Subgroups — C₆₇×#33613 #30712864

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3061	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2220	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2219	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2217	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2216	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#358×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#537×C₆₇	—	2	1	no	—
16	#536×C₆₇	—	2	1	no	—
17	#510×C₆₇	—	2	4	no	—
18	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
20	#504×C₆₇	—	2	4	no	—
21	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
25	#470×C₆₇	—	2	2	no	—
26	#468×C₆₇	—	2	2	no	—
27	#467×C₆₇	C₂²	1	2	yes	—
28	#467×C₆₇	C₂²	1	2	yes	—
29	#467×C₆₇	C₂²	1	2	yes	—
30	#466×C₆₇	C₂²	1	2	yes	—
31	#427×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
33	#419×C₆₇	—	2	1	no	—
34	#417×C₆₇	—	2	1	no	—
35	#367×C₆₇	C₂²	1	2	yes	—
36	#366×C₆₇	C₂²	1	2	yes	—
37	#358×C₆₇	C₂²	1	2	yes	—
38	#358×C₆₇	C₂²	1	2	yes	—
39	#358×C₆₇	C₂²	1	1	yes	characteristic
40	#358×C₆₇	C₂²	1	1	yes	characteristic
41	#357×C₆₇	C₂²	1	1	yes	characteristic
42	#357×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	#135×C₆₇	—	2	2	no	—
47	#135×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	2	no	—
51	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₆₇	—	2	1	no	—
54	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
55	#131×C₆₇	—	2	2	no	—
56	#131×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
58	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	—	4	1	no	—
62	#124×C₆₇	—	4	1	no	—
63	#123×C₆₇	—	4	2	no	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	2	no	—
66	C₂²⋊Q₈×C₆₇	—	2	2	no	—
67	C₂²⋊Q₈×C₆₇	—	4	1	no	—
68	#121×C₆₇	—	2	2	no	—
69	#121×C₆₇	—	4	1	no	—
70	#121×C₆₇	—	4	2	no	—
71	#117×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	Q₃₂×C₆₇	—	2	4	no	—
74	SD₃₂×C₆₇	—	2	4	no	—
75	M₃₂×C₆₇	—	2	4	no	—
76	M₃₂×C₆₇	D₄	1	4	yes	complemented
77	C₂×C₁₆×C₆₇	—	2	2	no	—
78	C₂×C₁₆×C₆₇	—	2	4	no	—
79	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
82	#107×C₆₇	—	2	1	no	—
83	#107×C₆₇	C₂³	1	2	yes	—
84	#107×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	C₂³	1	1	yes	characteristic
86	#106×C₆₇	—	2	1	no	—
87	#103×C₆₇	—	2	2	no	—
88	#103×C₆₇	—	2	2	no	—
89	#103×C₆₇	—	2	2	no	—
90	#102×C₆₇	—	2	2	no	—
91	#41×C₆₇	—	2	2	no	—
92	#41×C₆₇	—	2	2	no	—
93	#41×C₆₇	—	4	2	no	—
94	#41×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	4	1	no	—
98	C₂×Q₈×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	D₈	1	2	yes	—
105	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	Q₁₆×C₆₇	—	2	2	no	—
112	Q₁₆×C₆₇	—	2	1	no	—
113	Q₁₆×C₆₇	—	2	2	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	—	4	2	no	—
116	SD₁₆×C₆₇	—	4	2	no	—
117	D₈×C₆₇	—	2	1	no	—
118	D₈×C₆₇	—	2	1	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	M₁₆×C₆₇	—	2	4	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₆₇	#41	1	2	yes	—
127	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	4	1	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₂²⋊C₄×C₆₇	—	4	1	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	—	4	1	no	—
141	C₁₆×C₆₇	—	2	1	no	—
142	C₁₆×C₆₇	—	2	1	no	—
143	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	C₂×D₈	1	2	yes	—
146	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
147	Q₈×C₆₇	—	2	2	no	—
148	Q₈×C₆₇	—	4	1	no	—
149	Q₈×C₆₇	—	4	2	no	—
150	Q₈×C₆₇	—	4	2	no	—
151	Q₈×C₆₇	—	4	1	no	—
152	Q₈×C₆₇	—	8	2	no	—
153	D₄×C₆₇	—	2	4	no	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	4	1	no	—
