—
322	C₂²×C₄	—	8	2	no	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	8	2	no	—
347	C₄⋊C₄	—	8	2	no	—
348	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₄²	—	8	2	no	—
353	C₄²	—	8	1	no	—
354	C₄²	C₂×D₄×C₆₇	1	2	yes	—
355	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
356	C₄²	#41×C₆₇	1	2	yes	—
357	C₄²	#41×C₆₇	1	4	yes	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	1	no	—
362	C₂³	—	4	1	no	—
363	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
364	Q₈	—	8	4	no	—
365	D₄	—	2	2	no	—
366	D₄	—	2	4	no	—
367	D₄	C₄⋊Q₈×C₆₇	1	4	yes	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	8	4	no	—
389	C₂×C₄	—	8	8	no	—
390	C₂×C₄	#121×C₆₇	1	2	yes	—
391	C₂×C₄	#121×C₆₇	1	2	yes	—
392	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
393	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
394	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
396	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
397	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
398	C₈	—	4	1	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	1	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	4	4	no	—
406	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
407	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
408	C₂²	#576×C₆₇	1	2	yes	—
409	C₂²	#577×C₆₇	1	1	yes	characteristic
410	C₂²	#578×C₆₇	1	1	yes	characteristic
411	C₄	—	2	2	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	4	no	—
414	C₄	—	2	4	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	4	no	—
417	C₄	—	8	2	no	—
418	C₄	—	8	1	no	—
419	C₄	—	8	1	no	—
420	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
421	C₄	#532×C₆₇	1	1	yes	characteristic
422	C₄	#533×C₆₇	1	2	yes	—
423	C₂	—	4	4	no	—
424	C₂	C₆₇×#2857	1	1	yes	characteristic
425	C₂	C₆₇×#3041	1	1	yes	characteristic
426	C₂	C₆₇×#3046	1	1	yes	characteristic
427	C₂	C₆₇×#3081	1	2	yes	—
428	C₂	C₆₇×#3083	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2660	C₂	semidirect
5. C₆₇×#1989	C₂	semidirect
6. C₆₇×#1987	C₂	semidirect
7. C₆₇×#1949	C₂	semidirect
8. C₆₇×#1947	C₂	semidirect
9. C₆₇×#1843	C₂	semidirect
57. C₄×D₄×C₆₇	Q₈	semidirect
208. #33203	C₆₇	direct (N × H)
217. #2660	C₂×C₆₇	semidirect
219. #1989	C₂×C₆₇	semidirect
220. #1987	C₂×C₆₇	semidirect
221. #1949	C₂×C₆₇	semidirect
222. #1947	C₂×C₆₇	semidirect
223. #1843	C₂×C₆₇	semidirect
226. C₆₇	#33203	direct (N × H)
272. C₄×D₄	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#478×C₆₇	2
C₂×#474×C₆₇	2
C₆₇×#2660	2
C₆₇×#2275	2
C₆₇×#1989	2
C₆₇×#1987	2
C₆₇×#1949	2
C₆₇×#1947	2
C₆₇×#1843	2
C₆₇×#1804	2
C₆₇×#1802	2
#33203	67

Subgroups — C67×#33203 #30712454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33203 #30712454