Subgroups — C₆₇×#33170 #30712421

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#465×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1968	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#121×C₆₇	—	2	2	no	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#492×C₆₇	—	2	4	no	—
19	#465×C₆₇	—	2	16	no	—
20	C₂×#106×C₆₇	C₂²	1	4	yes	—
21	C₂×#103×C₆₇	C₂²	1	4	yes	—
22	C₂×#102×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₆₇	—	2	4	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#391×C₆₇	C₂²	1	2	yes	—
28	#390×C₆₇	—	2	2	no	—
29	#390×C₆₇	C₂²	1	2	yes	—
30	#389×C₆₇	—	2	2	no	—
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#380×C₆₇	—	2	2	no	—
34	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×D₄×C₆₇	—	2	2	no	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×SD₁₆×C₆₇	—	2	8	no	—
41	C₂×SD₁₆×C₆₇	—	2	8	no	—
42	C₂×SD₁₆×C₆₇	—	2	8	no	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	#127×C₆₇	—	2	2	no	—
51	#127×C₆₇	—	2	2	no	—
52	C₂²⋊Q₈×C₆₇	—	2	8	no	—
53	#121×C₆₇	—	2	8	no	—
54	#121×C₆₇	—	4	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	D₄	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	#106×C₆₇	—	2	8	no	—
70	#103×C₆₇	—	2	8	no	—
71	#102×C₆₇	—	2	8	no	—
72	C₄×C₈×C₆₇	—	2	4	no	—
73	C₄×C₈×C₆₇	—	2	2	no	—
74	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
75	#95×C₆₇	—	2	4	no	—
76	C₂⁴×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	2	4	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×D₄×C₆₇	—	2	4	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	8	no	—
88	C₂×D₄×C₆₇	—	4	4	no	—
89	C₂×D₄×C₆₇	—	4	4	no	—
90	C₂×D₄×C₆₇	—	4	4	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	SD₁₆	1	8	yes	complemented
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	SD₁₆×C₆₇	—	4	16	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	16	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₆₇	#41	1	4	yes	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	4	4	no	—
116	C₄⋊C₄×C₆₇	—	4	4	no	—
117	C₄⋊C₄×C₆₇	—	4	4	no	—
118	C₄⋊C₄×C₆₇	—	4	4	no	—
119	C₂²⋊C₄×C₆₇	—	4	4	no	—
120	C₂²⋊C₄×C₆₇	—	4	4	no	—
121	C₂²⋊C₄×C₆₇	—	4	8	no	—
122	C₄²×C₆₇	—	2	4	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	—	2	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₄²×C₆₇	#41	1	2	yes	—
127	C₂³×C₆₇	—	2	1	no	—
128	C₂³×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	4	no	—
135	C₂³×C₆₇	#127	1	2	yes	—
136	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
137	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	4	4	no	—
139	Q₈×C₆₇	—	4	4	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	4	4	no	—
142	D₄×C₆₇	—	8	8	no	—
143	D₄×C₆₇	—	8	8	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	16	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	#121	1	4	yes	—
159	C₂×C₄×C₆₇	#127	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
161	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
162	C₂×C₄×C₆₇	#135	1	4	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₈×C₆₇	—	2	4	no	—
168	C₈×C₆₇	#121	1	8	yes	complemented
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	4	no	—
172	C₂²×C₆₇	—	2	4	no	—
173	C₂²×C₆₇	—	8	2	no	—
174	C₂²×C₆₇	—	8	2	no	—
175	C₂²×C₆₇	—	8	2	no	—
176	C₂²×C₆₇	—	8	4	no	—
177	C₂²×C₆₇	—	8	4	no	—
178	C₂²×C₆₇	#492	1	4	yes	complemented
179	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
180	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