156	D₄×C₆₇	—	4	1	no	—
157	D₄×C₆₇	—	8	2	no	—
158	D₄×C₆₇	—	16	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	8	2	no	—
167	C₂×C₄×C₆₇	—	8	2	no	—
168	C₂×C₄×C₆₇	—	16	1	no	—
169	C₂×C₄×C₆₇	#121	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
171	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#135	1	2	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	#121	1	2	yes	—
176	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	8	1	no	—
184	C₂²×C₆₇	—	8	1	no	—
185	C₂²×C₆₇	—	8	1	no	—
186	C₂²×C₆₇	#466	1	2	yes	—
187	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	8	1	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	—	8	1	no	—
192	C₄×C₆₇	—	8	1	no	—
193	C₄×C₆₇	—	8	2	no	—
194	C₄×C₆₇	#466	1	2	yes	—
195	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
197	#33613	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	4	2	no	—
200	C₂×C₆₇	—	16	1	no	—
201	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
202	C₂×C₆₇	#3416	1	1	yes	characteristic
203	C₂×C₆₇	#3417	1	1	yes	characteristic
204	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
205	C₂×#504	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#3061	C₂×C₆₇	1	1	yes	characteristic
207	#3059	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#2220	C₂×C₆₇	1	2	yes	complemented
209	#2219	C₂×C₆₇	1	2	yes	complemented
210	#2217	C₂×C₆₇	1	2	yes	complemented
211	#2216	C₂×C₆₇	1	2	yes	complemented
212	#2159	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#2138	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₂×#358	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₆₇	#33613	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
219	#537	—	2	1	no	—
220	#536	—	2	1	no	—
221	#510	—	2	4	no	—
222	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
223	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
224	#504	—	2	4	no	—
225	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
226	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
227	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
228	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
229	#470	—	2	2	no	—
230	#468	—	2	2	no	—
231	#467	C₂²×C₆₇	1	2	yes	—
232	#467	C₂²×C₆₇	1	2	yes	—
233	#467	C₂²×C₆₇	1	2	yes	—
234	#466	C₂²×C₆₇	1	2	yes	—
235	#427	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
237	#419	—	2	1	no	—
238	#417	—	2	1	no	—
239	#367	C₂²×C₆₇	1	2	yes	—
240	#366	C₂²×C₆₇	1	2	yes	—
241	#358	C₂²×C₆₇	1	2	yes	—
242	#358	C₂²×C₆₇	1	2	yes	—
243	#358	C₂²×C₆₇	1	1	yes	characteristic
244	#358	C₂²×C₆₇	1	1	yes	characteristic
245	#357	C₂²×C₆₇	1	1	yes	characteristic
246	#357	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#41	—	2	1	no	—
248	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
249	C₂²×Q₈	—	2	1	no	—
250	#135	—	2	2	no	—
251	#135	—	2	2	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	2	1	no	—
254	C₂×Q₁₆	—	2	2	no	—
255	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
256	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
257	C₂×SD₁₆	—	2	1	no	—
258	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
259	#131	—	2	2	no	—
260	#131	—	2	2	no	—
261	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
262	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
264	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	4	1	no	—
266	#124	—	4	1	no	—
267	#123	—	4	2	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	2	2	no	—
270	C₂²⋊Q₈	—	2	2	no	—
271	C₂²⋊Q₈	—	4	1	no	—
272	#121	—	2	2	no	—
273	#121	—	4	1	no	—
274	#121	—	4	2	no	—
275	#117	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	Q₃₂	—	2	4	no	—
