Subgroups — C₆₇×#17330 #30696581

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2965	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1454	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1453	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1451	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1450	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
13	#446×C₆₇	C₄	1	2	yes	—
14	C₈×Q₈×C₆₇	C₄	1	2	yes	—
15	#436×C₆₇	C₄	1	2	yes	—
16	#435×C₆₇	C₄	1	4	yes	—
17	C₈×D₄×C₆₇	C₄	1	2	yes	—
18	C₈×D₄×C₆₇	C₄	1	4	yes	—
19	#433×C₆₇	C₄	1	4	yes	—
20	#431×C₆₇	C₂²	1	2	yes	—
21	#431×C₆₇	C₂²	1	2	yes	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	#431×C₆₇	C₂²	1	2	yes	—
24	#431×C₆₇	C₂²	1	1	yes	characteristic
25	#431×C₆₇	C₂²	1	2	yes	—
26	#424×C₆₇	C₄	1	2	yes	—
27	#408×C₆₇	C₄	1	2	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#322×C₆₇	C₂²	1	1	yes	characteristic
35	#322×C₆₇	C₂²	1	1	yes	characteristic
36	#322×C₆₇	C₂²	1	1	yes	characteristic
37	#322×C₆₇	C₂²	1	1	yes	characteristic
38	#322×C₆₇	C₂²	1	2	yes	—
39	#322×C₆₇	C₂²	1	2	yes	—
40	#322×C₆₇	C₂²	1	2	yes	—
41	#322×C₆₇	C₂²	1	2	yes	—
42	C₈²×C₆₇	C₂²	1	2	yes	—
43	C₈²×C₆₇	C₂²	1	2	yes	—
44	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
58	#117×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
62	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
65	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
66	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
73	#97×C₆₇	C₂×C₄	1	2	yes	—
74	#97×C₆₇	C₂×C₄	1	4	yes	—
75	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂³	1	2	yes	—
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	#41×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	C₄²	1	1	yes	characteristic
90	C₂×D₄×C₆₇	C₄²	1	2	yes	—
91	C₂×D₄×C₆₇	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	M₁₆×C₆₇	—	2	4	no	—
99	M₁₆×C₆₇	—	4	4	no	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	8	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
114	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
115	C₄⋊C₄×C₆₇	C₄²	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	C₄²	1	1	yes	characteristic
117	C₂²⋊C₄×C₆₇	C₄²	1	2	yes	—
118	C₂²⋊C₄×C₆₇	C₄²	1	2	yes	—
119	C₂²⋊C₄×C₆₇	C₄²	1	2	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
121	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
122	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
128	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	2	no	—
130	D₄×C₆₇	—	2	4	no	—
131	D₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#131	1	4	yes	—
144	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₈×C₆₇	—	2	2	no	—
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	—	2	4	no	—
150	C₈×C₆₇	—	2	2	no	—
151	C₈×C₆₇	—	2	4	no	—
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	2	4	no	—
155	C₈×C₆₇	#131	1	4	yes	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	#405	1	2	yes	—
162	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	#405	1	2	yes	—
169	C₄×C₆₇	C₂×#131	1	2	yes	—
170	C₄×C₆₇	C₂×#131	1	2	yes	—
171	C₄×C₆₇	#568	1	2	yes	—
172	#17330	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	2	2	no	—
174	C₂×C₆₇	—	4	2	no	—
175	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
176	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
177	C₂×C₆₇	#2875	1	1	yes	characteristic
178	#2965	C₂×C₆₇	1	1	yes	characteristic
179	C₈×#41	C₂×C₆₇	1	1	yes	characteristic
180	#2922	C₂×C₆₇	1	1	yes	characteristic
181	#1454	C₂×C₆₇	1	2	yes	complemented
182	#1453	C₂×C₆₇	1	2	yes	complemented
183	#1451	C₂×C₆₇	1	2	yes	complemented
184	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#1450	C₂×C₆₇	1	2	yes	complemented
186	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
187	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
188	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₆₇	#17330	1	1	yes	characteristic, direct product factor, Sylow
190	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
191	#446	C₄×C₆₇	1	2	yes	—
192	C₈×Q₈	C₄×C₆₇	1	2	yes	—
193	#436	C₄×C₆₇	1	2	yes	—
194	#435	C₄×C₆₇	1	4	yes	—
195	C₈×D₄	C₄×C₆₇	1	2	yes	—
196	C₈×D₄	C₄×C₆₇	1	4	yes	—
197	#433	C₄×C₆₇	1	4	yes	—
198	#431	C₂²×C₆₇	1	2	yes	—
199	#431	C₂²×C₆₇	1	2	yes	—
200	#431	C₂²×C₆₇	1	1	yes	characteristic
201	#431	C₂²×C₆₇	1	2	yes	—
202	#431	C₂²×C₆₇	1	1	yes	characteristic
203	#431	C₂²×C₆₇	1	2	yes	—
204	#424	C₄×C₆₇	1	2	yes	—
205	#408	C₄×C₆₇	1	2	yes	—
206	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
207	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
208	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
212	#322	C₂²×C₆₇	1	1	yes	characteristic
213	#322	C₂²×C₆₇	1	1	yes	characteristic
214	#322	C₂²×C₆₇	1	1	yes	characteristic
215	#322	C₂²×C₆₇	1	1	yes	characteristic
216	#322	C₂²×C₆₇	1	2	yes	—
217	#322	C₂²×C₆₇	1	2	yes	—
218	#322	C₂²×C₆₇	1	2	yes	—
219	#322	C₂²×C₆₇	1	2	yes	—
220	C₈²	C₂²×C₆₇	1	2	yes	—
221	C₈²	C₂²×C₆₇	1	2	yes	—
222	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
223	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