181	C₂²×C₆₇	#532	1	1	yes	characteristic
182	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
183	C₂²×C₆₇	C₂×#135	1	2	yes	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	4	no	—
186	C₄×C₆₇	—	8	2	no	—
187	C₄×C₆₇	—	8	2	no	—
188	C₄×C₆₇	—	8	2	no	—
189	C₄×C₆₇	#465	1	8	yes	complemented
190	C₄×C₆₇	C₂×#121	1	2	yes	—
191	C₄×C₆₇	#532	1	2	yes	—
192	#33170	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	2	4	no	—
194	C₂×C₆₇	—	16	2	no	—
195	C₂×C₆₇	#2856	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#465	1	2	yes	—
197	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
198	C₂×C₆₇	#3147	1	1	yes	characteristic
199	C₂×C₆₇	#3167	1	2	yes	—
200	C₂×#492	C₂×C₆₇	1	1	yes	characteristic, complemented
201	C₂×#465	C₂×C₆₇	1	4	yes	complemented
202	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#2856	C₂×C₆₇	1	1	yes	characteristic
205	#1968	C₂×C₆₇	1	4	yes	complemented
206	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₆₇	#33170	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
211	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
212	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
214	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
215	C₂×#121	—	2	2	no	—
216	C₂×#121	C₂²×C₆₇	1	2	yes	—
217	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
218	#492	—	2	4	no	—
219	#465	—	2	16	no	—
220	C₂×#106	C₂²×C₆₇	1	4	yes	—
221	C₂×#103	C₂²×C₆₇	1	4	yes	—
222	C₂×#102	C₂²×C₆₇	1	4	yes	—
223	C₂×C₄×C₈	—	2	4	no	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	#391	C₂²×C₆₇	1	2	yes	—
228	#390	—	2	2	no	—
229	#390	C₂²×C₆₇	1	2	yes	—
230	#389	—	2	2	no	—
231	#384	C₂²×C₆₇	1	1	yes	characteristic
232	#384	C₂²×C₆₇	1	1	yes	characteristic
233	#380	—	2	2	no	—
234	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
235	C₂²×D₄	—	2	2	no	—
236	C₂²×D₄	—	2	2	no	—
237	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
238	C₂³×C₄	D₄×C₆₇	1	2	yes	—
239	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
240	C₂×SD₁₆	—	2	8	no	—
241	C₂×SD₁₆	—	2	8	no	—
242	C₂×SD₁₆	—	2	8	no	—
243	C₂²×C₈	—	2	4	no	—
244	C₂²×C₈	—	2	4	no	—
245	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
246	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
247	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
248	C₄⋊Q₈	—	2	2	no	—
249	C₄⋊Q₈	—	2	2	no	—
250	#127	—	2	2	no	—
251	#127	—	2	2	no	—
252	C₂²⋊Q₈	—	2	8	no	—
253	#121	—	2	8	no	—
254	#121	—	4	8	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
261	C₂×C₂²⋊C₄	—	2	2	no	—
262	C₂×C₂²⋊C₄	—	2	4	no	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₄²	—	2	2	no	—
265	C₂×C₄²	—	2	2	no	—
266	C₂×C₄²	D₄×C₆₇	1	2	yes	—
267	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	#106	—	2	8	no	—
270	#103	—	2	8	no	—
271	#102	—	2	8	no	—
272	C₄×C₈	—	2	4	no	—
273	C₄×C₈	—	2	2	no	—
274	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
275	#95	—	2	4	no	—
276	C₂⁴	—	2	2	no	—
277	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂×Q₈	—	2	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	4	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	4	no	—
286	C₂×D₄	—	4	4	no	—
287	C₂×D₄	—	4	8	no	—
288	C₂×D₄	—	4	4	no	—
289	C₂×D₄	—	4	4	no	—
290	C₂×D₄	—	4	4	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	SD₁₆×C₆₇	1	8	yes	complemented
300	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
305	SD₁₆	—	4	16	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	16	no	—
309	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