278	SD₃₂	—	2	4	no	—
279	M₃₂	—	2	4	no	—
280	M₃₂	D₄×C₆₇	1	4	yes	complemented
281	C₂×C₁₆	—	2	2	no	—
282	C₂×C₁₆	—	2	4	no	—
283	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
285	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
286	#107	—	2	1	no	—
287	#107	C₂³×C₆₇	1	1	yes	characteristic
288	#107	C₂³×C₆₇	1	1	yes	characteristic
289	#107	C₂³×C₆₇	1	2	yes	—
290	#106	—	2	1	no	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#103	—	2	2	no	—
294	#102	—	2	2	no	—
295	#41	—	2	2	no	—
296	#41	—	2	2	no	—
297	#41	—	4	2	no	—
298	#41	—	4	2	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	4	2	no	—
304	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	D₈×C₆₇	1	2	yes	—
309	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	Q₁₆	—	2	2	no	—
316	Q₁₆	—	2	1	no	—
317	Q₁₆	—	2	2	no	—
318	Q₁₆	—	2	1	no	—
319	Q₁₆	—	4	2	no	—
320	SD₁₆	—	4	2	no	—
321	D₈	—	2	1	no	—
322	D₈	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
330	C₂×C₈	#41×C₆₇	1	2	yes	—
331	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	1	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	—	4	2	no	—
344	C₄²	—	4	1	no	—
345	C₁₆	—	2	1	no	—
346	C₁₆	—	2	1	no	—
347	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
348	C₂³	—	4	1	no	—
349	C₂³	C₂×D₈×C₆₇	1	2	yes	—
350	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	Q₈	—	4	1	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	8	2	no	—
357	D₄	—	2	4	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	1	no	—
360	D₄	—	4	1	no	—
361	D₄	—	8	2	no	—
362	D₄	—	16	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	2	no	—
371	C₂×C₄	—	8	2	no	—
372	C₂×C₄	—	16	1	no	—
373	C₂×C₄	#121×C₆₇	1	2	yes	—
374	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
375	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
376	C₂×C₄	#135×C₆₇	1	2	yes	—
377	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	#121×C₆₇	1	2	yes	—
380	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	8	1	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	#466×C₆₇	1	2	yes	—
391	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	8	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	8	1	no	—
397	C₄	—	8	1	no	—
398	C₄	#466×C₆₇	1	2	yes	—
399	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
400	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	4	2	no	—
403	C₂	—	16	1	no	—
404	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
405	C₂	C₆₇×#3416	1	1	yes	characteristic
406	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#41×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₆₇	C₂	semidirect
2. C₂×#504×C₆₇	C₂	semidirect
4. C₆₇×#3059	C₂	semidirect
5. C₆₇×#2220	C₂	semidirect
6. C₆₇×#2219	C₂	semidirect
7. C₆₇×#2217	C₂	semidirect
8. C₆₇×#2216	C₂	semidirect
9. C₆₇×#2159	C₂	semidirect
10. C₆₇×#2138	C₂	semidirect
11. C₂×#358×C₆₇	C₂	semidirect
76. M₃₂×C₆₇	D₄	semidirect
197. #33613	C₆₇	direct (N × H)
204. C₂×#510	C₂×C₆₇	semidirect
205. C₂×#504	C₂×C₆₇	semidirect
207. #3059	C₂×C₆₇	semidirect
208. #2220	C₂×C₆₇	semidirect
209. #2219	C₂×C₆₇	semidirect
210. #2217	C₂×C₆₇	semidirect
211. #2216	C₂×C₆₇	semidirect
212. #2159	C₂×C₆₇	semidirect
213. #2138	C₂×C₆₇	semidirect
214. C₂×#358	C₂×C₆₇	semidirect
215. C₆₇	#33613	direct (N × H)
280. M₃₂	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510×C₆₇	2
C₂×#504×C₆₇	2
C₆₇×#3061	2
C₆₇×#3059	2
C₆₇×#2220	2
C₆₇×#2219	2
C₆₇×#2217	2
C₆₇×#2216	2
C₆₇×#2159	2
C₆₇×#2138	2
C₂×#358×C₆₇	2
#33613	67

Subgroups — C67×#33613 #30712864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33613 #30712864