224	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
225	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
226	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
227	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
228	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
229	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
230	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
232	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
233	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
234	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
235	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
236	#117	C₂×C₄×C₆₇	1	2	yes	—
237	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
239	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
240	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
241	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
242	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
243	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
244	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
245	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
246	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
247	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
248	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
249	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
250	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
251	#97	C₂×C₄×C₆₇	1	2	yes	—
252	#97	C₂×C₄×C₆₇	1	4	yes	—
253	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
254	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
255	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
256	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
257	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₆₇	1	2	yes	—
259	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₆₇	1	2	yes	—
261	C₄×C₈	C₂³×C₆₇	1	2	yes	—
262	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
266	#41	—	2	4	no	—
267	C₂×Q₈	C₄²×C₆₇	1	1	yes	characteristic
268	C₂×D₄	C₄²×C₆₇	1	2	yes	—
269	C₂×D₄	C₄²×C₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	4	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	8	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
283	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
284	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
285	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
286	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
289	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
290	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
291	C₄⋊C₄	C₄²×C₆₇	1	2	yes	—
292	C₄⋊C₄	C₄²×C₆₇	1	2	yes	—
293	C₄⋊C₄	C₄²×C₆₇	1	1	yes	characteristic
294	C₄⋊C₄	C₄²×C₆₇	1	1	yes	characteristic
295	C₂²⋊C₄	C₄²×C₆₇	1	2	yes	—
296	C₂²⋊C₄	C₄²×C₆₇	1	2	yes	—
297	C₂²⋊C₄	C₄²×C₆₇	1	2	yes	—
298	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
299	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
300	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
301	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
302	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
303	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
304	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
305	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
306	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
317	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
319	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
320	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
321	C₂×C₄	#131×C₆₇	1	4	yes	—
322	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	4	no	—
328	C₈	—	2	2	no	—
329	C₈	—	2	4	no	—
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	#131×C₆₇	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#405×C₆₇	1	2	yes	—
340	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	#405×C₆₇	1	2	yes	—
347	C₄	C₂×#131×C₆₇	1	2	yes	—
348	C₄	C₂×#131×C₆₇	1	2	yes	—
349	C₄	#568×C₆₇	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
353	C₂	C₄×#131×C₆₇	1	1	yes	characteristic
354	C₂	C₆₇×#2875	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₆₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₄²
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1454	C₂	semidirect
5. C₆₇×#1453	C₂	semidirect
6. C₆₇×#1451	C₂	semidirect
7. C₆₇×#1451	C₂	semidirect
8. C₆₇×#1450	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
10. C₈×M₁₆×C₆₇	C₂	semidirect
11. C₂×#322×C₆₇	C₂	semidirect
172. #17330	C₆₇	direct (N × H)
181. #1454	C₂×C₆₇	semidirect
182. #1453	C₂×C₆₇	semidirect
183. #1451	C₂×C₆₇	semidirect
184. #1451	C₂×C₆₇	semidirect
185. #1450	C₂×C₆₇	semidirect
186. #1450	C₂×C₆₇	semidirect
187. C₈×M₁₆	C₂×C₆₇	semidirect
188. C₂×#322	C₂×C₆₇	semidirect
189. C₆₇	#17330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2965	2
C₈×#41×C₆₇	2
C₆₇×#2922	2
C₆₇×#1454	2
C₆₇×#1453	2
C₆₇×#1451	2
C₆₇×#1450	2
C₈×M₁₆×C₆₇	2
C₂×#322×C₆₇	2
#17330	67

Subgroups — C67×#17330 #30696581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17330 #30696581