310	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
311	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
312	C₂×C₈	#41×C₆₇	1	4	yes	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	4	4	no	—
316	C₄⋊C₄	—	4	4	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	—	4	4	no	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	—	4	8	no	—
322	C₄²	—	2	4	no	—
323	C₄²	—	2	1	no	—
324	C₄²	—	2	1	no	—
325	C₄²	C₂×D₄×C₆₇	1	2	yes	—
326	C₄²	#41×C₆₇	1	2	yes	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	4	no	—
334	C₂³	—	4	4	no	—
335	C₂³	#127×C₆₇	1	2	yes	—
336	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
337	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
338	Q₈	—	4	4	no	—
339	Q₈	—	4	4	no	—
340	D₄	—	4	4	no	—
341	D₄	—	4	4	no	—
342	D₄	—	8	8	no	—
343	D₄	—	8	8	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	16	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	#121×C₆₇	1	4	yes	—
359	C₂×C₄	#127×C₆₇	1	2	yes	—
360	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
361	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
362	C₂×C₄	#135×C₆₇	1	4	yes	—
363	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
365	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
367	C₈	—	2	4	no	—
368	C₈	#121×C₆₇	1	8	yes	complemented
369	C₂²	—	2	4	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	8	4	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	4	no	—
378	C₂²	#492×C₆₇	1	4	yes	complemented
379	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
380	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
381	C₂²	#532×C₆₇	1	1	yes	characteristic
382	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
383	C₂²	C₂×#135×C₆₇	1	2	yes	—
384	C₄	—	2	4	no	—
385	C₄	—	2	2	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	2	no	—
389	C₄	#465×C₆₇	1	8	yes	complemented
390	C₄	C₂×#121×C₆₇	1	2	yes	—
391	C₄	#532×C₆₇	1	2	yes	—
392	C₂	—	2	4	no	—
393	C₂	—	16	2	no	—
394	C₂	C₆₇×#2856	1	1	yes	characteristic
395	C₂	C₂×#465×C₆₇	1	2	yes	—
396	C₂	C₂×#492×C₆₇	1	1	yes	characteristic
397	C₂	C₆₇×#3147	1	1	yes	characteristic
398	C₂	C₆₇×#3167	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
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
1. C₂×#492×C₆₇	C₂	semidirect
2. C₂×#465×C₆₇	C₂	semidirect
3. C₂²×C₄×C₈×C₆₇	C₂	semidirect
4. C₆₇×#2857	C₂	semidirect
6. C₆₇×#1968	C₂	semidirect
7. C₆₇×#1957	C₂	semidirect
8. C₆₇×#1956	C₂	semidirect
9. C₆₇×#1844	C₂	semidirect
45. C₂²×C₈×C₆₇	D₄	semidirect
74. C₄×C₈×C₆₇	D₄	semidirect
99. C₂²×C₄×C₆₇	SD₁₆	semidirect
168. C₈×C₆₇	#121	semidirect
178. C₂²×C₆₇	#492	semidirect
189. C₄×C₆₇	#465	semidirect
192. #33170	C₆₇	direct (N × H)
200. C₂×#492	C₂×C₆₇	semidirect
201. C₂×#465	C₂×C₆₇	semidirect
202. C₂²×C₄×C₈	C₂×C₆₇	semidirect
203. #2857	C₂×C₆₇	semidirect
205. #1968	C₂×C₆₇	semidirect
206. #1957	C₂×C₆₇	semidirect
207. #1956	C₂×C₆₇	semidirect
208. #1844	C₂×C₆₇	semidirect
209. C₆₇	#33170	direct (N × H)
245. C₂²×C₈	D₄×C₆₇	semidirect
274. C₄×C₈	D₄×C₆₇	semidirect
299. C₂²×C₄	SD₁₆×C₆₇	semidirect
368. C₈	#121×C₆₇	semidirect
378. C₂²	#492×C₆₇	semidirect
389. C₄	#465×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₆₇	2
C₂×#465×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2857	2
C₆₇×#2856	2
C₆₇×#1968	2
C₆₇×#1957	2
C₆₇×#1956	2
C₆₇×#1844	2
#33170	67

Subgroups — C67×#33170 #30712421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33170 #30712421